Publications of Jianfeng Lu    :chronological  alphabetical  combined listing:

%% Papers Published   
@article{fds374395,
   Author = {Li, X and Pura, J and Allen, A and Owzar, K and Lu, J and Harms, M and Xie,
             J},
   Title = {DYNATE: Localizing rare-variant association regions via
             multiple testing embedded in an aggregation
             tree.},
   Journal = {Genet Epidemiol},
   Volume = {48},
   Number = {1},
   Pages = {42-55},
   Year = {2024},
   Month = {February},
   url = {http://dx.doi.org/10.1002/gepi.22542},
   Abstract = {Rare-variants (RVs) genetic association studies enable
             researchers to uncover the variation in phenotypic traits
             left unexplained by common variation. Traditional
             single-variant analysis lacks power; thus, researchers have
             developed various methods to aggregate the effects of RVs
             across genomic regions to study their collective impact.
             Some existing methods utilize a static delineation of
             genomic regions, often resulting in suboptimal effect
             aggregation, as neutral subregions within the test region
             will result in an attenuation of signal. Other methods use
             varying windows to search for signals but often result in
             long regions containing many neutral RVs. To pinpoint short
             genomic regions enriched for disease-associated RVs, we
             developed a novel method, DYNamic Aggregation TEsting
             (DYNATE). DYNATE dynamically and hierarchically aggregates
             smaller genomic regions into larger ones and performs
             multiple testing for disease associations with a controlled
             weighted false discovery rate. DYNATE's main advantage lies
             in its strong ability to identify short genomic regions
             highly enriched for disease-associated RVs. Extensive
             numerical simulations demonstrate the superior performance
             of DYNATE under various scenarios compared with existing
             methods. We applied DYNATE to an amyotrophic lateral
             sclerosis study and identified a new gene, EPG5, harboring
             possibly pathogenic mutations.},
   Doi = {10.1002/gepi.22542},
   Key = {fds374395}
}

@article{fds374248,
   Author = {Jing, Y and Chen, J and Li, L and Lu, J},
   Title = {A Machine Learning Framework for Geodesics Under Spherical
             Wasserstein–Fisher–Rao Metric and Its Application for
             Weighted Sample Generation},
   Journal = {Journal of Scientific Computing},
   Volume = {98},
   Number = {1},
   Year = {2024},
   Month = {January},
   url = {http://dx.doi.org/10.1007/s10915-023-02396-y},
   Abstract = {Wasserstein–Fisher–Rao (WFR) distance is a family of
             metrics to gauge the discrepancy of two Radon measures,
             which takes into account both transportation and weight
             change. Spherical WFR distance is a projected version of WFR
             distance for probability measures so that the space of Radon
             measures equipped with WFR can be viewed as metric cone over
             the space of probability measures with spherical WFR.
             Compared to the case for Wasserstein distance, the
             understanding of geodesics under the spherical WFR is less
             clear and still an ongoing research focus. In this paper, we
             develop a deep learning framework to compute the geodesics
             under the spherical WFR metric, and the learned geodesics
             can be adopted to generate weighted samples. Our approach is
             based on a Benamou–Brenier type dynamic formulation for
             spherical WFR. To overcome the difficulty in enforcing the
             boundary constraint brought by the weight change, a
             Kullback–Leibler divergence term based on the inverse map
             is introduced into the cost function. Moreover, a new
             regularization term using the particle velocity is
             introduced as a substitute for the Hamilton–Jacobi
             equation for the potential in dynamic formula. When used for
             sample generation, our framework can be beneficial for
             applications with given weighted samples, especially in the
             Bayesian inference, compared to sample generation with
             previous flow models.},
   Doi = {10.1007/s10915-023-02396-y},
   Key = {fds374248}
}

@article{fds374205,
   Author = {Wang, Z and Zhang, Z and Lu, J and Li, Y},
   Title = {Coordinate Descent Full Configuration Interaction for
             Excited States.},
   Journal = {Journal of chemical theory and computation},
   Volume = {19},
   Number = {21},
   Pages = {7731-7739},
   Year = {2023},
   Month = {November},
   url = {http://dx.doi.org/10.1021/acs.jctc.3c00452},
   Abstract = {An efficient excited state method, named xCDFCI, in the
             configuration interaction framework is proposed. xCDFCI
             extends the unconstrained nonconvex optimization problem in
             coordinate descent full configuration interaction (CDFCI) to
             a multicolumn version for low-lying excited states
             computation. The optimization problem is addressed via a
             tailored coordinate descent method. In each iteration, a
             determinant is selected based on an approximated gradient,
             and coefficients of all states associated with the selected
             determinant are updated. A deterministic compression is
             applied to limit memory usage. We test xCDFCI applied to
             H<sub>2</sub>O and N<sub>2</sub> molecules under the cc-pVDZ
             basis set. For both systems, five low-lying excited states
             in the same symmetry sector are calculated, together with
             the ground state. xCDFCI also produces accurate binding
             curves of the carbon dimer in the cc-pVDZ basis with
             chemical accuracy, where the ground state and four excited
             states in the same symmetry sector are benchmarked.},
   Doi = {10.1021/acs.jctc.3c00452},
   Key = {fds374205}
}

@article{fds372698,
   Author = {Cao, Y and Lu, J and Wang, L},
   Title = {On Explicit L2 -Convergence Rate Estimate for
             Underdamped Langevin Dynamics},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {247},
   Number = {5},
   Year = {2023},
   Month = {October},
   url = {http://dx.doi.org/10.1007/s00205-023-01922-4},
   Abstract = {We provide a refined explicit estimate of the exponential
             decay rate of underdamped Langevin dynamics in the L2
             distance, based on a framework developed in Albritton et al.
             (Variational methods for the kinetic Fokker–Planck
             equation, arXiv arXiv:1902.04037 , 2019). To achieve this,
             we first prove a Poincaré-type inequality with a Gibbs
             measure in space and a Gaussian measure in momentum. Our
             estimate provides a more explicit and simpler expression of
             the decay rate; moreover, when the potential is convex with
             a Poincaré constant m≪ 1 , our estimate shows the decay
             rate of O(m) after optimizing the choice of the friction
             coefficient, which is much faster than m for the overdamped
             Langevin dynamics.},
   Doi = {10.1007/s00205-023-01922-4},
   Key = {fds372698}
}

@article{fds370609,
   Author = {Lu, J and Wu, Y and Xiang, Y},
   Title = {Score-based Transport Modeling for Mean-Field Fokker-Planck
             Equations},
   Volume = {503},
   Year = {2023},
   Month = {April},
   url = {http://dx.doi.org/10.1016/j.jcp.2024.112859},
   Abstract = {We use the score-based transport modeling method to solve
             the mean-field Fokker-Planck equations, which we call MSBTM.
             We establish an upper bound on the time derivative of the
             Kullback-Leibler (KL) divergence to MSBTM numerical
             estimation from the exact solution, thus validates the MSBTM
             approach. Besides, we provide an error analysis for the
             algorithm. In numerical experiments, we study three types of
             mean-field Fokker-Planck equation and their corresponding
             dynamics of particles in interacting systems. The MSBTM
             algorithm is numerically validated through qualitative and
             quantitative comparison between the MSBTM solutions, the
             results of integrating the associated stochastic
             differential equation and the analytical solutions if
             available.},
   Doi = {10.1016/j.jcp.2024.112859},
   Key = {fds370609}
}

@article{fds369850,
   Author = {Wang, M and Lu, J},
   Title = {Neural Network-Based Variational Methods for Solving
             Quadratic Porous Medium Equations in High
             Dimensions},
   Journal = {Communications in Mathematics and Statistics},
   Volume = {11},
   Number = {1},
   Pages = {21-57},
   Year = {2023},
   Month = {March},
   url = {http://dx.doi.org/10.1007/s40304-023-00339-5},
   Abstract = {In this paper, we propose and study neural network-based
             methods for solutions of high-dimensional quadratic porous
             medium equation (QPME). Three variational formulations of
             this nonlinear PDE are presented: a strong formulation and
             two weak formulations. For the strong formulation, the
             solution is directly parameterized with a neural network and
             optimized by minimizing the PDE residual. It can be proved
             that the convergence of the optimization problem guarantees
             the convergence of the approximate solution in the L1 sense.
             The weak formulations are derived following (Brenier in
             Examples of hidden convexity in nonlinear PDEs, 2020) which
             characterizes the very weak solutions of QPME. Specifically
             speaking, the solutions are represented with intermediate
             functions who are parameterized with neural networks and are
             trained to optimize the weak formulations. Extensive
             numerical tests are further carried out to investigate the
             pros and cons of each formulation in low and high
             dimensions. This is an initial exploration made along the
             line of solving high-dimensional nonlinear PDEs with neural
             network-based methods, which we hope can provide some useful
             experience for future investigations.},
   Doi = {10.1007/s40304-023-00339-5},
   Key = {fds369850}
}

@article{fds369337,
   Author = {Bierman, J and Li, Y and Lu, J},
   Title = {Improving the Accuracy of Variational Quantum Eigensolvers
             with Fewer Qubits Using Orbital Optimization.},
   Journal = {Journal of chemical theory and computation},
   Volume = {19},
   Number = {3},
   Pages = {790-798},
   Year = {2023},
   Month = {February},
   url = {http://dx.doi.org/10.1021/acs.jctc.2c00895},
   Abstract = {Near-term quantum computers will be limited in the number of
             qubits on which they can process information as well as the
             depth of the circuits that they can coherently carry out. To
             date, experimental demonstrations of algorithms such as the
             Variational Quantum Eigensolver (VQE) have been limited to
             small molecules using minimal basis sets for this reason. In
             this work we propose incorporating an orbital optimization
             scheme into quantum eigensolvers wherein a parametrized
             partial unitary transformation is applied to the basis
             functions set in order to reduce the number of qubits
             required for a given problem. The optimal transformation is
             found by minimizing the ground state energy with respect to
             this partial unitary matrix. Through numerical simulations
             of small molecules up to 16 spin orbitals, we demonstrate
             that this method has the ability to greatly extend the
             capabilities of near-term quantum computers with regard to
             the electronic structure problem. We find that VQE paired
             with orbital optimization consistently achieves lower ground
             state energies than traditional VQE when using the same
             number of qubits and even frequently achieves lower ground
             state energies than VQE methods using more
             qubits.},
   Doi = {10.1021/acs.jctc.2c00895},
   Key = {fds369337}
}

@article{fds368436,
   Author = {Cai, Z and Lu, J and Yang, S},
   Title = {NUMERICAL ANALYSIS FOR INCHWORM MONTE CARLO METHOD: SIGN
             PROBLEM AND ERROR GROWTH},
   Journal = {Mathematics of Computation},
   Volume = {92},
   Number = {341},
   Pages = {1141-1209},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.1090/MCOM/3785},
   Abstract = {We consider the numerical analysis of the inchworm Monte
             Carlo method, which is proposed recently to tackle the
             numerical sign problem for open quantum systems. We focus on
             the growth of the numerical error with respect to the
             simulation time, for which the inchworm Monte Carlo method
             shows a flatter curve than the direct application of Monte
             Carlo method to the classical Dyson series. To better
             understand the underlying mechanism of the inchworm Monte
             Carlo method, we distinguish two types of exponential error
             growth, which are known as the numerical sign problem and
             the error amplification. The former is due to the fast
             growth of variance in the stochastic method, which can be
             observed from the Dyson series, and the latter comes from
             the evolution of the numerical solution. Our analysis
             demonstrates that the technique of partial resummation can
             be considered as a tool to balance these two types of error,
             and the inchworm Monte Carlo method is a successful case
             where the numerical sign problem is effectively suppressed
             by such means. We first demonstrate our idea in the context
             of ordinary differential equations, and then provide
             complete analysis for the inchworm Monte Carlo method.
             Several numerical experiments are carried out to verify our
             theoretical results.},
   Doi = {10.1090/MCOM/3785},
   Key = {fds368436}
}

@article{fds370310,
   Author = {Chen, Z and Lu, J and Lu, Y and Zhou, S},
   Title = {A REGULARITY THEORY FOR STATIC SCHRÖDINGER EQUATIONS ON R
             d IN SPECTRAL BARRON SPACES},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {55},
   Number = {1},
   Pages = {557-570},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.1137/22M1478719},
   Abstract = {Spectral Barron spaces have received considerable interest
             recently, as it is the natural function space for
             approximation theory of two-layer neural networks with a
             dimension-free convergence rate. In this paper, we study the
             regularity of solutions to the whole-space static
             Schrödinger equation in spectral Barron spaces. We prove
             that if the source of the equation lies in the spectral
             Barron space B s(R d) and the potential function admitting a
             nonnegative lower bound decomposes as a positive constant
             plus a function in B s(R d), then the solution lies in the
             spectral Barron space B s+2(R d).},
   Doi = {10.1137/22M1478719},
   Key = {fds370310}
}

@article{fds371889,
   Author = {Chen, Z and Lu, J and Qian, H and Wang, X and Yin, W},
   Title = {HeteRSGD: Tackling Heterogeneous Sampling Costs via Optimal
             Reweighted Stochastic Gradient Descent},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {206},
   Pages = {10732-10781},
   Year = {2023},
   Month = {January},
   Abstract = {One implicit assumption in current stochastic gradient
             descent (SGD) algorithms is the identical cost for sampling
             each component function of the finite-sum objective.
             However, there are applications where the costs differ
             substantially, for which SGD schemes with uniform sampling
             invoke a high sampling load. We investigate the use of
             importance sampling (IS) as a cost saver in this setting, in
             contrast to its traditional use for variance reduction. The
             key ingredient is a novel efficiency metric for IS that
             advocates low sampling costs while penalizing high gradient
             variances. We then propose HeteRSGD, an SGD scheme that
             performs gradient sampling according to optimal probability
             weights stipulated by the metric, and establish theories on
             its optimal asymptotic and finite-time convergence rates
             among all possible IS-based SGD schemes. We show that the
             relative efficiency gain of HeteRSGD can be arbitrarily
             large regardless of the problem dimension and number of
             components. Our theoretical results are validated
             numerically for both convex and nonconvex
             problems.},
   Key = {fds371889}
}

@article{fds371890,
   Author = {Lee, H and Lu, J and Tan, Y},
   Title = {Convergence of score-based generative modeling for general
             data distributions},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {201},
   Pages = {946-985},
   Year = {2023},
   Month = {January},
   Abstract = {Score-based generative modeling (SGM) has grown to be a
             hugely successful method for learning to generate samples
             from complex data distributions such as that of images and
             audio. It is based on evolving an SDE that transforms white
             noise into a sample from the learned distribution, using
             estimates of the score function, or gradient log-pdf.
             Previous convergence analyses for these methods have
             suffered either from strong assumptions on the data
             distribution or exponential dependencies, and hence fail to
             give efficient guarantees for the multimodal and non-smooth
             distributions that arise in practice and for which good
             empirical performance is observed. We consider a popular
             kind of SGM—denoising diffusion models—and give
             polynomial convergence guarantees for general data
             distributions, with no assumptions related to functional
             inequalities or smoothness. Assuming L2-accurate score
             estimates, we obtain Wasserstein distance guarantees for any
             distribution of bounded support or sufficiently decaying
             tails, as well as TV guarantees for distributions with
             further smoothness assumptions.},
   Key = {fds371890}
}

@article{fds372260,
   Author = {Chen, Z and Li, Y and Lu, J},
   Title = {ON THE GLOBAL CONVERGENCE OF RANDOMIZED COORDINATE GRADIENT
             DESCENT FOR NONCONVEX OPTIMIZATION*},
   Journal = {SIAM Journal on Optimization},
   Volume = {33},
   Number = {2},
   Pages = {713-738},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.1137/21M1460375},
   Abstract = {In this work, we analyze the global convergence property of
             a coordinate gradient descent with random choice of
             coordinates and stepsizes for nonconvex optimization
             problems. Under generic assumptions, we prove that the
             algorithm iterate will almost surely escape strict saddle
             points of the objective function. As a result, the algorithm
             is guaranteed to converge to local minima if all saddle
             points are strict. Our proof is based on viewing the
             coordinate descent algorithm as a nonlinear random dynamical
             system and a quantitative finite block analysis of its
             linearization around saddle points.},
   Doi = {10.1137/21M1460375},
   Key = {fds372260}
}

@article{fds372774,
   Author = {Sachs, M and Sen, D and Lu, J and Dunson, D},
   Title = {Posterior Computation with the Gibbs Zig-Zag
             Sampler},
   Journal = {Bayesian Analysis},
   Volume = {18},
   Number = {3},
   Pages = {909-927},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.1214/22-BA1319},
   Abstract = {An intriguing new class of piecewise deterministic Markov
             processes (PDMPs) has recently been proposed as an
             alternative to Markov chain Monte Carlo (MCMC). We propose a
             new class of PDMPs termed Gibbs zig-zag samplers, which
             allow parameters to be updated in blocks with a zig-zag
             sampler applied to certain parameters and traditional
             MCMC-style updates to others. We demonstrate the flexibility
             of this framework on posterior sampling for logistic models
             with shrinkage priors for high-dimensional regression and
             random effects, and provide conditions for geometric
             ergodicity and the validity of a central limit
             theorem.},
   Doi = {10.1214/22-BA1319},
   Key = {fds372774}
}

@article{fds372815,
   Author = {Huang, H and Landsberg, JM and Lu, J},
   Title = {GEOMETRY OF BACKFLOW TRANSFORMATION ANSATZE FOR QUANTUM
             MANY-BODY FERMIONIC WAVEFUNCTIONS},
   Journal = {Communications in Mathematical Sciences},
   Volume = {21},
   Number = {5},
   Pages = {1447-1453},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2023.v21.n5.a12},
   Abstract = {Wave function ansatze based on the backflow transformation
             are widely used to parametrize anti-symmetric multivariable
             functions for many-body quantum problems. We study the
             geometric aspects of such ansatze, in particular we show
             that in general totally antisymmetric polynomials cannot be
             efficiently represented by backflow transformation ansatze
             at least in the category of polynomials. In fact, if there
             are N particles in the system, one needs a linear
             combination of at least O(N3N−3) determinants to represent
             a generic totally antisymmetric polynomial. Our proof is
             based on bounding the dimension of the source of the ansatze
             from above and bounding the dimension of the target from
             below.},
   Doi = {10.4310/CMS.2023.v21.n5.a12},
   Key = {fds372815}
}

@article{fds373339,
   Author = {Bal, G and Becker, S and Drouot, A and Kammerer, CF and Lu, J and Watson,
             AB},
   Title = {EDGE STATE DYNAMICS ALONG CURVED INTERFACES},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {55},
   Number = {5},
   Pages = {4219-4254},
   Year = {2023},
   Month = {January},
   url = {http://dx.doi.org/10.1137/22M1489708},
   Abstract = {We study the propagation of wavepackets along weakly curved
             interfaces between topologically distinct media. Our
             Hamiltonian is an adiabatic modulation of Dirac operators
             omnipresent in the topological insulators literature. Using
             explicit formulas for straight edges, we construct a family
             of solutions that propagates, for long times,
             unidirectionally and dispersion-free along the curved edge.
             We illustrate our results through various numerical
             simulations.},
   Doi = {10.1137/22M1489708},
   Key = {fds373339}
}

@article{fds373537,
   Author = {Zhang, S and Lu, J and Zhao, H},
   Title = {On Enhancing Expressive Power via Compositions of Single
             Fixed-Size ReLU Network},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {202},
   Pages = {41452-41487},
   Year = {2023},
   Month = {January},
   Abstract = {This paper explores the expressive power of deep neural
             networks through the framework of function compositions. We
             demonstrate that the repeated compositions of a single
             fixed-size ReLU network exhibit surprising expressive power,
             despite the limited expressive capabilities of the
             individual network itself. Specifically, we prove by
             construction that L2◦g◦r◦L1 can approximate
             1-Lipschitz continuous functions on [0, 1]d with an error
             O(r−1/d), where g is realized by a fixed-size ReLU
             network, L1 and L2 are two affine linear maps matching the
             dimensions, and g◦r denotes the r-times composition of g.
             Furthermore, we extend such a result to generic continuous
             functions on [0, 1]d with the approximation error
             characterized by the modulus of continuity. Our results
             reveal that a continuous-depth network generated via a
             dynamical system has immense approximation power even if its
             dynamics function is time-independent and realized by a
             fixed-size ReLU network.},
   Key = {fds373537}
}

@article{fds373538,
   Author = {Chen, H and Lee, H and Lu, J},
   Title = {Improved Analysis of Score-based Generative Modeling:
             User-Friendly Bounds under Minimal Smoothness
             Assumptions},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {202},
   Pages = {5367-5382},
   Year = {2023},
   Month = {January},
   Abstract = {We give an improved theoretical analysis of score-based
             generative modeling. Under a score estimate with small L2
             error (averaged across timesteps), we provide efficient
             convergence guarantees for any data distribution with
             second-order moment, by either employing early stopping or
             assuming a smoothness condition on the score function of the
             data distribution. Our result does not rely on any
             log-concavity or functional inequality assumption and has a
             logarithmic dependence on the smoothness. In particular, we
             show that under only a finite second moment condition,
             approximating the following in reverse KL divergence in
             ϵ-accuracy can be done in (equation presented)Õ (
             dlog(1ϵ/δ) ) steps: 1) the variance-δ Gaussian
             perturbation of any data distribution; 2) data distributions
             with 1/δ-smooth score functions. Our analysis also provides
             a quantitative comparison between different discrete
             approximations and may guide the choice of discretization
             points in practice.},
   Key = {fds373538}
}

@article{fds373539,
   Author = {Agazzi, A and Lu, J and Mukherjee, S},
   Title = {Global optimality of Elman-type RNNs in the mean-field
             regime},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {202},
   Pages = {196-227},
   Year = {2023},
   Month = {January},
   Abstract = {We analyze Elman-type Recurrent Reural Networks (RNNs) and
             their training in the mean-field regime. Specifically, we
             show convergence of gradient descent training dynamics of
             the RNN to the corresponding mean-field formulation in the
             large width limit. We also show that the fixed points of the
             limiting infinite-width dynamics are globally optimal, under
             some assumptions on the initialization of the weights. Our
             results establish optimality for feature-learning with wide
             RNNs in the mean-field regime.},
   Key = {fds373539}
}

@article{fds373540,
   Author = {Marwah, T and Lipton, ZC and Lu, J and Risteski, A},
   Title = {Neural Network Approximations of PDEs Beyond Linearity: A
             Representational Perspective},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {202},
   Pages = {24139-24172},
   Year = {2023},
   Month = {January},
   Abstract = {A burgeoning line of research leverages deep neural networks
             to approximate the solutions to high dimensional PDEs,
             opening lines of theoretical inquiry focused on explaining
             how it is that these models appear to evade the curse of
             dimensionality. However, most prior theoretical analyses
             have been limited to linear PDEs. In this work, we take a
             step towards studying the representational power of neural
             networks for approximating solutions to nonlinear PDEs. We
             focus on a class of PDEs known as nonlinear elliptic
             variational PDEs, whose solutions minimize an Euler-Lagrange
             energy functional E(u) = RΩ L(x, u(x), ∇u(x)) −
             f(x)u(x)dx. We show that if composing a function with Barron
             norm b with partial derivatives of L produces a function of
             Barron norm at most BLbp, the solution to the PDE can be
             ϵ-approximated in the L2 sense by a function with Barron
             norm O ( (dBL)max{p log(1/ϵ),plog(1/ϵ) }). By a classical
             result due to (Barron, 1993), this correspondingly bounds
             the size of a 2-layer neural network needed to approximate
             the solution. Treating p, ϵ, BL as constants, this quantity
             is polynomial in dimension, thus showing neural networks can
             evade the curse of dimensionality. Our proof technique
             involves neurally simulating (preconditioned) gradient in an
             appropriate Hilbert space, which converges exponentially
             fast to the solution of the PDE, and such that we can bound
             the increase of the Barron norm at each iterate. Our results
             subsume and substantially generalize analogous prior results
             for linear elliptic PDEs over a unit hypercube.},
   Key = {fds373540}
}

@article{fds367255,
   Author = {Holst, M and Hu, H and Lu, J and Marzuola, JL and Song, D and Weare,
             J},
   Title = {Symmetry Breaking and the Generation of Spin Ordered
             Magnetic States in Density Functional Theory Due to Dirac
             Exchange for a Hydrogen Molecule},
   Journal = {Journal of Nonlinear Science},
   Volume = {32},
   Number = {6},
   Year = {2022},
   Month = {December},
   url = {http://dx.doi.org/10.1007/s00332-022-09845-2},
   Abstract = {We study symmetry breaking in the mean field solutions to
             the electronic structure problem for the 2 electron hydrogen
             molecule within the Kohn Sham (KS) local spin density
             functional theory with Dirac exchange (the XLDA model). This
             simplified model shows behavior related to that of the (KS)
             spin density functional theory (SDFT) predictions in
             condensed matter and molecular systems. The KS solutions to
             the constrained SDFT variation problem undergo spontaneous
             symmetry breaking leading to the formation of spin ordered
             states as the relative strength of the non-convex exchange
             term increases. Numerically, we observe that with increases
             in the internuclear bond length, the molecular ground state
             changes from a paramagnetic state (spin delocalized) to an
             antiferromagnetic (spin localized) ground state and a
             symmetric delocalized (spin delocalized) excited state. We
             further characterize the limiting behavior of the minimizer
             when the strength of the exchange term goes to infinity both
             analytically and numerically. This leads to further
             bifurcations and highly localized states with varying
             character. Finite element numerical results provide support
             for the formal conjectures. Various solution classes are
             found to be numerically stable. However, for changes in the
             R parameter, numerical Hessian calculations demonstrate that
             these are stationary but not stable solutions.},
   Doi = {10.1007/s00332-022-09845-2},
   Key = {fds367255}
}

@article{fds367406,
   Author = {Craig, K and Liu, JG and Lu, J and Marzuola, JL and Wang,
             L},
   Title = {A proximal-gradient algorithm for crystal surface
             evolution},
   Journal = {Numerische Mathematik},
   Volume = {152},
   Number = {3},
   Pages = {631-662},
   Year = {2022},
   Month = {November},
   url = {http://dx.doi.org/10.1007/s00211-022-01320-0},
   Abstract = {As a counterpoint to recent numerical methods for crystal
             surface evolution, which agree well with microscopic
             dynamics but suffer from significant stiffness that prevents
             simulation on fine spatial grids, we develop a new numerical
             method based on the macroscopic partial differential
             equation, leveraging its formal structure as the gradient
             flow of the total variation energy, with respect to a
             weighted H- 1 norm. This gradient flow structure relates to
             several metric space gradient flows of recent interest,
             including 2-Wasserstein flows and their generalizations to
             nonlinear mobilities. We develop a novel semi-implicit time
             discretization of the gradient flow, inspired by the
             classical minimizing movements scheme (known as the JKO
             scheme in the 2-Wasserstein case). We then use a primal dual
             hybrid gradient (PDHG) method to compute each element of the
             semi-implicit scheme. In one dimension, we prove convergence
             of the PDHG method to the semi-implicit scheme, under
             general integrability assumptions on the mobility and its
             reciprocal. Finally, by taking finite difference
             approximations of our PDHG method, we arrive at a fully
             discrete numerical algorithm, with iterations that converge
             at a rate independent of the spatial discretization: in
             particular, the convergence properties do not deteriorate as
             we refine our spatial grid. We close with several numerical
             examples illustrating the properties of our method,
             including facet formation at local maxima, pinning at local
             minima, and convergence as the spatial and temporal
             discretizations are refined.},
   Doi = {10.1007/s00211-022-01320-0},
   Key = {fds367406}
}

@article{fds364046,
   Author = {Cai, Z and Lu, J and Yang, S},
   Title = {Fast algorithms of bath calculations in simulations of
             quantum system-bath dynamics},
   Journal = {Computer Physics Communications},
   Volume = {278},
   Year = {2022},
   Month = {September},
   url = {http://dx.doi.org/10.1016/j.cpc.2022.108417},
   Abstract = {We present fast algorithms for the summation of Dyson series
             and the inchworm Monte Carlo method for quantum systems that
             are coupled with harmonic baths. The algorithms are based on
             evolving the integro-differential equations where the most
             expensive part comes from the computation of bath influence
             functionals. To accelerate the computation, we design fast
             algorithms based on reusing the bath influence functionals
             computed in the previous time steps to reduce the number of
             calculations. It is proven that the proposed fast algorithms
             reduce the number of such calculations by a factor of O(N),
             where N is the total number of time steps. Numerical
             experiments are carried out to show the efficiency of the
             method and to verify the theoretical results.},
   Doi = {10.1016/j.cpc.2022.108417},
   Key = {fds364046}
}

@article{fds361220,
   Author = {Barthel, T and Lu, J and Friesecke, G},
   Title = {On the closedness and geometry of tensor network state
             sets},
   Journal = {Letters in Mathematical Physics},
   Volume = {112},
   Number = {4},
   Year = {2022},
   Month = {August},
   url = {http://dx.doi.org/10.1007/s11005-022-01552-z},
   Abstract = {Tensor network states (TNS) are a powerful approach for the
             study of strongly correlated quantum matter. The curse of
             dimensionality is addressed by parametrizing the many-body
             state in terms of a network of partially contracted tensors.
             These tensors form a substantially reduced set of effective
             degrees of freedom. In practical algorithms, functionals
             like energy expectation values or overlaps are optimized
             over certain sets of TNS. Concerning algorithmic stability,
             it is important whether the considered sets are closed
             because, otherwise, the algorithms may approach a boundary
             point that is outside the TNS set and tensor elements
             diverge. We discuss the closedness and geometries of TNS
             sets, and we propose regularizations for optimization
             problems on non-closed TNS sets. We show that sets of matrix
             product states (MPS) with open boundary conditions, tree
             tensor network states, and the multiscale entanglement
             renormalization ansatz are always closed, whereas sets of
             translation-invariant MPS with periodic boundary conditions
             (PBC), heterogeneous MPS with PBC, and projected entangled
             pair states are generally not closed. The latter is done
             using explicit examples like the W state, states that we
             call two-domain states, and fine-grained versions
             thereof.},
   Doi = {10.1007/s11005-022-01552-z},
   Key = {fds361220}
}

@article{fds364955,
   Author = {Bierman, J and Li, Y and Lu, J},
   Title = {Quantum Orbital Minimization Method for Excited States
             Calculation on a Quantum Computer.},
   Journal = {Journal of chemical theory and computation},
   Volume = {18},
   Number = {8},
   Pages = {4674-4689},
   Year = {2022},
   Month = {August},
   url = {http://dx.doi.org/10.1021/acs.jctc.2c00218},
   Abstract = {We propose a quantum-classical hybrid variational algorithm,
             the quantum orbital minimization method (qOMM), for
             obtaining the ground state and low-lying excited states of a
             Hermitian operator. Given parametrized ansatz circuits
             representing eigenstates, qOMM implements quantum circuits
             to represent the objective function in the orbital
             minimization method and adopts a classical optimizer to
             minimize the objective function with respect to the
             parameters in ansatz circuits. The objective function has an
             orthogonality constraint implicitly embedded, which allows
             qOMM to apply a different ansatz circuit to each input
             reference state. We carry out numerical simulations that
             seek to find excited states of H<sub>2</sub>, LiH, and a toy
             model consisting of four hydrogen atoms arranged in a square
             lattice in the STO-3G basis with UCCSD ansatz circuits.
             Comparing the numerical results with existing excited states
             methods, qOMM is less prone to getting stuck in local minima
             and can achieve convergence with more shallow ansatz
             circuits.},
   Doi = {10.1021/acs.jctc.2c00218},
   Key = {fds364955}
}

@article{fds361802,
   Author = {Lu, J and Steinerberger, S},
   Title = {Neural collapse under cross-entropy loss},
   Journal = {Applied and Computational Harmonic Analysis},
   Volume = {59},
   Pages = {224-241},
   Year = {2022},
   Month = {July},
   url = {http://dx.doi.org/10.1016/j.acha.2021.12.011},
   Abstract = {We consider the variational problem of cross-entropy loss
             with n feature vectors on a unit hypersphere in Rd. We prove
             that when d≥n−1, the global minimum is given by the
             simplex equiangular tight frame, which justifies the neural
             collapse behavior. We also prove that, as n→∞ with fixed
             d, the minimizing points will distribute uniformly on the
             hypersphere and show a connection with the frame potential
             of Benedetto & Fickus.},
   Doi = {10.1016/j.acha.2021.12.011},
   Key = {fds361802}
}

@article{fds364047,
   Author = {Lu, J and Wang, L},
   Title = {Complexity of zigzag sampling algorithm for strongly
             log-concave distributions},
   Journal = {Statistics and Computing},
   Volume = {32},
   Number = {3},
   Year = {2022},
   Month = {June},
   url = {http://dx.doi.org/10.1007/s11222-022-10109-y},
   Abstract = {We study the computational complexity of zigzag sampling
             algorithm for strongly log-concave distributions. The zigzag
             process has the advantage of not requiring time
             discretization for implementation, and that each proposed
             bouncing event requires only one evaluation of partial
             derivative of the potential, while its convergence rate is
             dimension independent. Using these properties, we prove that
             the zigzag sampling algorithm achieves ε error in
             chi-square divergence with a computational cost equivalent
             to O(κ2d12(log1ε)32) gradient evaluations in the regime
             κ≪dlogd under a warm start assumption, where κ is the
             condition number and d is the dimension.},
   Doi = {10.1007/s11222-022-10109-y},
   Key = {fds364047}
}

@article{fds364048,
   Author = {Pescia, G and Han, J and Lovato, A and Lu, J and Carleo,
             G},
   Title = {Neural-network quantum states for periodic systems in
             continuous space},
   Journal = {Physical Review Research},
   Volume = {4},
   Number = {2},
   Year = {2022},
   Month = {June},
   url = {http://dx.doi.org/10.1103/PhysRevResearch.4.023138},
   Abstract = {We introduce a family of neural quantum states for the
             simulation of strongly interacting systems in the presence
             of spatial periodicity. Our variational state is
             parametrized in terms of a permutationally invariant part
             described by the Deep Sets neural-network architecture. The
             input coordinates to the Deep Sets are periodically
             transformed such that they are suitable to directly describe
             periodic bosonic systems. We show example applications to
             both one- and two-dimensional interacting quantum gases with
             Gaussian interactions, as well as to He4 confined in a
             one-dimensional geometry. For the one-dimensional systems we
             find very precise estimations of the ground-state energies
             and the radial distribution functions of the particles. In
             two dimensions we obtain good estimations of the
             ground-state energies, comparable to results obtained from
             more conventional methods.},
   Doi = {10.1103/PhysRevResearch.4.023138},
   Key = {fds364048}
}

@article{fds365175,
   Author = {Chen, K and Chen, S and Li, Q and Lu, J and Wright, S},
   Title = {Low-Rank Approximation for Multiscale PDEs},
   Journal = {Notices of the American Mathematical Society},
   Volume = {69},
   Number = {6},
   Pages = {901-913},
   Year = {2022},
   Month = {June},
   url = {http://dx.doi.org/10.1090/noti2488},
   Doi = {10.1090/noti2488},
   Key = {fds365175}
}

@article{fds363886,
   Author = {Lu, J and Wang, L},
   Title = {ON EXPLICIT L2-CONVERGENCE RATE ESTIMATE FOR
             PIECEWISE DETERMINISTIC MARKOV PROCESSES IN MCMC
             ALGORITHMS},
   Journal = {Annals of Applied Probability},
   Volume = {32},
   Number = {2},
   Pages = {1333-1361},
   Year = {2022},
   Month = {April},
   url = {http://dx.doi.org/10.1214/21-AAP1710},
   Abstract = {We establish L2-exponential convergence rate for three
             popular piecewise deterministic Markov processes for
             sampling: the randomized Hamiltonian Monte Carlo method, the
             zigzag process and the bouncy particle sampler. Our analysis
             is based on a variational framework for hypocoercivity,
             which combines a Poincaré-type inequality in time-augmented
             state space and a standard L2 energy estimate. Our analysis
             provides explicit convergence rate estimates, which are more
             quantitative than existing results.},
   Doi = {10.1214/21-AAP1710},
   Key = {fds363886}
}

@article{fds362039,
   Author = {Lu, J and Stubbs, KD and Watson, AB},
   Title = {Existence and Computation of Generalized Wannier Functions
             for Non-Periodic Systems in Two Dimensions and
             Higher},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {243},
   Number = {3},
   Pages = {1269-1323},
   Year = {2022},
   Month = {March},
   url = {http://dx.doi.org/10.1007/s00205-021-01721-9},
   Abstract = {Exponentially-localized Wannier functions (ELWFs) are an
             orthonormal basis of the Fermi projection of a material
             consisting of functions which decay exponentially fast away
             from their maxima. When the material is insulating and
             crystalline, conditions which guarantee existence of ELWFs
             in dimensions one, two, and three are well-known, and
             methods for constructing ELWFs numerically are
             well-developed. We consider the case where the material is
             insulating but not necessarily crystalline, where much less
             is known. In one spatial dimension, Kivelson and
             Nenciu-Nenciu have proved ELWFs can be constructed as the
             eigenfunctions of a self-adjoint operator acting on the
             Fermi projection. In this work, we identify an assumption
             under which we can generalize the Kivelson–Nenciu–Nenciu
             result to two dimensions and higher. Under this assumption,
             we prove that ELWFs can be constructed as the eigenfunctions
             of a sequence of self-adjoint operators acting on the Fermi
             projection.},
   Doi = {10.1007/s00205-021-01721-9},
   Key = {fds362039}
}

@article{fds360558,
   Author = {Lu, J and Murphey, C and Steinerberger, S},
   Title = {Fast Localization of Eigenfunctions via Smoothed
             Potentials},
   Journal = {Journal of Scientific Computing},
   Volume = {90},
   Number = {1},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.1007/s10915-021-01682-x},
   Abstract = {We study the problem of predicting highly localized
             low-lying eigenfunctions (- Δ + V) ϕ= λϕ in bounded
             domains Ω ⊂ Rd for rapidly varying potentials V. Filoche
             and Mayboroda introduced the function 1/u, where (- Δ + V)
             u= 1 , as a suitable regularization of V from whose minima
             one can predict the location of eigenfunctions with high
             accuracy. We proposed a fast method that produces a
             landscapes that is exceedingly similar, can be used for the
             same purposes and can be computed very efficiently: the
             computation time on an n× n grid, for example, is merely
             O(n2log n) , the cost of two FFTs.},
   Doi = {10.1007/s10915-021-01682-x},
   Key = {fds360558}
}

@article{fds362442,
   Author = {Lu, J and Marzuola, JL and Watson, AB},
   Title = {DEFECT RESONANCES OF TRUNCATED CRYSTAL STRUCTURES},
   Journal = {SIAM Journal on Applied Mathematics},
   Volume = {82},
   Number = {1},
   Pages = {49-74},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.1137/21M1415601},
   Abstract = {Defects in the atomic structure of crystalline materials may
             spawn electronic bound states, known as defect states, which
             decay rapidly away from the defect. Simplified models of
             defect states typically assume the defect is surrounded on
             all sides by an infinite perfectly crystalline material. In
             reality the surrounding structure must be finite, and in
             certain contexts the structure can be small enough that edge
             effects are significant. In this work we investigate these
             edge effects and prove the following result. Suppose that a
             one-dimensional infinite crystalline material hosting a
             positive energy defect state is truncated a distance M from
             the defect. Then, for sufficiently large M, there exists a
             resonance exponentially close (in M) to the bound state
             eigenvalue. It follows that the truncated structure hosts a
             metastable state with an exponentially long lifetime. Our
             methods allow both the resonance frequency and associated
             resonant state to be computed to all orders in e - M. We
             expect this result to be of particular interest in the
             context of photonic crystals, where defect states are used
             for wave-guiding and structures are relatively small.
             Finally, under a mild additional assumption we prove that if
             the defect state has negative energy, then the truncated
             structure hosts a bound state with exponentially close
             energy.},
   Doi = {10.1137/21M1415601},
   Key = {fds362442}
}

@article{fds362815,
   Author = {Lu, J and Zhang, Z and Zhou, Z},
   Title = {Bloch dynamics with second order Berry phase
             correction},
   Journal = {Asymptotic Analysis},
   Volume = {128},
   Number = {1},
   Pages = {55-84},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.3233/ASY-211697},
   Abstract = {We derive the semiclassical Bloch dynamics with second-order
             Berry phase correction in the presence of the slow-varying
             scalar potential as perturbation. Our mathematical
             derivation is based on a two-scale WKB asymptotic analysis.
             For a uniform external electric field, the
             bi-characteristics system after a positional shift
             introduced by Berry connections agrees with the recent
             result in previous works. Moreover, for the case with a
             linear external electric field, we show that the extra terms
             arising in the bi-characteristics system after the
             positional shift are also gauge independent.},
   Doi = {10.3233/ASY-211697},
   Key = {fds362815}
}

@article{fds364049,
   Author = {Han, J and Li, Y and Lin, L and Lu, J and Zhang, J and Zhang,
             L},
   Title = {UNIVERSAL APPROXIMATION OF SYMMETRIC AND ANTI-SYMMETRIC
             FUNCTIONS},
   Journal = {Communications in Mathematical Sciences},
   Volume = {20},
   Number = {5},
   Pages = {1397-1408},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2022.v20.n5.a8},
   Abstract = {We consider universal approximations of symmetric and
             anti-symmetric functions, which are important for
             applications in quantum physics, as well as other scientific
             and engineering computations. We give constructive
             approximations with explicit bounds on the number of
             parameters with respect to the dimension and the target
             accuracy ϵ. While the approximation still suffers from the
             curse of dimensionality, to the best of our knowledge, these
             are the first results in the literature with explicit error
             bounds for functions with symmetry or anti-symmetry
             constraints},
   Doi = {10.4310/CMS.2022.v20.n5.a8},
   Key = {fds364049}
}

@article{fds367647,
   Author = {Chen, S and Li, Q and Lu, J and Wright, SJ},
   Title = {MANIFOLD LEARNING AND NONLINEAR HOMOGENIZATION},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {20},
   Number = {3},
   Pages = {1093-1126},
   Year = {2022},
   Month = {January},
   url = {http://dx.doi.org/10.1137/20M1377771},
   Abstract = {We describe an efficient domain decomposition-based
             framework for nonlinear multiscale PDE problems. The
             framework is inspired by manifold learning techniques and
             exploits the tangent spaces spanned by the nearest neighbors
             to compress local solution manifolds. Our framework is
             applied to a semilinear elliptic equation with oscillatory
             media and a nonlinear radiative transfer equation; in both
             cases, significant improvements in efficacy are observed.
             This new method does not rely on a detailed analytical
             understanding of multiscale PDEs, such as their asymptotic
             limits, and thus is more versatile for general multiscale
             problems.},
   Doi = {10.1137/20M1377771},
   Key = {fds367647}
}

@article{fds370034,
   Author = {Lu, Y and Chen, H and Lu, J and Ying, L and Blanchet,
             J},
   Title = {MACHINE LEARNING FOR ELLIPTIC PDES: FAST RATE GENERALIZATION
             BOUND, NEURAL SCALING LAW AND MINIMAX OPTIMALITY},
   Journal = {ICLR 2022 - 10th International Conference on Learning
             Representations},
   Year = {2022},
   Month = {January},
   Abstract = {In this paper, we study the statistical limits of deep
             learning techniques for solving elliptic partial
             differential equations (PDEs) from random samples using the
             Deep Ritz Method (DRM) and Physics-Informed Neural Networks
             (PINNs). To simplify the problem, we focus on a prototype
             elliptic PDE: the Schrödinger equation on a hypercube with
             zero Dirichlet boundary condition, which is applied in
             quantum-mechanical systems. We establish upper and lower
             bounds for both methods, which improve upon concurrently
             developed upper bounds for this problem via a fast rate
             generalization bound. We discover that the current Deep Ritz
             Method is sub-optimal and propose a modified version of it.
             We also prove that PINN and the modified version of DRM can
             achieve minimax optimal bounds over Sobolev spaces.
             Empirically, following recent work which has shown that the
             deep model accuracy will improve with growing training sets
             according to a power law, we supply computational
             experiments to show similar-behavior of dimension dependent
             power law for deep PDE solvers.},
   Key = {fds370034}
}

@article{fds371624,
   Author = {Lee, H and Lu, J and Tan, Y},
   Title = {Convergence for score-based generative modeling with
             polynomial complexity},
   Journal = {Advances in Neural Information Processing
             Systems},
   Volume = {35},
   Year = {2022},
   Month = {January},
   ISBN = {9781713871088},
   Abstract = {Score-based generative modeling (SGM) is a highly successful
             approach for learning a probability distribution from data
             and generating further samples. We prove the first
             polynomial convergence guarantees for the core mechanic
             behind SGM: drawing samples from a probability density p
             given a score estimate (an estimate of ∇ln p) that is
             accurate in L2(p). Compared to previous works, we do not
             incur error that grows exponentially in time or that suffers
             from a curse of dimensionality. Our guarantee works for any
             smooth distribution and depends polynomially on its
             log-Sobolev constant. Using our guarantee, we give a
             theoretical analysis of score-based generative modeling,
             which transforms white-noise input into samples from a
             learned data distribution given score estimates at different
             noise scales. Our analysis gives theoretical grounding to
             the observation that an annealed procedure is required in
             practice to generate good samples, as our proof depends
             essentially on using annealing to obtain a warm start at
             each step. Moreover, we show that a predictor-corrector
             algorithm gives better convergence than using either portion
             alone.},
   Key = {fds371624}
}

@article{fds355603,
   Author = {Lu, J and Otto, F},
   Title = {Optimal Artificial Boundary Condition for Random Elliptic
             Media},
   Journal = {Foundations of Computational Mathematics},
   Volume = {21},
   Number = {6},
   Pages = {1643-1702},
   Year = {2021},
   Month = {December},
   url = {http://dx.doi.org/10.1007/s10208-021-09492-1},
   Abstract = {We are given a uniformly elliptic coefficient field that we
             regard as a realization of a stationary and finite-range
             ensemble of coefficient fields. Given a right-hand side
             supported in a ball of size ℓ≫ 1 and of vanishing
             average, we are interested in an algorithm to compute the
             solution near the origin, just using the knowledge of the
             given realization of the coefficient field in some large box
             of size L≫ ℓ. More precisely, we are interested in the
             most seamless artificial boundary condition on the boundary
             of the computational domain of size L. Motivated by the
             recently introduced multipole expansion in random media, we
             propose an algorithm. We rigorously establish an error
             estimate on the level of the gradient in terms of L≫
             ℓ≫ 1 , using recent results in quantitative stochastic
             homogenization. More precisely, our error estimate has an a
             priori and an a posteriori aspect: with a priori
             overwhelming probability, the prefactor can be bounded by a
             constant that is computable without much further effort, on
             the basis of the given realization in the box of size L. We
             also rigorously establish that the order of the error
             estimate in both L and ℓ is optimal, where in this paper
             we focus on the case of d= 2. This amounts to a lower bound
             on the variance of the quantity of interest when conditioned
             on the coefficients inside the computational domain, and
             relies on the deterministic insight that a sensitivity
             analysis with respect to a defect commutes with stochastic
             homogenization. Finally, we carry out numerical experiments
             that show that this optimal convergence rate already sets in
             at only moderately large L, and that more naive boundary
             conditions perform worse both in terms of rate and
             prefactor.},
   Doi = {10.1007/s10208-021-09492-1},
   Key = {fds355603}
}

@article{fds361454,
   Author = {Chen, K and Chen, S and Li, Q and Lu, J and Wright, SJ},
   Title = {Low-rank approximation for multiscale PDEs},
   Year = {2021},
   Month = {November},
   Abstract = {Historically, analysis for multiscale PDEs is largely
             unified while numerical schemes tend to be
             equation-specific. In this paper, we propose a unified
             framework for computing multiscale problems through random
             sampling. This is achieved by incorporating randomized SVD
             solvers and manifold learning techniques to numerically
             reconstruct the low-rank features of multiscale PDEs. We use
             multiscale radiative transfer equation and elliptic equation
             with rough media to showcase the application of this
             framework.},
   Key = {fds361454}
}

@article{fds361455,
   Author = {Huang, H and Landsberg, JM and Lu, J},
   Title = {Geometry of backflow transformation ansatz for quantum
             many-body fermionic wavefunctions},
   Year = {2021},
   Month = {November},
   Abstract = {Wave function ansatz based on the backflow transformation
             are widely used to parametrize anti-symmetric multivariable
             functions for many-body quantum problems. We study the
             geometric aspects of such ansatz, in particular we show that
             in general totally antisymmetric polynomials cannot be
             efficiently represented by backflow transformation ansatz at
             least in the category of polynomials. In fact, one needs a
             linear combination of at least $O(N^{3N-3})$ determinants to
             represent a generic totally antisymmetric polynomial. Our
             proof is based on bounding the dimension of the source of
             the ansatz from above and bounding the dimension of the
             target from below.},
   Key = {fds361455}
}

@article{fds350227,
   Author = {Cheng, C and Daubechies, I and Dym, N and Lu, J},
   Title = {Stable phase retrieval from locally stable and conditionally
             connected measurements},
   Journal = {Applied and Computational Harmonic Analysis},
   Volume = {55},
   Pages = {440-465},
   Year = {2021},
   Month = {November},
   url = {http://dx.doi.org/10.1016/j.acha.2021.07.001},
   Abstract = {In this paper, we study the stability of phase retrieval
             problems via a family of locally stable phase retrieval
             frame measurements in Banach spaces, which we call
             “locally stable and conditionally connected” (LSCC)
             measurement schemes. For any signal f in the Banach space,
             we associate it with a weighted graph Gf, defined by the
             LSCC measurement scheme, and show that the phase
             retrievability of the signal f is determined by the
             connectivity of Gf. We quantify the phase retrieval
             stability of the signal by two common measures of graph
             connectivity: The Cheeger constant for real-valued signals,
             and algebraic connectivity for complex-valued signals. We
             then use our results to study the stability of two phase
             retrieval models. In the first model, we study a
             finite-dimensional phase retrieval problem from locally
             supported measurements such as the windowed Fourier
             transform. We show that signals “without large holes”
             are phase retrievable, and that for such signals in Rd the
             phase retrieval stability constant grows proportionally to
             d1/2, while in Cd it grows proportionally to d. The second
             model we consider is an infinite-dimensional phase retrieval
             problem in a shift-invariant space. In infinite-dimension
             spaces, even phase retrievable signals can have the Cheeger
             constant being zero, and hence have an infinite stability
             constant. We give an example of signals with monotone
             polynomial decay which has the Cheeger constant being zero,
             and an example with exponential decay which has a strictly
             positive Cheeger constant.},
   Doi = {10.1016/j.acha.2021.07.001},
   Key = {fds350227}
}

@article{fds361456,
   Author = {Lu, J and Otto, F and Wang, L},
   Title = {Optimal artificial boundary conditions based on second-order
             correctors for three dimensional random elliptic
             media},
   Year = {2021},
   Month = {September},
   Abstract = {We are interested in numerical algorithms for computing the
             electrical field generated by a charge distribution
             localized on scale $\ell$ in an infinite heterogeneous
             medium, in a situation where the medium is only known in a
             box of diameter $L\gg\ell$ around the support of the charge.
             We propose a boundary condition that with overwhelming
             probability is (near) optimal with respect to scaling in
             terms of $\ell$ and $L$, in the setting where the medium is
             a sample from a stationary ensemble with a finite range of
             dependence (set to be unity and with the assumption that
             $\ell \gg 1$). The boundary condition is motivated by
             quantitative stochastic homogenization that allows for a
             multipole expansion [BGO20]. This work extends [LO21], the
             algorithm in which is optimal in two dimension, and thus we
             need to take quadrupoles, next to dipoles, into account.
             This in turn relies on stochastic estimates of second-order,
             next to first-order, correctors. These estimates are
             provided for finite range ensembles under consideration,
             based on an extension of the semi-group approach of
             [GO15].},
   Key = {fds361456}
}

@article{fds367407,
   Author = {Ding, Z and Li, Q and Lu, J},
   Title = {ENSEMBLE KALMAN INVERSION FOR NONLINEAR PROBLEMS: WEIGHTS,
             CONSISTENCY, AND VARIANCE BOUNDS},
   Journal = {Foundations of Data Science},
   Volume = {3},
   Number = {3},
   Pages = {371-411},
   Year = {2021},
   Month = {September},
   url = {http://dx.doi.org/10.3934/fods.2020018},
   Abstract = {Ensemble Kalman Inversion (EnKI) [23] and Ensemble Square
             Root Filter (EnSRF) [36] are popular sampling methods for
             obtaining a target posterior distribution. They can be seem
             as one step (the analysis step) in the data assimilation
             method Ensemble Kalman Filter [17, 3]. Despite their
             popularity, they are, however, not unbiased when the forward
             map is nonlinear [12, 16, 25]. Important Sampling (IS), on
             the other hand, obtains the unbiased sampling at the expense
             of large variance of weights, leading to slow convergence of
             high moments. We propose WEnKI and WEnSRF, the weighted
             versions of EnKI and EnSRF in this paper. It follows the
             same gradient flow as that of EnKI/EnSRF with weight
             corrections. Compared to the classical methods, the new
             methods are unbiased, and compared with IS, the method has
             bounded weight variance. Both properties will be proved
             rigorously in this paper. We further discuss the stability
             of the underlying Fokker-Planck equation. This partially
             explains why EnKI, despite being inconsistent, performs well
             occasionally in nonlinear settings. Numerical evidence will
             be demonstrated at the end.},
   Doi = {10.3934/fods.2020018},
   Key = {fds367407}
}

@article{fds358292,
   Author = {Li, L and Goodrich, C and Yang, H and Phillips, KR and Jia, Z and Chen, H and Wang, L and Zhong, J and Liu, A and Lu, J and Shuai, J and Brenner, MP and Spaepen, F and Aizenberg, J},
   Title = {Microscopic origins of the crystallographically preferred
             growth in evaporation-induced colloidal crystals.},
   Journal = {Proceedings of the National Academy of Sciences of the
             United States of America},
   Volume = {118},
   Number = {32},
   Pages = {e2107588118},
   Year = {2021},
   Month = {August},
   url = {http://dx.doi.org/10.1073/pnas.2107588118},
   Abstract = {Unlike crystalline atomic and ionic solids, texture
             development due to crystallographically preferred growth in
             colloidal crystals is less studied. Here we investigate the
             underlying mechanisms of the texture evolution in an
             evaporation-induced colloidal assembly process through
             experiments, modeling, and theoretical analysis. In this
             widely used approach to obtain large-area colloidal
             crystals, the colloidal particles are driven to the meniscus
             via the evaporation of a solvent or matrix precursor
             solution where they close-pack to form a face-centered cubic
             colloidal assembly. Via two-dimensional large-area
             crystallographic mapping, we show that the initial crystal
             orientation is dominated by the interaction of particles
             with the meniscus, resulting in the expected coalignment of
             the close-packed direction with the local meniscus geometry.
             By combining with crystal structure analysis at a
             single-particle level, we further reveal that, at the later
             stage of self-assembly, however, the colloidal crystal
             undergoes a gradual rotation facilitated by geometrically
             necessary dislocations (GNDs) and achieves a large-area
             uniform crystallographic orientation with the close-packed
             direction perpendicular to the meniscus and parallel to the
             growth direction. Classical slip analysis, finite
             element-based mechanical simulation, computational colloidal
             assembly modeling, and continuum theory unequivocally show
             that these GNDs result from the tensile stress field along
             the meniscus direction due to the constrained shrinkage of
             the colloidal crystal during drying. The generation of GNDs
             with specific slip systems within individual grains leads to
             crystallographic rotation to accommodate the mechanical
             stress. The mechanistic understanding reported here can be
             utilized to control crystallographic features of colloidal
             assemblies, and may provide further insights into
             crystallographically preferred growth in synthetic,
             biological, and geological crystals.},
   Doi = {10.1073/pnas.2107588118},
   Key = {fds358292}
}

@article{fds356406,
   Author = {An, D and Cheng, SY and Head-Gordon, T and Lin, L and Lu,
             J},
   Title = {Convergence of stochastic-extended Lagrangian molecular
             dynamics method for polarizable force field
             simulation},
   Journal = {Journal of Computational Physics},
   Volume = {438},
   Year = {2021},
   Month = {August},
   url = {http://dx.doi.org/10.1016/j.jcp.2021.110338},
   Abstract = {Extended Lagrangian molecular dynamics (XLMD) is a general
             method for performing molecular dynamics simulations using
             quantum and classical many-body potentials. Recently several
             new XLMD schemes have been proposed and tested on several
             classes of many-body polarization models such as induced
             dipoles or Drude charges, by creating an auxiliary set of
             these same degrees of freedom that are reversibly integrated
             through time. This gives rise to a singularly perturbed
             Hamiltonian system that provides a good approximation to the
             time evolution of the real mutual polarization field. To
             further improve upon the accuracy of the XLMD dynamics in
             the context of classical polarizable force field simulation,
             and to potentially extend it to other many-body potentials,
             we introduce a stochastic modification which leads to a set
             of singularly perturbed Langevin equations with degenerate
             noise. We prove that the resulting Stochastic-XLMD converges
             to the accurate dynamics, and the convergence rate is both
             sharp and is independent of the accuracy of the initial
             polarization field. We carefully study the scaling of the
             damping factor and numerical noise for efficient numerical
             simulation for Stochastic-XLMD, and we demonstrate the
             effectiveness of the method for water molecules described by
             a polarizable force field.},
   Doi = {10.1016/j.jcp.2021.110338},
   Key = {fds356406}
}

@article{fds362600,
   Author = {Lu, J and Stubbs, KD},
   Title = {Algebraic localization of Wannier functions implies Chern
             triviality in non-periodic insulators},
   Year = {2021},
   Month = {July},
   Abstract = {For gapped periodic systems (insulators), it has been
             established that the insulator is topologically trivial
             (i.e., its Chern number is equal to $0$) if and only if its
             Fermi projector admits an orthogonal basis with finite
             second moment (i.e., all basis elements satisfy $\int
             |\boldsymbol{x}|^2 |w(\boldsymbol{x})|^2
             \,\textrm{d}{\boldsymbol{x}} < \infty$). In this paper, we
             extend one direction of this result to non-periodic gapped
             systems. In particular, we show that the existence of an
             orthogonal basis with slightly more decay ($\int
             |\boldsymbol{x}|^{2+\epsilon} |w(\boldsymbol{x})|^2
             \,\textrm{d}{\boldsymbol{x}} < \infty$ for any $\epsilon >
             0$) is a sufficient condition to conclude that the Chern
             marker, the natural generalization of the Chern number,
             vanishes.},
   Key = {fds362600}
}

@article{fds361668,
   Author = {Chen, Z and Lu, J and Lu, Y},
   Title = {On the Representation of Solutions to Elliptic PDEs in
             Barron Spaces},
   Year = {2021},
   Month = {June},
   Abstract = {Numerical solutions to high-dimensional partial differential
             equations (PDEs) based on neural networks have seen exciting
             developments. This paper derives complexity estimates of the
             solutions of $d$-dimensional second-order elliptic PDEs in
             the Barron space, that is a set of functions admitting the
             integral of certain parametric ridge function against a
             probability measure on the parameters. We prove under some
             appropriate assumptions that if the coefficients and the
             source term of the elliptic PDE lie in Barron spaces, then
             the solution of the PDE is $\epsilon$-close with respect to
             the $H^1$ norm to a Barron function. Moreover, we prove
             dimension-explicit bounds for the Barron norm of this
             approximate solution, depending at most polynomially on the
             dimension $d$ of the PDE. As a direct consequence of the
             complexity estimates, the solution of the PDE can be
             approximated on any bounded domain by a two-layer neural
             network with respect to the $H^1$ norm with a
             dimension-explicit convergence rate.},
   Key = {fds361668}
}

@article{fds352859,
   Author = {Khoo, Y and Lu, J and Ying, L},
   Title = {Solving parametric PDE problems with artificial neural
             networks},
   Journal = {European Journal of Applied Mathematics},
   Volume = {32},
   Number = {3},
   Pages = {421-435},
   Year = {2021},
   Month = {June},
   url = {http://dx.doi.org/10.1017/S0956792520000182},
   Abstract = {The curse of dimensionality is commonly encountered in
             numerical partial differential equations (PDE), especially
             when uncertainties have to be modelled into the equations as
             random coefficients. However, very often the variability of
             physical quantities derived from PDE can be captured by a
             few features on the space of the coefficient fields. Based
             on such observation, we propose using neural network to
             parameterise the physical quantity of interest as a function
             of input coefficients. The representability of such quantity
             using a neural network can be justified by viewing the
             neural network as performing time evolution to find the
             solutions to the PDE. We further demonstrate the simplicity
             and accuracy of the approach through notable examples of
             PDEs in engineering and physics.},
   Doi = {10.1017/S0956792520000182},
   Key = {fds352859}
}

@article{fds357910,
   Author = {Yang, S and Cai, Z and Lu, J},
   Title = {Inclusion-exclusion principle for open quantum systems with
             bosonic bath},
   Journal = {New Journal of Physics},
   Volume = {23},
   Number = {6},
   Year = {2021},
   Month = {June},
   url = {http://dx.doi.org/10.1088/1367-2630/ac02e1},
   Abstract = {We present two fast algorithms which apply
             inclusion-exclusion principle to sum over the bosonic
             diagrams in bare diagrammatic quantum Monte Carlo and
             inchworm Monte Carlo method, respectively. In the case of
             inchworm Monte Carlo, the proposed fast algorithm gives an
             extension to the work [2018 Inclusion-exclusion principle
             for many-body diagrammatics Phys. Rev. B 98 115152] from
             fermionic to bosonic systems. We prove that the proposed
             fast algorithms reduce the computational complexity from
             double factorial to exponential. Numerical experiments are
             carried out to verify the theoretical results and to compare
             the efficiency of the methods.},
   Doi = {10.1088/1367-2630/ac02e1},
   Key = {fds357910}
}

@article{fds361669,
   Author = {Bal, G and Becker, S and Drouot, A and Kammerer, CF and Lu, J and Watson,
             A},
   Title = {Edge state dynamics along curved interfaces},
   Year = {2021},
   Month = {June},
   Abstract = {We study the propagation of wavepackets along weakly curved
             interfaces between topologically distinct media. Our
             Hamiltonian is an adiabatic modulation of Dirac operators
             omnipresent in the topological insulators literature. Using
             explicit formulas for straight edges, we construct a family
             of solutions that propagates, for long times,
             unidirectionally and dispersion-free along the curved edge.
             We illustrate our results through various numerical
             simulations.},
   Key = {fds361669}
}

@article{fds361670,
   Author = {Lu, J and Lu, Y},
   Title = {A Priori Generalization Error Analysis of Two-Layer Neural
             Networks for Solving High Dimensional Schrödinger
             Eigenvalue Problems},
   Year = {2021},
   Month = {May},
   Abstract = {This paper analyzes the generalization error of two-layer
             neural networks for computing the ground state of the
             Schr\"odinger operator on a $d$-dimensional hypercube. We
             prove that the convergence rate of the generalization error
             is independent of the dimension $d$, under the a priori
             assumption that the ground state lies in a spectral Barron
             space. We verify such assumption by proving a new regularity
             estimate for the ground state in the spectral Barron space.
             The later is achieved by a fixed point argument based on the
             Krein-Rutman theorem.},
   Key = {fds361670}
}

@article{fds349489,
   Author = {Lu, J and Steinerberger, S},
   Title = {Optimal Trapping for Brownian Motion: a Nonlinear Analogue
             of the Torsion Function},
   Journal = {Potential Analysis},
   Volume = {54},
   Number = {4},
   Pages = {687-698},
   Year = {2021},
   Month = {April},
   url = {http://dx.doi.org/10.1007/s11118-020-09845-5},
   Abstract = {We study the problem of maximizing the expected lifetime of
             drift diffusion in a bounded domain. More formally, we
             consider the PDE−Δu+b(x)⋅∇u=1inΩ subject to
             Dirichlet boundary conditions for ∥b∥L∞ fixed. We show
             that, in any given C2 −domain Ω, the vector field
             maximizing the expected lifetime is (nonlinearly) coupled to
             the solution and satisfies b=−∥b∥L∞∇u/|∇u| which
             reduces the problem to the study of the nonlinear PDE− Δ
             u− b⋅ | ∇ u| = 1 , where b=∥b∥L∞ is a constant.
             We believe that this PDE is a natural and interesting
             nonlinear analogue of the torsion function (b = 0). We prove
             that, for fixed volume, ∥∇u∥L1 and ∥Δu∥L1 are
             maximized if Ω is the ball (the ball is also known to
             maximize ∥u∥Lp for p ≥ 1 from a result of Hamel &
             Russ).},
   Doi = {10.1007/s11118-020-09845-5},
   Key = {fds349489}
}

@article{fds356125,
   Author = {Coffman, AJ and Lu, J and Subotnik, JE},
   Title = {A grid-free approach for simulating sweep and cyclic
             voltammetry.},
   Journal = {The Journal of chemical physics},
   Volume = {154},
   Number = {16},
   Pages = {161101},
   Year = {2021},
   Month = {April},
   url = {http://dx.doi.org/10.1063/5.0044156},
   Abstract = {We present a computational approach to simulate linear sweep
             and cyclic voltammetry experiments that does not require a
             discretized grid in space to quantify diffusion. By using a
             Green's function solution coupled to a standard implicit
             ordinary differential equation solver, we are able to
             simulate current and redox species concentrations using only
             a small grid in time. As a result, where benchmarking is
             possible, we find that the current method is faster than
             (and quantitatively identical to) established techniques.
             The present algorithm should help open the door for studying
             adsorption effects in inner sphere electrochemistry.},
   Doi = {10.1063/5.0044156},
   Key = {fds356125}
}

@article{fds355983,
   Author = {Thicke, K and Watson, AB and Lu, J},
   Title = {Computing edge states without hard truncation},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {43},
   Number = {2},
   Pages = {B323-B353},
   Year = {2021},
   Month = {March},
   url = {http://dx.doi.org/10.1137/19M1282696},
   Abstract = {We present a numerical method which accurately computes the
             discrete spectrum and associated bound states of
             semi-infinite Hamiltonians which model electronic “edge”
             states localized at boundaries of one- and two-dimensional
             crystalline materials. The problem is nontrivial since
             arbitrarily large finite “hard” (Dirichlet) truncations
             of the Hamiltonian in the infinite bulk direction tend to
             produce spurious bound states partially supported at the
             truncation. Our method, which overcomes this difficulty, is
             to compute the Green’s function of the semi-infinite
             Hamiltonian by imposing an appropriate boundary condition in
             the bulk direction; then, the spectral data is recovered via
             Riesz projection. We demonstrate our method’s
             effectiveness by studies of edge states at a graphene
             zig-zag edge in the presence of defects modeled both by a
             discrete tight-binding model and a continuum PDE model under
             finite difference discretization. Our method may also be
             used to study states localized at domain wall-type edges in
             one- and two-dimensional materials where the edge
             Hamiltonian is infinite in both directions; we demonstrate
             this for the case of a tight-binding model of distinct
             honeycomb structures joined along a zig-zag edge. We expect
             our method to be useful for designing novel devices based on
             precise wave-guiding by edge states.},
   Doi = {10.1137/19M1282696},
   Key = {fds355983}
}

@article{fds361340,
   Author = {Cao, Y and Lu, J},
   Title = {Structure-preserving numerical schemes for Lindblad
             equations},
   Year = {2021},
   Month = {March},
   Abstract = {We study a family of structure-preserving deterministic
             numerical schemes for Lindblad equations, and carry out
             detailed error analysis and absolute stability analysis.
             Both error and absolute stability analysis are validated by
             numerical examples.},
   Key = {fds361340}
}

@article{fds355602,
   Author = {Stubbs, KD and Watson, AB and Lu, J},
   Title = {Iterated projected position algorithm for constructing
             exponentially localized generalized Wannier functions for
             periodic and nonperiodic insulators in two dimensions and
             higher},
   Journal = {Physical Review B},
   Volume = {103},
   Number = {7},
   Year = {2021},
   Month = {February},
   url = {http://dx.doi.org/10.1103/PhysRevB.103.075125},
   Abstract = {Localized bases play an important role in understanding
             electronic structure. In periodic insulators, a natural
             choice of localized basis is given by the Wannier functions
             which depend on a choice of unitary transform known as a
             gauge transformation. Over the past few decades, there have
             been many works that have focused on optimizing the choice
             of the gauge so that the corresponding Wannier functions are
             maximally localized or reflect some symmetry of the
             underlying system. In this work, we consider fully
             nonperiodic materials where the usual Wannier functions are
             not well defined and gauge optimization is impractical. To
             tackle the problem of calculating exponentially localized
             generalized Wannier functions in both periodic and
             nonperiodic systems, we discuss the 'iterated projected
             position (IPP)"algorithm. The IPP algorithm is based on
             matrix diagonalization and therefore unlike
             optimization-based approaches, it does not require
             initialization and cannot get stuck at a local minimum.
             Furthermore, the IPP algorithm is guaranteed by a rigorous
             analysis to produce exponentially localized functions under
             certain mild assumptions. We numerically demonstrate that
             the IPP algorithm can be used to calculate exponentially
             localized bases for the Haldane model, the Kane-Mele model
             (in both Z2 invariant even and Z2 invariant odd phases), and
             the px+ipy model on a quasicrystal lattice.},
   Doi = {10.1103/PhysRevB.103.075125},
   Key = {fds355602}
}

@article{fds361341,
   Author = {Lu, J and Stubbs, KD},
   Title = {Algebraic localization implies exponential localization in
             non-periodic insulators},
   Year = {2021},
   Month = {January},
   Abstract = {Exponentially-localized Wannier functions are a basis of the
             Fermi projection of a Hamiltonian consisting of functions
             which decay exponentially fast in space. In two and three
             spatial dimensions, it is well understood for periodic
             insulators that exponentially-localized Wannier functions
             exist if and only if there exists an orthonormal basis for
             the Fermi projection with finite second moment (i.e. all
             basis elements satisfy $\int |\boldsymbol{x}|^2
             |w(\boldsymbol{x})|^2 \,\text{d}{\boldsymbol{x}} < \infty$).
             In this work, we establish a similar result for non-periodic
             insulators in two spatial dimensions. In particular, we
             prove that if there exists an orthonormal basis for the
             Fermi projection which satisfies $\int |\boldsymbol{x}|^{5 +
             \epsilon} |w(\boldsymbol{x})|^2 \,\text{d}{\boldsymbol{x}} <
             \infty$ for some $\epsilon > 0$ then there also exists an
             orthonormal basis for the Fermi projection which decays
             exponentially fast in space. This result lends support to
             the Localization Dichotomy Conjecture for non-periodic
             systems recently proposed by Marcelli, Monaco, Moscolari,
             and Panati},
   Key = {fds361341}
}

@article{fds361342,
   Author = {Lu, J and Lu, Y and Wang, M},
   Title = {A Priori Generalization Analysis of the Deep Ritz Method for
             Solving High Dimensional Elliptic Equations},
   Year = {2021},
   Month = {January},
   Abstract = {This paper concerns the a priori generalization analysis of
             the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular
             neural-network-based method for solving high dimensional
             partial differential equations. We derive the generalization
             error bounds of two-layer neural networks in the framework
             of the DRM for solving two prototype elliptic PDEs: Poisson
             equation and static Schr\"odinger equation on the
             $d$-dimensional unit hypercube. Specifically, we prove that
             the convergence rates of generalization errors are
             independent of the dimension $d$, under the a priori
             assumption that the exact solutions of the PDEs lie in a
             suitable low-complexity space called spectral Barron space.
             Moreover, we give sufficient conditions on the forcing term
             and the potential function which guarantee that the
             solutions are spectral Barron functions. We achieve this by
             developing a new solution theory for the PDEs on the
             spectral Barron space, which can be viewed as an analog of
             the classical Sobolev regularity theory for
             PDEs.},
   Key = {fds361342}
}

@article{fds361343,
   Author = {Chen, Z and Li, Y and Lu, J},
   Title = {On the global convergence of randomized coordinate gradient
             descent for non-convex optimization},
   Year = {2021},
   Month = {January},
   Abstract = {In this work, we analyze the global convergence property of
             coordinate gradient descent with random choice of
             coordinates and stepsizes for non-convex optimization
             problems. Under generic assumptions, we prove that the
             algorithm iterate will almost surely escape strict saddle
             points of the objective function. As a result, the algorithm
             is guaranteed to converge to local minima if all saddle
             points are strict. Our proof is based on viewing coordinate
             descent algorithm as a nonlinear random dynamical system and
             a quantitative finite block analysis of its linearization
             around saddle points.},
   Key = {fds361343}
}

@article{fds358293,
   Author = {Li, L and Lu, J and Mattingly, JC and Wang, L},
   Title = {Numerical Methods For Stochastic Differential Equations
             Based On Gaussian Mixture},
   Journal = {Communications in Mathematical Sciences},
   Volume = {19},
   Number = {6},
   Pages = {1549-1577},
   Publisher = {International Press of Boston},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2021.v19.n6.a5},
   Abstract = {We develop in this work a numerical method for stochastic
             differential equations (SDEs) with weak second-order
             accuracy based on Gaussian mixture. Unlike conventional
             higher order schemes for SDEs based on Itô-Taylor expansion
             and iterated Itô integrals, the scheme we propose
             approximates the probability measure μ(Xn+1|Xn =xn) using a
             mixture of Gaussians. The solution at the next time step
             Xn+1 is drawn from the Gaussian mixture with complexity
             linear in dimension d. This provides a new strategy to
             construct efflcient high weak order numerical schemes for
             SDEs},
   Doi = {10.4310/CMS.2021.v19.n6.a5},
   Key = {fds358293}
}

@article{fds360559,
   Author = {Zhou, M and Han, J and Lu, J},
   Title = {ACTOR-CRITIC METHOD FOR HIGH DIMENSIONAL STATIC
             HAMILTON-JACOBI-BELLMAN PARTIAL DIFFERENTIAL EQUATIONS BASED
             ON NEURAL NETWORKS},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {43},
   Number = {6},
   Pages = {A4043-A4066},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.1137/21M1402303},
   Abstract = {We propose a novel numerical method for high dimensional
             Hamilton-Jacobi-Bellman (HJB) type elliptic partial
             differential equations (PDEs). The HJB PDEs, reformulated as
             optimal control problems, are tackled by the actor-critic
             framework inspired by reinforcement learning, based on
             neural network parametrization of the value and control
             functions. Within the actor-critic framework, we employ a
             policy gradient approach to improve the control, while for
             the value function, we derive a variance reduced
             least-squares temporal difference method using stochastic
             calculus. To numerically discretize the stochastic control
             problem, we employ an adaptive step size scheme to improve
             the accuracy near the domain boundary. Numerical examples up
             to 20 spatial dimensions including the linear quadratic
             regulators, the stochastic Van der Pol oscillators, the
             diffusive Eikonal equations, and fully nonlinear elliptic
             PDEs derived from a regulator problem are presented to
             validate the effectiveness of our proposed
             method.},
   Doi = {10.1137/21M1402303},
   Key = {fds360559}
}

@article{fds359801,
   Author = {Lu, J and Shen, Z and Yang, H and Zhang, S},
   Title = {DEEP NETWORK APPROXIMATION FOR SMOOTH FUNCTIONS},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {53},
   Number = {5},
   Pages = {5465-5506},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.1137/20M134695X},
   Abstract = {\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . This paper
             establishes the optimal approximation error characterization
             of deep rectified linear unit (ReLU) networks for smooth
             functions in terms of both width and depth simultaneously.
             To that end, we first prove that multivariate polynomials
             can be approximated by deep ReLU networks of width \scrO (N)
             and depth \scrO (L) with an approximation error \scrO (N -
             L). Through local Taylor expansions and their deep ReLU
             network approximations, we show that deep ReLU networks of
             width \scrO (N ln N) and depth \scrO (Lln L) can approximate
             f \in Cs([0, 1]d) with a nearly optimal approximation error
             \scrO (\| f\| Cs([0,1]d)N -2s/dL -2s/d). Our estimate is
             nonasymptotic in the sense that it is valid for arbitrary
             width and depth specified by N \in \BbbN + and L \in \BbbN
             +, respectively.},
   Doi = {10.1137/20M134695X},
   Key = {fds359801}
}

@article{fds359229,
   Author = {Cao, Y and Lu, J and Wang, L},
   Title = {COMPLEXITY OF RANDOMIZED ALGORITHMS FOR UNDERDAMPED LANGEVIN
             DYNAMICS*},
   Journal = {Communications in Mathematical Sciences},
   Volume = {19},
   Number = {7},
   Pages = {1827-1853},
   Publisher = {International Press of Boston},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.4310/CMS.2021.v19.n7.a4},
   Abstract = {We establish an information complexity lower bound of
             randomized algorithms for simulating underdamped Langevin
             dynamics. More specifically, we prove that the worst L2
             strong error is of order (Formula Presented), for solving a
             family of d-dimensional underdamped Langevin dynamics, by
             any randomized algorithm with only N queries to rU, the
             driving Brownian motion and its weighted integration,
             respectively. The lower bound we establish matches the upper
             bound for the randomized midpoint method recently proposed
             by Shen and Lee [NIPS 2019], in terms of both parameters N
             and d.},
   Doi = {10.4310/CMS.2021.v19.n7.a4},
   Key = {fds359229}
}

@article{fds359091,
   Author = {Khoo, Y and Lu, J and Ying, L},
   Title = {Efficient construction of tensor ring representations from
             sampling},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {19},
   Number = {3},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1154382},
   Abstract = {In this paper we propose an efficient method to compress a
             high dimensional function into a tensor ring format, based
             on alternating least squares (ALS). Since the function has
             size exponential in d, where d is the number of dimensions,
             we propose an efficient sampling scheme to obtain O(d)
             important samples in order to learn the tensor ring.
             Furthermore, we devise an initialization method for ALS that
             allows fast convergence in practice. Numerical examples show
             that to approximate a function with similar accuracy, the
             tensor ring format provided by the proposed method has fewer
             parameters than the tensor-train format and also better
             respects the structure of the original function.},
   Doi = {10.1137/17M1154382},
   Key = {fds359091}
}

@article{fds361344,
   Author = {Loring, TA and Lu, J and Watson, AB},
   Title = {Locality of the windowed local density of
             states},
   Journal = {Discussion Contributions 10th Vienna Conference on
             Mathematical Modelling, volume 17. ARGESIM,
             2022},
   Year = {2021},
   Month = {January},
   Abstract = {We introduce a generalization of local density of states
             which is "windowed" with respect to position and energy,
             called the windowed local density of states (wLDOS). This
             definition generalizes the usual LDOS in the sense that the
             usual LDOS is recovered in the limit where the position
             window captures individual sites and the energy window is a
             delta distribution. We prove that the wLDOS is local in the
             sense that it can be computed up to arbitrarily small error
             using spatial truncations of the system Hamiltonian. Using
             this result we prove that the wLDOS is well-defined and
             computable for infinite systems satisfying some natural
             assumptions. We finally present numerical computations of
             the wLDOS at the edge and in the bulk of a "Fibonacci SSH
             model", a one-dimensional non-periodic model with
             topological edge states.},
   Key = {fds361344}
}

@article{fds356985,
   Author = {Chen, K and Li, Q and Lu, J and Wright, SJ},
   Title = {A low-rank schwarz method for radiative transfer equation
             with heterogeneous scattering coefficient},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {19},
   Number = {2},
   Pages = {775-801},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.1137/19M1276327},
   Abstract = {Random sampling has been used to find low-rank structure and
             to build fast direct solvers for multiscale partial
             differential equations of various types. In this work, we
             design an accelerated Schwarz method for radiative transfer
             equations that makes use of approximate local solution maps
             constructed offline via a random sampling strategy.
             Numerical examples demonstrate the accuracy, robustness, and
             efficiency of the proposed approach.},
   Doi = {10.1137/19M1276327},
   Key = {fds356985}
}

@article{fds363839,
   Author = {Chen, Z and Lu, J and Lu, Y},
   Title = {On the Representation of Solutions to Elliptic PDEs in
             Barron Spaces},
   Journal = {Advances in Neural Information Processing
             Systems},
   Volume = {8},
   Pages = {6454-6465},
   Year = {2021},
   Month = {January},
   ISBN = {9781713845393},
   Abstract = {Numerical solutions to high-dimensional partial differential
             equations (PDEs) based on neural networks have seen exciting
             developments. This paper derives complexity estimates of the
             solutions of d-dimensional second-order elliptic PDEs in the
             Barron space, that is a set of functions admitting the
             integral of certain parametric ridge function against a
             probability measure on the parameters. We prove under some
             appropriate assumptions that if the coefficients and the
             source term of the elliptic PDE lie in Barron spaces, then
             the solution of the PDE is ǫ-close with respect to the H1
             norm to a Barron function. Moreover, we prove
             dimension-explicit bounds for the Barron norm of this
             approximate solution, depending at most polynomially on the
             dimension d of the PDE. As a direct consequence of the
             complexity estimates, the solution of the PDE can be
             approximated on any bounded domain by a two-layer neural
             network with respect to the H1 norm with a
             dimension-explicit convergence rate.},
   Key = {fds363839}
}

@article{fds371435,
   Author = {Ge, R and Lee, H and Lu, J and Risteski, A},
   Title = {Efficient sampling from the Bingham distribution},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {132},
   Pages = {673-685},
   Year = {2021},
   Month = {January},
   Abstract = {We give a algorithm for exact sampling from the Bingham
             distribution p(x) ∝ exp(x⊺Ax) on the sphere Sd-1 with
             expected runtime of poly(d, λmax(A) - λmin(A)). The
             algorithm is based on rejection sampling, where the proposal
             distribution is a polynomial approximation of the pdf, and
             can be sampled from by explicitly evaluating integrals of
             polynomials over the sphere. Our algorithm gives exact
             samples, assuming exact computation of an inverse function
             of a polynomial. This is in contrast with Markov Chain Monte
             Carlo algorithms, which are not known to enjoy rapid mixing
             on this problem, and only give approximate samples. As a
             direct application, we use this to sample from the posterior
             distribution of a rank-1 matrix inference problem in
             polynomial time.},
   Key = {fds371435}
}

@article{fds371436,
   Author = {Lu, J and Lu, Y and Wang, M},
   Title = {A Priori Generalization Analysis of the Deep Ritz Method for
             Solving High Dimensional Elliptic Partial Differential
             Equations},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {134},
   Pages = {3196-3241},
   Year = {2021},
   Month = {January},
   Abstract = {This paper concerns the a priori generalization analysis of
             the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular
             neural-network-based method for solving high dimensional
             partial differential equations. We derive the generalization
             error bounds of two-layer neural networks in the framework
             of the DRM for solving two prototype elliptic PDEs: Poisson
             equation and static Schrödinger equation on the
             d-dimensional unit hypercube. Specifically, we prove that
             the convergence rates of generalization errors are
             independent of the dimension d, under the a priori
             assumption that the exact solutions of the PDEs lie in a
             suitable low-complexity space called spectral Barron space.
             Moreover, we give sufficient conditions on the forcing term
             and the potential function which guarantee that the
             solutions are spectral Barron functions. We achieve this by
             developing a new solution theory for the PDEs on the
             spectral Barron space, which can be viewed as an analog of
             the classical Sobolev regularity theory for
             PDEs.},
   Key = {fds371436}
}

@article{fds371891,
   Author = {Agazzi, A and Lu, J},
   Title = {Temporal-difference learning with nonlinear function
             approximation: lazy training and mean field
             regimes},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {145},
   Pages = {37-74},
   Year = {2021},
   Month = {January},
   Abstract = {We discuss the approximation of the value function for
             infinite-horizon discounted Markov Reward Processes (MRP)
             with wide neural networks trained with the
             Temporal-Difference (TD) learning algorithm. We first
             consider this problem under a certain scaling of the
             approximating function, leading to a regime called lazy
             training. In this regime, which arises naturally, implicit
             in the initialization of the neural network, the parameters
             of the model vary only slightly during the learning process,
             resulting in approximately linear behavior of the model.
             Both in the under- and over-parametrized frameworks, we
             prove exponential convergence to local, respectively global
             minimizers of the TD learning algorithm in the lazy training
             regime. We then compare the above scaling with the
             alternative mean-field scaling, where the approximately
             linear behavior of the model is lost. In this nonlinear,
             mean-field regime we prove that all fixed points of the
             dynamics in parameter space are global minimizers. We
             finally give examples of our convergence results in the case
             of models that diverge if trained with non-lazy TD
             learning.},
   Key = {fds371891}
}

@article{fds375331,
   Author = {Gao, Y and Katsevich, AE and Liu, JG and Lu, J and Marzuola,
             JL},
   Title = {ANALYSIS OF A FOURTH-ORDER EXPONENTIAL PDE ARISING FROM A
             CRYSTAL SURFACE JUMP PROCESS WITH METROPOLIS-TYPE TRANSITION
             RATES},
   Journal = {Pure and Applied Analysis},
   Volume = {3},
   Number = {4},
   Pages = {595-612},
   Year = {2021},
   Month = {January},
   url = {http://dx.doi.org/10.2140/paa.2021.3.595},
   Abstract = {We analytically and numerically study a fourth-order PDE
             modeling rough crystal surface diffusion on the macroscopic
             level. We discuss existence of solutions globally in time
             and long-time dynamics for the PDE model. The PDE,
             originally derived by Katsevich is the continuum limit of a
             microscopic model of the surface dynamics, given by a Markov
             jump process with Metropolis-type transition rates. We
             outline the convergence argument, which depends on a
             simplifying assumption on the local equilibrium measure that
             is valid in the high-temperature regime. We provide
             numerical evidence for the convergence of the microscopic
             model to the PDE in this regime.},
   Doi = {10.2140/paa.2021.3.595},
   Key = {fds375331}
}

@article{fds376400,
   Author = {Ding, Z and Li, Q and Lu, J and Wright, SJ},
   Title = {Random Coordinate Underdamped Langevin Monte
             Carlo},
   Journal = {Proceedings of Machine Learning Research},
   Volume = {130},
   Pages = {2701-2709},
   Year = {2021},
   Month = {January},
   Abstract = {The Underdamped Langevin Monte Carlo (ULMC) is a popular
             Markov chain Monte Carlo sampling method. It requires the
             computation of the full gradient of the log-density at each
             iteration, an expensive operation if the dimension of the
             problem is high. We propose a sampling method called Random
             Coordinate ULMC (RC-ULMC), which selects a single coordinate
             at each iteration to be updated and leaves the other
             coordinates untouched. We investigate the computational
             complexity of RC-ULMC and compare it with the classical ULMC
             for strongly log-concave probability distributions. We show
             that RC-ULMC is always cheaper than the classical ULMC, with
             a significant cost reduction when the problem is highly
             skewed and high dimensional. Our complexity bound for
             RC-ULMC is also tight in terms of dimension
             dependence.},
   Key = {fds376400}
}

@article{fds358751,
   Author = {Ding, Z and Li, Q and Lu, J and Wright, SJ},
   Title = {Random Coordinate Underdamped Langevin Monte
             Carlo},
   Journal = {24TH INTERNATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE AND
             STATISTICS (AISTATS)},
   Volume = {130},
   Pages = {2701-2709},
   Year = {2021},
   Abstract = {The Underdamped Langevin Monte Carlo (ULMC) is a popular
             Markov chain Monte Carlo sampling method. It requires the
             computation of the full gradient of the log-density at each
             iteration, an expensive operation if the dimension of the
             problem is high. We propose a sampling method called Random
             Coordinate ULMC (RC-ULMC), which selects a single coordinate
             at each iteration to be updated and leaves the other
             coordinates untouched. We investigate the computational
             complexity of RC-ULMC and compare it with the classical ULMC
             for strongly log-concave probability distributions. We show
             that RC-ULMC is always cheaper than the classical ULMC, with
             a significant cost reduction when the problem is highly
             skewed and high dimensional. Our complexity bound for
             RC-ULMC is also tight in terms of dimension
             dependence.},
   Key = {fds358751}
}

@article{fds376399,
   Author = {Li, L and Goodrich, C and Yang, H and Phillips, KR and Jia, Z and Chen, H and Wang, L and Zhong, J and Liu, A and Lu, J and Shuai, J and Brenner, MP and Spaepen, F and Aizenberg, J},
   Title = {Microscopic origins of the crystallographically preferred
             growth in evaporation-induced colloidal crystals},
   Journal = {PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE
             UNITED STATES OF AMERICA},
   Volume = {118},
   Number = {32},
   Year = {2021},
   url = {http://dx.doi.org/10.1073/pnas.2107588118},
   Doi = {10.1073/pnas.2107588118},
   Key = {fds376399}
}

@article{fds361585,
   Author = {Lu, J and Wang, L},
   Title = {Complexity of zigzag sampling algorithm for strongly
             log-concave distributions},
   Journal = {Stat Comput},
   Volume = {32},
   Pages = {48},
   Year = {2020},
   Month = {December},
   Abstract = {We study the computational complexity of zigzag sampling
             algorithm for strongly log-concave distributions. The zigzag
             process has the advantage of not requiring time
             discretization for implementation, and that each proposed
             bouncing event requires only one evaluation of partial
             derivative of the potential, while its convergence rate is
             dimension independent. Using these properties, we prove that
             the zigzag sampling algorithm achieves $\varepsilon$ error
             in chi-square divergence with a computational cost
             equivalent to $O\bigl(\kappa^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$
             gradient evaluations in the regime $\kappa \ll \frac{d}{\log
             d}$ under a warm start assumption, where $\kappa$ is the
             condition number and $d$ is the dimension.},
   Key = {fds361585}
}

@article{fds352641,
   Author = {Han, J and Lu, J and Zhou, M},
   Title = {Solving high-dimensional eigenvalue problems using deep
             neural networks: A diffusion Monte Carlo like
             approach},
   Journal = {Journal of Computational Physics},
   Volume = {423},
   Year = {2020},
   Month = {December},
   url = {http://dx.doi.org/10.1016/j.jcp.2020.109792},
   Abstract = {We propose a new method to solve eigenvalue problems for
             linear and semilinear second order differential operators in
             high dimensions based on deep neural networks. The
             eigenvalue problem is reformulated as a fixed point problem
             of the semigroup flow induced by the operator, whose
             solution can be represented by Feynman-Kac formula in terms
             of forward-backward stochastic differential equations. The
             method shares a similar spirit with diffusion Monte Carlo
             but augments a direct approximation to the eigenfunction
             through neural-network ansatz. The criterion of fixed point
             provides a natural loss function to search for parameters
             via optimization. Our approach is able to provide accurate
             eigenvalue and eigenfunction approximations in several
             numerical examples, including Fokker-Planck operator and the
             linear and nonlinear Schrödinger operators in high
             dimensions.},
   Doi = {10.1016/j.jcp.2020.109792},
   Key = {fds352641}
}

@article{fds352857,
   Author = {Lu, J and Lu, Y and Zhou, Z},
   Title = {Continuum limit and preconditioned Langevin sampling of the
             path integral molecular dynamics},
   Journal = {Journal of Computational Physics},
   Volume = {423},
   Year = {2020},
   Month = {December},
   url = {http://dx.doi.org/10.1016/j.jcp.2020.109788},
   Abstract = {We investigate the continuum limit that the number of beads
             goes to infinity in the ring polymer representation of
             thermal averages. Studying the continuum limit of the
             trajectory sampling equation sheds light on possible
             preconditioning techniques for sampling ring polymer
             configurations with large number of beads. We propose two
             preconditioned Langevin sampling dynamics, which are shown
             to have improved stability and sampling accuracy. We present
             a careful mode analysis of the preconditioned dynamics and
             show their connections to the normal mode, the staging
             coordinate and the Matsubara mode representation for ring
             polymers. In the case where the potential is quadratic, we
             show that the continuum limit of the preconditioned mass
             modified Langevin dynamics converges to its equilibrium
             exponentially fast, which suggests that the finite
             dimensional counterpart has a dimension-independent
             convergence rate. In addition, the preconditioning
             techniques can be naturally applied to the multi-level
             quantum systems in the nonadiabatic regime, which are
             compatible with various numerical approaches.},
   Doi = {10.1016/j.jcp.2020.109788},
   Key = {fds352857}
}

@article{fds361586,
   Author = {Lu, J and Steinerberger, S},
   Title = {Neural Collapse with Cross-Entropy Loss},
   Year = {2020},
   Month = {December},
   Abstract = {We consider the variational problem of cross-entropy loss
             with $n$ feature vectors on a unit hypersphere in
             $\mathbb{R}^d$. We prove that when $d \geq n - 1$, the
             global minimum is given by the simplex equiangular tight
             frame, which justifies the neural collapse behavior. We also
             prove that as $n \rightarrow \infty$ with fixed $d$, the
             minimizing points will distribute uniformly on the
             hypersphere and show a connection with the frame potential
             of Benedetto & Fickus.},
   Key = {fds361586}
}

@article{fds354126,
   Author = {Sen, D and Sachs, M and Lu, J and Dunson, DB},
   Title = {Efficient posterior sampling for high-dimensional imbalanced
             logistic regression.},
   Journal = {Biometrika},
   Volume = {107},
   Number = {4},
   Pages = {1005-1012},
   Publisher = {Oxford University Press (OUP)},
   Year = {2020},
   Month = {December},
   url = {http://dx.doi.org/10.1093/biomet/asaa035},
   Abstract = {Classification with high-dimensional data is of widespread
             interest and often involves dealing with imbalanced data.
             Bayesian classification approaches are hampered by the fact
             that current Markov chain Monte Carlo algorithms for
             posterior computation become inefficient as the number
             [Formula: see text] of predictors or the number [Formula:
             see text] of subjects to classify gets large, because of the
             increasing computational time per step and worsening mixing
             rates. One strategy is to employ a gradient-based sampler to
             improve mixing while using data subsamples to reduce the
             per-step computational complexity. However, the usual
             subsampling breaks down when applied to imbalanced data.
             Instead, we generalize piecewise-deterministic Markov chain
             Monte Carlo algorithms to include importance-weighted and
             mini-batch subsampling. These maintain the correct
             stationary distribution with arbitrarily small subsamples
             and substantially outperform current competitors. We provide
             theoretical support for the proposed approach and
             demonstrate its performance gains in simulated data examples
             and an application to cancer data.},
   Doi = {10.1093/biomet/asaa035},
   Key = {fds354126}
}

@article{fds348786,
   Author = {Cai, Z and Lu, J and Yang, S},
   Title = {Inchworm Monte Carlo Method for Open Quantum
             Systems},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {73},
   Number = {11},
   Pages = {2430-2472},
   Year = {2020},
   Month = {November},
   url = {http://dx.doi.org/10.1002/cpa.21888},
   Abstract = {We investigate in this work a recently proposed diagrammatic
             quantum Monte Carlo method—the inchworm Monte Carlo
             method—for open quantum systems. We establish its validity
             rigorously based on resummation of Dyson series. Moreover,
             we introduce an integro-differential equation formulation
             for open quantum systems, which illuminates the mathematical
             structure of the inchworm algorithm. This new formulation
             leads to an improvement of the inchworm algorithm by
             introducing classical deterministic time-integration
             schemes. The numerical method is validated by applications
             to the spin-boson model. © 2020 Wiley Periodicals,
             Inc.},
   Doi = {10.1002/cpa.21888},
   Key = {fds348786}
}

@article{fds350485,
   Author = {Yu, VWZ and Campos, C and Dawson, W and García, A and Havu, V and Hourahine, B and Huhn, WP and Jacquelin, M and Jia, W and Keçeli, M and Laasner, R and Li, Y and Lin, L and Lu, J and Moussa, J and Roman, JE and Vázquez-Mayagoitia, Á and Yang, C and Blum, V},
   Title = {ELSI — An open infrastructure for electronic structure
             solvers},
   Journal = {Computer Physics Communications},
   Volume = {256},
   Pages = {107459-107459},
   Publisher = {Elsevier BV},
   Year = {2020},
   Month = {November},
   url = {http://dx.doi.org/10.1016/j.cpc.2020.107459},
   Abstract = {Routine applications of electronic structure theory to
             molecules and periodic systems need to compute the electron
             density from given Hamiltonian and, in case of
             non-orthogonal basis sets, overlap matrices. System sizes
             can range from few to thousands or, in some examples,
             millions of atoms. Different discretization schemes (basis
             sets) and different system geometries (finite non-periodic
             vs. infinite periodic boundary conditions) yield matrices
             with different structures. The ELectronic Structure
             Infrastructure (ELSI) project provides an open-source
             software interface to facilitate the implementation and
             optimal use of high-performance solver libraries covering
             cubic scaling eigensolvers, linear scaling
             density-matrix-based algorithms, and other reduced scaling
             methods in between. In this paper, we present recent
             improvements and developments inside ELSI, mainly covering
             (1) new solvers connected to the interface, (2) matrix
             layout and communication adapted for parallel calculations
             of periodic and/or spin-polarized systems, (3) routines for
             density matrix extrapolation in geometry optimization and
             molecular dynamics calculations, and (4) general utilities
             such as parallel matrix I/O and JSON output. The ELSI
             interface has been integrated into four electronic structure
             code projects (DFTB+, DGDFT, FHI-aims, SIESTA), allowing us
             to rigorously benchmark the performance of the solvers on an
             equal footing. Based on results of a systematic set of
             large-scale benchmarks performed with Kohn–Sham
             density-functional theory and density-functional
             tight-binding theory, we identify factors that strongly
             affect the efficiency of the solvers, and propose a decision
             layer that assists with the solver selection process.
             Finally, we describe a reverse communication interface
             encoding matrix-free iterative solver strategies that are
             amenable, e.g., for use with planewave basis sets. Program
             summary: Program title: ELSI Interface CPC Library link to
             program files: http://dx.doi.org/10.17632/473mbbznrs.1
             Licensing provisions: BSD 3-clause Programming language:
             Fortran 2003, with interface to C/C++ External
             routines/libraries: BLACS, BLAS, BSEPACK (optional),
             EigenExa (optional), ELPA, FortJSON, LAPACK, libOMM, MPI,
             MAGMA (optional), MUMPS (optional), NTPoly, ParMETIS
             (optional), PETSc (optional), PEXSI, PT-SCOTCH (optional),
             ScaLAPACK, SLEPc (optional), SuperLU_DIST Nature of problem:
             Solving the electronic structure from given Hamiltonian and
             overlap matrices in electronic structure calculations.
             Solution method: ELSI provides a unified software interface
             to facilitate the use of various electronic structure
             solvers including cubic scaling dense eigensolvers, linear
             scaling density matrix methods, and other
             approaches.},
   Doi = {10.1016/j.cpc.2020.107459},
   Key = {fds350485}
}

@article{fds352858,
   Author = {Lu, J and Steinerberger, S},
   Title = {Synchronization of Kuramoto oscillators in dense
             networks},
   Journal = {Nonlinearity},
   Volume = {33},
   Number = {11},
   Pages = {5905-5918},
   Year = {2020},
   Month = {November},
   url = {http://dx.doi.org/10.1088/1361-6544/ab9baa},
   Abstract = {We study synchronization properties of systems of Kuramoto
             oscillators. The problem can also be understood as a
             question about the properties of an energy landscape created
             by a graph. More formally, let G = (V, E) be a connected
             graph and (ai j)ni, j=1 denotes its adjacency matrix. Let
             the function f : Tn → R n be given by f(θ1, . . ., θn) =
             P ai j cos(θi − θ j). This function has a global i, j=1
             maximum when θi = θ for all 1 6 i 6 n. It is known that if
             every vertex is connected to at least µ(n − 1) other
             vertices for µ sufficiently large, then every local maximum
             is global. Taylor proved this for µ > 0.9395 and Ling, Xu &
             Bandeira improved this to µ > 0.7929. We give a slight
             improvement to µ > 0.7889. Townsend, Stillman & Strogatz
             suggested that the critical value might be µc =
             0.75.},
   Doi = {10.1088/1361-6544/ab9baa},
   Key = {fds352858}
}

@article{fds353874,
   Author = {Li, Y and Cheng, X and Lu, J},
   Title = {Butterfly-net: Optimal function representation based on
             convolutional neural networks},
   Journal = {Communications in Computational Physics},
   Volume = {28},
   Number = {5},
   Pages = {1838-1885},
   Publisher = {Global Science Press},
   Year = {2020},
   Month = {November},
   url = {http://dx.doi.org/10.4208/CICP.OA-2020-0214},
   Abstract = {Deep networks, especially convolutional neural networks
             (CNNs), have been successfully applied in various areas of
             machine learning as well as to challenging problems in other
             scientific and engineering fields. This paper introduces
             Butterfly-net, a low-complexity CNN with structured and
             sparse cross-channel connections, together with a Butterfly
             initialization strategy for a family of networks.
             Theoretical analysis of the approximation power of
             Butterfly-net to the Fourier representation of input data
             shows that the error decays exponentially as the depth
             increases. Combining Butterfly-net with a fully connected
             neural network, a large class of problems are proved to be
             well approximated with network complexity depending on the
             effective frequency bandwidth instead of the input
             dimension. Regular CNN is covered as a special case in our
             analysis. Numerical experiments validate the analytical
             results on the approximation of Fourier kernels and energy
             functionals of Poisson's equations. Moreover, all
             experiments support that training from Butterfly
             initialization outperforms training from random
             initialization. Also, adding the remaining cross-channel
             connections, although significantly increases the parameter
             number, does not much improve the post-training accuracy and
             is more sensitive to data distribution.},
   Doi = {10.4208/CICP.OA-2020-0214},
   Key = {fds353874}
}

@article{fds361587,
   Author = {Chen, S and Li, Q and Lu, J and Wright, SJ},
   Title = {Manifold Learning and Nonlinear Homogenization},
   Year = {2020},
   Month = {November},
   Abstract = {We describe an efficient domain decomposition-based
             framework for nonlinear multiscale PDE problems. The
             framework is inspired by manifold learning techniques and
             exploits the tangent spaces spanned by the nearest neighbors
             to compress local solution manifolds. Our framework is
             applied to a semilinear elliptic equation with oscillatory
             media and a nonlinear radiative transfer equation; in both
             cases, significant improvements in efficacy are observed.
             This new method does not rely on detailed analytical
             understanding of the multiscale PDEs, such as their
             asymptotic limits, and thus is more versatile for general
             multiscale problems.},
   Key = {fds361587}
}

@article{fds352986,
   Author = {Agazzi, A and Lu, J},
   Title = {Global optimality of softmax policy gradient with single
             hidden layer neural networks in the mean-field
             regime},
   Volume = {abs/2010.11858},
   Publisher = {OpenReview.net},
   Year = {2020},
   Month = {October},
   Abstract = {We study the problem of policy optimization for
             infinite-horizon discounted Markov Decision Processes with
             softmax policy and nonlinear function approximation trained
             with policy gradient algorithms. We concentrate on the
             training dynamics in the mean-field regime, modeling e.g.,
             the behavior of wide single hidden layer neural networks,
             when exploration is encouraged through entropy
             regularization. The dynamics of these models is established
             as a Wasserstein gradient flow of distributions in parameter
             space. We further prove global optimality of the fixed
             points of this dynamics under mild conditions on their
             initialization.},
   Key = {fds352986}
}

@article{fds361588,
   Author = {Ding, Z and Li, Q and Lu, J and Wright, SJ},
   Title = {Random Coordinate Langevin Monte Carlo},
   Year = {2020},
   Month = {October},
   Abstract = {Langevin Monte Carlo (LMC) is a popular Markov chain Monte
             Carlo sampling method. One drawback is that it requires the
             computation of the full gradient at each iteration, an
             expensive operation if the dimension of the problem is high.
             We propose a new sampling method: Random Coordinate LMC
             (RC-LMC). At each iteration, a single coordinate is randomly
             selected to be updated by a multiple of the partial
             derivative along this direction plus noise, and all other
             coordinates remain untouched. We investigate the total
             complexity of RC-LMC and compare it with the classical LMC
             for log-concave probability distributions. When the gradient
             of the log-density is Lipschitz, RC-LMC is less expensive
             than the classical LMC if the log-density is highly skewed
             for high dimensional problems, and when both the gradient
             and the Hessian of the log-density are Lipschitz, RC-LMC is
             always cheaper than the classical LMC, by a factor
             proportional to the square root of the problem dimension. In
             the latter case, our estimate of complexity is sharp with
             respect to the dimension.},
   Key = {fds361588}
}

@article{fds351553,
   Author = {Li, Y and Lu, J},
   Title = {Optimal Orbital Selection for Full Configuration Interaction
             (OptOrbFCI): Pursuing the Basis Set Limit under a
             Budget.},
   Journal = {Journal of chemical theory and computation},
   Volume = {16},
   Number = {10},
   Pages = {6207-6221},
   Year = {2020},
   Month = {October},
   url = {http://dx.doi.org/10.1021/acs.jctc.0c00613},
   Abstract = {Full configuration interaction (FCI) solvers are limited to
             small basis sets due to their expensive computational costs.
             An optimal orbital selection for FCI (OptOrbFCI) is proposed
             to boost the power of existing FCI solvers to pursue the
             basis set limit under a computational budget. The
             optimization problem coincides with that of the complete
             active space SCF method (CASSCF), while OptOrbFCI is
             algorithmically quite different. OptOrbFCI effectively finds
             an optimal rotation matrix via solving a constrained
             optimization problem directly to compress the orbitals of
             large basis sets to one with a manageable size, conducts FCI
             calculations only on rotated orbital sets, and produces a
             variational ground-state energy and its wave function.
             Coupled with coordinate descent full configuration
             interaction (CDFCI), we demonstrate the efficiency and
             accuracy of the method on the carbon dimer and nitrogen
             dimer under basis sets up to cc-pV5Z. We also benchmark the
             binding curve of the nitrogen dimer under the cc-pVQZ basis
             set with 28 selected orbitals, which provide consistently
             lower ground-state energies than the FCI results under the
             cc-pVDZ basis set. The dissociation energy in this case is
             found to be of higher accuracy.},
   Doi = {10.1021/acs.jctc.0c00613},
   Key = {fds351553}
}

@article{fds361589,
   Author = {Ge, R and Lee, H and Lu, J and Risteski, A},
   Title = {Efficient sampling from the Bingham distribution},
   Journal = {Algorithmic Learning Theory. PMLR, 2021},
   Year = {2020},
   Month = {September},
   Abstract = {We give a algorithm for exact sampling from the Bingham
             distribution $p(x)\propto \exp(x^\top A x)$ on the sphere
             $\mathcal S^{d-1}$ with expected runtime of
             $\operatorname{poly}(d, \lambda_{\max}(A)-\lambda_{\min}(A))$.
             The algorithm is based on rejection sampling, where the
             proposal distribution is a polynomial approximation of the
             pdf, and can be sampled from by explicitly evaluating
             integrals of polynomials over the sphere. Our algorithm
             gives exact samples, assuming exact computation of an
             inverse function of a polynomial. This is in contrast with
             Markov Chain Monte Carlo algorithms, which are not known to
             enjoy rapid mixing on this problem, and only give
             approximate samples. As a direct application, we use this to
             sample from the posterior distribution of a rank-1 matrix
             inference problem in polynomial time.},
   Key = {fds361589}
}

@article{fds349981,
   Author = {Li, W and Lu, J and Wang, L},
   Title = {Fisher information regularization schemes for Wasserstein
             gradient flows},
   Journal = {Journal of Computational Physics},
   Volume = {416},
   Year = {2020},
   Month = {September},
   url = {http://dx.doi.org/10.1016/j.jcp.2020.109449},
   Abstract = {We propose a variational scheme for computing Wasserstein
             gradient flows. The scheme builds upon the
             Jordan–Kinderlehrer–Otto framework with the
             Benamou-Brenier's dynamic formulation of the quadratic
             Wasserstein metric and adds a regularization by the Fisher
             information. This regularization can be derived in terms of
             energy splitting and is closely related to the Schrödinger
             bridge problem. It improves the convexity of the variational
             problem and automatically preserves the non-negativity of
             the solution. As a result, it allows us to apply sequential
             quadratic programming to solve the sub-optimization problem.
             We further save the computational cost by showing that no
             additional time interpolation is needed in the underlying
             dynamic formulation of the Wasserstein-2 metric, and
             therefore, the dimension of the problem is vastly reduced.
             Several numerical examples, including porous media equation,
             nonlinear Fokker-Planck equation, aggregation diffusion
             equation, and Derrida-Lebowitz-Speer-Spohn equation, are
             provided. These examples demonstrate the simplicity and
             stableness of the proposed scheme.},
   Doi = {10.1016/j.jcp.2020.109449},
   Key = {fds349981}
}

@article{fds350518,
   Author = {Gao, Y and Liu, JG and Lu, J and Marzuola, JL},
   Title = {Analysis of a continuum theory for broken bond crystal
             surface models with evaporation and deposition
             effects},
   Journal = {Nonlinearity},
   Volume = {33},
   Number = {8},
   Pages = {3816-3845},
   Year = {2020},
   Month = {August},
   url = {http://dx.doi.org/10.1088/1361-6544/ab853d},
   Abstract = {We study a 4th order degenerate parabolic PDE model in
             one-dimension with a 2nd order correction modeling the
             evolution of a crystal surface under the influence of both
             thermal fluctuations and evaporation/deposition effects.
             First, we provide a non-rigorous derivation of the PDE from
             an atomistic model using variations on kinetic Monte Carlo
             rates proposed by the last author with Weare [Marzuola J L
             and Weare J 2013 Phys. Rev. E 88 032403]. Then, we prove the
             existence of a global in time weak solution for the PDE by
             regularizing the equation in a way that allows us to apply
             the tools of Bernis-Friedman [Bernis F and Friedman A 1990
             J. Differ. Equ. 83 179-206]. The methods developed here can
             be applied to a large number of 4th order degenerate PDE
             models. In an appendix, we also discuss the global smooth
             solution with small data in the Weiner algebra framework
             following recent developments using tools of the second
             author with Robert Strain [Liu J G and Strain R M 2019
             Interfaces Free Boundaries 21 51-86].},
   Doi = {10.1088/1361-6544/ab853d},
   Key = {fds350518}
}

@article{fds361285,
   Author = {Lu, J and Wang, L},
   Title = {On explicit $L^2$-convergence rate estimate for piecewise
             deterministic Markov processes in MCMC algorithms},
   Journal = {Ann. Appl. Probab. 32(2): 1333-1361 (April
             2022)},
   Year = {2020},
   Month = {July},
   Abstract = {We establish $L^2$-exponential convergence rate for three
             popular piecewise deterministic Markov processes for
             sampling: the randomized Hamiltonian Monte Carlo method, the
             zigzag process, and the bouncy particle sampler. Our
             analysis is based on a variational framework for
             hypocoercivity, which combines a Poincar\'{e}-type
             inequality in time-augmented state space and a standard
             $L^2$ energy estimate. Our analysis provides explicit
             convergence rate estimates, which are more quantitative than
             existing results.},
   Key = {fds361285}
}

@article{fds361590,
   Author = {Craig, K and Liu, J-G and Lu, J and Marzuola, JL and Wang,
             L},
   Title = {A Proximal-Gradient Algorithm for Crystal Surface
             Evolution},
   Year = {2020},
   Month = {June},
   Abstract = {As a counterpoint to recent numerical methods for crystal
             surface evolution, which agree well with microscopic
             dynamics but suffer from significant stiffness that prevents
             simulation on fine spatial grids, we develop a new numerical
             method based on the macroscopic partial differential
             equation, leveraging its formal structure as the gradient
             flow of the total variation energy, with respect to a
             weighted $H^{-1}$ norm. This gradient flow structure relates
             to several metric space gradient flows of recent interest,
             including 2-Wasserstein flows and their generalizations to
             nonlinear mobilities. We develop a novel semi-implicit time
             discretization of the gradient flow, inspired by the
             classical minimizing movements scheme (known as the JKO
             scheme in the 2-Wasserstein case). We then use a primal dual
             hybrid gradient (PDHG) method to compute each element of the
             semi-implicit scheme. In one dimension, we prove convergence
             of the PDHG method to the semi-implicit scheme, under
             general integrability assumptions on the mobility and its
             reciprocal. Finally, by taking finite difference
             approximations of our PDHG method, we arrive at a fully
             discrete numerical algorithm, with iterations that converge
             at a rate independent of the spatial discretization: in
             particular, the convergence properties do not deteriorate as
             we refine our spatial grid. We close with several numerical
             examples illustrating the properties of our method,
             including facet formation at local maxima, pinning at local
             minima, and convergence as the spatial and temporal
             discretizations are refined.},
   Key = {fds361590}
}

@article{fds361591,
   Author = {Lu, J and Marzuola, JL and Watson, AB},
   Title = {Defect resonances of truncated crystal structures},
   Journal = {SIAM J. Appl. Math 82},
   Volume = {1},
   Pages = {49-74},
   Year = {2020},
   Month = {June},
   Abstract = {Defects in the atomic structure of crystalline materials may
             spawn electronic bound states, known as \emph{defect
             states}, which decay rapidly away from the defect.
             Simplified models of defect states typically assume the
             defect is surrounded on all sides by an infinite perfectly
             crystalline material. In reality the surrounding structure
             must be finite, and in certain contexts the structure can be
             small enough that edge effects are significant. In this work
             we investigate these edge effects and prove the following
             result. Suppose that a one-dimensional infinite crystalline
             material hosting a positive energy defect state is truncated
             a distance $M$ from the defect. Then, for sufficiently large
             $M$, there exists a resonance \emph{exponentially close} (in
             $M$) to the bound state eigenvalue. It follows that the
             truncated structure hosts a metastable state with an
             exponentially long lifetime. Our methods allow both the
             resonance frequency and associated resonant state to be
             computed to all orders in $e^{-M}$. We expect this result to
             be of particular interest in the context of photonic
             crystals, where defect states are used for wave-guiding and
             structures are relatively small. Finally, under a mild
             additional assumption we prove that if the defect state has
             negative energy then the truncated structure hosts a bound
             state with exponentially-close energy.},
   Key = {fds361591}
}

@article{fds361592,
   Author = {Cai, Z and Lu, J and Yang, S},
   Title = {Numerical analysis for inchworm Monte Carlo method: Sign
             problem and error growth},
   Year = {2020},
   Month = {June},
   Abstract = {We consider the numerical analysis of the inchworm Monte
             Carlo method, which is proposed recently to tackle the
             numerical sign problem for open quantum systems. We focus on
             the growth of the numerical error with respect to the
             simulation time, for which the inchworm Monte Carlo method
             shows a flatter curve than the direct application of Monte
             Carlo method to the classical Dyson series. To better
             understand the underlying mechanism of the inchworm Monte
             Carlo method, we distinguish two types of exponential error
             growth, which are known as the numerical sign problem and
             the error amplification. The former is due to the fast
             growth of variance in the stochastic method, which can be
             observed from the Dyson series, and the latter comes from
             the evolution of the numerical solution. Our analysis
             demonstrates that the technique of partial resummation can
             be considered as a tool to balance these two types of error,
             and the inchwormMonte Carlo method is a successful case
             where the numerical sign problem is effectively suppressed
             by such means. We first demonstrate our idea in the context
             of ordinary differential equations, and then provide
             complete analysis for the inchworm Monte Carlo method.
             Several numerical experiments are carried out to verify our
             theoretical results.},
   Key = {fds361592}
}

@article{fds350226,
   Author = {Ge, R and Lee, H and Lu, J},
   Title = {Estimating normalizing constants for log-concave
             distributions: Algorithms and lower bounds},
   Journal = {Proceedings of the Annual ACM Symposium on Theory of
             Computing},
   Pages = {579-586},
   Year = {2020},
   Month = {June},
   url = {http://dx.doi.org/10.1145/3357713.3384289},
   Abstract = {Estimating the normalizing constant of an unnormalized
             probability distribution has important applications in
             computer science, statistical physics, machine learning, and
             statistics. In this work, we consider the problem of
             estimating the normalizing constant Z=gg.,d e-f(x) dx to
             within a multiplication factor of 1 ± ϵ for a μ-strongly
             convex and L-smooth function f, given query access to f(x)
             and g‡ f(x). We give both algorithms and lowerbounds for
             this problem. Using an annealing algorithm combined with a
             multilevel Monte Carlo method based on underdamped Langevin
             dynamics, we show that O(d4/3κ + d7/6κ7/6/ϵ2) queries to
             g‡ f are sufficient, where κ= L / μ is the condition
             number. Moreover, we provide an information theoretic
             lowerbound, showing that at least d1-o(1)/ϵ2-o(1) queries
             are necessary. This provides a first nontrivial lowerbound
             for the problem.},
   Doi = {10.1145/3357713.3384289},
   Key = {fds350226}
}

@article{fds350519,
   Author = {Nishimura, A and Dunson, DB and Lu, J},
   Title = {Discontinuous Hamiltonian Monte Carlo for discrete
             parameters and discontinuous likelihoods},
   Journal = {Biometrika},
   Volume = {107},
   Number = {2},
   Pages = {365-380},
   Year = {2020},
   Month = {June},
   url = {http://dx.doi.org/10.1093/biomet/asz083},
   Abstract = {Hamiltonian Monte Carlo has emerged as a standard tool for
             posterior computation. In this article we present an
             extension that can efficiently explore target distributions
             with discontinuous densities. Our extension in particular
             enables efficient sampling from ordinal parameters through
             the embedding of probability mass functions into continuous
             spaces. We motivate our approach through a theory of
             discontinuous Hamiltonian dynamics and develop a
             corresponding numerical solver. The proposed solver is the
             first of its kind, with a remarkable ability to exactly
             preserve the Hamiltonian. We apply our algorithm to
             challenging posterior inference problems to demonstrate its
             wide applicability and competitive performance.},
   Doi = {10.1093/biomet/asz083},
   Key = {fds350519}
}

@article{fds348705,
   Author = {Li, Y and Lu, J and Mao, A},
   Title = {Variational training of neural network approximations of
             solution maps for physical models},
   Journal = {Journal of Computational Physics},
   Volume = {409},
   Pages = {109338-109338},
   Publisher = {Elsevier BV},
   Year = {2020},
   Month = {May},
   url = {http://dx.doi.org/10.1016/j.jcp.2020.109338},
   Abstract = {A novel solve-training framework is proposed to train neural
             network in representing low dimensional solution maps of
             physical models. Solve-training framework uses the neural
             network as the ansatz of the solution map and trains the
             network variationally via loss functions from the underlying
             physical models. Solve-training framework avoids expensive
             data preparation in the traditional supervised training
             procedure, which prepares labels for input data, and still
             achieves effective representation of the solution map
             adapted to the input data distribution. The efficiency of
             solve-training framework is demonstrated through obtaining
             solution maps for linear and nonlinear elliptic equations,
             and maps from potentials to ground states of linear and
             nonlinear Schrödinger equations.},
   Doi = {10.1016/j.jcp.2020.109338},
   Key = {fds348705}
}

@article{fds365309,
   Author = {Sachs, M and Sen, D and Lu, J and Dunson, D},
   Title = {Posterior computation with the Gibbs zig-zag
             sampler},
   Year = {2020},
   Month = {April},
   Abstract = {Markov chain Monte Carlo (MCMC) sampling algorithms have
             dominated the literature on posterior computation. However,
             MCMC faces substantial hurdles in performing efficient
             posterior sampling for challenging Bayesian models,
             particularly in high-dimensional and large data settings.
             Motivated in part by such hurdles, an intriguing new class
             of piecewise deterministic Markov processes (PDMPs) has
             recently been proposed as an alternative to MCMC. One of the
             most popular types of PDMPs is known as the zig-zag (ZZ)
             sampler. Such algorithms require a computational upper bound
             in a Poisson thinning step, with performance improving for
             tighter bounds. In order to facilitate scaling to larger
             classes of problems, we propose a general class of Gibbs
             zig-zag (GZZ) samplers. GZZ allows parameters to be updated
             in blocks with ZZ applied to certain parameters and
             traditional MCMC style updates to others. This provides a
             flexible framework to combine PDMPs with the rich literature
             on MCMC algorithms. We prove appealing theoretical
             properties of GZZ and demonstrate it on posterior sampling
             for logistic models with shrinkage priors for
             high-dimensional regression and random effects.},
   Key = {fds365309}
}

@article{fds361710,
   Author = {Gao, Y and Katsevich, AE and Liu, J-G and Lu, J and Marzuola,
             JL},
   Title = {Analysis of a fourth order exponential PDE arising from a
             crystal surface jump process with Metropolis-type transition
             rates},
   Journal = {Pure Appl. Analysis},
   Volume = {3},
   Pages = {595-612},
   Year = {2020},
   Month = {March},
   Abstract = {We analytically and numerically study a fourth order PDE
             modeling rough crystal surface diffusion on the macroscopic
             level. We discuss existence of solutions globally in time
             and long time dynamics for the PDE model. The PDE,
             originally derived by the second author, is the continuum
             limit of a microscopic model of the surface dynamics, given
             by a Markov jump process with Metropolis type transition
             rates. We outline the convergence argument, which depends on
             a simplifying assumption on the local equilibrium measure
             that is valid in the high temperature regime. We provide
             numerical evidence for the convergence of the microscopic
             model to the PDE in this regime.},
   Key = {fds361710}
}

@article{fds361711,
   Author = {Lu, J and Stubbs, KD and Watson, AB},
   Title = {Existence and computation of generalized Wannier functions
             for non-periodic systems in two dimensions and
             higher},
   Journal = {Arch. Rational Mech. Anal. 243},
   Volume = {3},
   Pages = {1269-1323},
   Year = {2020},
   Month = {March},
   Abstract = {Exponentially-localized Wannier functions (ELWFs) are an
             orthonormal basis of the Fermi projection of a material
             consisting of functions which decay exponentially fast away
             from their maxima. When the material is insulating and
             crystalline, conditions which guarantee existence of ELWFs
             in dimensions one, two, and three are well-known, and
             methods for constructing the ELWFs numerically are
             well-developed. We consider the case where the material is
             insulating but not necessarily crystalline, where much less
             is known. In one spatial dimension, Kivelson and
             Nenciu-Nenciu have proved ELWFs can be constructed as the
             eigenfunctions of a self-adjoint operator acting on the
             Fermi projection. In this work, we identify an assumption
             under which we can generalize the Kivelson-Nenciu-Nenciu
             result to two dimensions and higher. Under this assumption,
             we prove that ELWFs can be constructed as the eigenfunctions
             of a sequence of self-adjoint operators acting on the Fermi
             projection. We conjecture that the assumption we make is
             equivalent to vanishing of topological obstructions to the
             existence of ELWFs in the special case where the material is
             crystalline. We numerically verify that our construction
             yields ELWFs in various cases where our assumption holds and
             provide numerical evidence for our conjecture.},
   Key = {fds361711}
}

@article{fds361712,
   Author = {Ding, Z and Li, Q and Lu, J},
   Title = {Ensemble Kalman Inversion for nonlinear problems: weights,
             consistency, and variance bounds},
   Year = {2020},
   Month = {March},
   Abstract = {Ensemble Kalman Inversion (EnKI) and Ensemble Square Root
             Filter (EnSRF) are popular sampling methods for obtaining a
             target posterior distribution. They can be seem as one step
             (the analysis step) in the data assimilation method Ensemble
             Kalman Filter. Despite their popularity, they are, however,
             not unbiased when the forward map is nonlinear. Important
             Sampling (IS), on the other hand, obtains the unbiased
             sampling at the expense of large variance of weights,
             leading to slow convergence of high moments. We propose
             WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in
             this paper. It follows the same gradient flow as that of
             EnKI/EnSRF with weight corrections. Compared to the
             classical methods, the new methods are unbiased, and
             compared with IS, the method has bounded weight variance.
             Both properties will be proved rigorously in this paper. We
             further discuss the stability of the underlying
             Fokker-Planck equation. This partially explains why EnKI,
             despite being inconsistent, performs well occasionally in
             nonlinear settings. Numerical evidence will be demonstrated
             at the end.},
   Key = {fds361712}
}

@article{fds345425,
   Author = {Lu, J and Sachs, M and Steinerberger, S},
   Title = {Quadrature Points via Heat Kernel Repulsion},
   Journal = {Constructive Approximation},
   Volume = {51},
   Number = {1},
   Pages = {27-48},
   Year = {2020},
   Month = {February},
   url = {http://dx.doi.org/10.1007/s00365-019-09471-4},
   Abstract = {We discuss the classical problem of how to pick N weighted
             points on a d-dimensional manifold so as to obtain a
             reasonable quadrature rule 1|M|∫Mf(x)dx≃∑n=1Naif(xi).This
             problem, naturally, has a long history; the purpose of our
             paper is to propose selecting points and weights so as to
             minimize the energy functional ∑i,j=1Naiajexp(-d(xi,xj)24t)→min,wheret∼N-2/d,d(x, y)
             is the geodesic distance, and d is the dimension of the
             manifold. This yields point sets that are theoretically
             guaranteed, via spectral theoretic properties of the
             Laplacian - Δ , to have good properties. One nice aspect is
             that the energy functional is universal and independent of
             the underlying manifold; we show several numerical
             examples.},
   Doi = {10.1007/s00365-019-09471-4},
   Key = {fds345425}
}

@article{fds361690,
   Author = {Kovalsky, SZ and Aigerman, N and Daubechies, I and Kazhdan, M and Lu, J and Steinerberger, S},
   Title = {Non-Convex Planar Harmonic Maps},
   Year = {2020},
   Month = {January},
   Abstract = {We formulate a novel characterization of a family of
             invertible maps between two-dimensional domains. Our work
             follows two classic results: The Rad\'o-Kneser-Choquet (RKC)
             theorem, which establishes the invertibility of harmonic
             maps into a convex planer domain; and Tutte's embedding
             theorem for planar graphs - RKC's discrete counterpart -
             which proves the invertibility of piecewise linear maps of
             triangulated domains satisfying a discrete-harmonic
             principle, into a convex planar polygon. In both theorems,
             the convexity of the target domain is essential for ensuring
             invertibility. We extend these characterizations, in both
             the continuous and discrete cases, by replacing convexity
             with a less restrictive condition. In the continuous case,
             Alessandrini and Nesi provide a characterization of
             invertible harmonic maps into non-convex domains with a
             smooth boundary by adding additional conditions on
             orientation preservation along the boundary. We extend their
             results by defining a condition on the normal derivatives
             along the boundary, which we call the cone condition; this
             condition is tractable and geometrically intuitive, encoding
             a weak notion of local invertibility. The cone condition
             enables us to extend Alessandrini and Nesi to the case of
             harmonic maps into non-convex domains with a
             piecewise-smooth boundary. In the discrete case, we use an
             analog of the cone condition to characterize invertible
             discrete-harmonic piecewise-linear maps of triangulations.
             This gives an analog of our continuous results and
             characterizes invertible discrete-harmonic maps in terms of
             the orientation of triangles incident on the
             boundary.},
   Key = {fds361690}
}

@article{fds348624,
   Author = {Lu, J and Steinerberger, S},
   Title = {A dimension-free hermite-hadamard inequality via gradient
             estimates for the torsion function},
   Journal = {Proceedings of the American Mathematical
             Society},
   Volume = {148},
   Number = {2},
   Pages = {673-679},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1090/proc/14843},
   Abstract = {Let Ω ⊂ Rn be a convex domain, and let f : Ω → R be a
             subharmonic function, Δf ≥ 0, which satisfies f ≥ 0 on
             the boundary ∂Ω. Then (Formula Presented) Our proof is
             based on a new gradient estimate for the torsion function,
             Δu = -1 with Dirichlet boundary conditions, which is of
             independent interest.},
   Doi = {10.1090/proc/14843},
   Key = {fds348624}
}

@article{fds350797,
   Author = {Chen, K and Li, Q and Lu, J and Wright, SJ},
   Title = {Randomized sampling for basis function construction in
             generalized finite element methods},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {18},
   Number = {2},
   Pages = {1153-1177},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1137/18M1166432},
   Abstract = {In the framework of generalized finite element methods for
             elliptic equations with rough coefficients, efficiency and
             accuracy of the numerical method depend critically on the
             use of appropriate basis functions. This work explores
             several random sampling strategies that construct
             approximations to the optimal set of basis functions of a
             given dimension, and proposes a quantitative criterion to
             analyze and compare these sampling strategies. Numerical
             evidence shows that the best results are achieved by two
             strategies, Random Gaussian and Smooth Boundary
             sampling.},
   Doi = {10.1137/18M1166432},
   Key = {fds350797}
}

@article{fds349488,
   Author = {Lu, J and Wang, Z},
   Title = {The full configuration interaction quantum monte carlo
             method through the lens of inexact power
             iteration},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {42},
   Number = {1},
   Pages = {B1-B29},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1137/18M1166626},
   Abstract = {In this paper, we propose a general analysis framework for
             inexact power iteration, which can be used to efficiently
             solve high-dimensional eigenvalue problems arising from
             quantum many-body problems. Under this framework, we
             establish the convergence theorems for several recently
             proposed randomized algorithms, including full configuration
             interaction quantum Monte Carlo and fast randomized
             iteration. The analysis is consistent with numerical
             experiments for physical systems such as the Hubbard model
             and small chemical molecules. We also compare the algorithms
             both in convergence analysis and numerical
             results.},
   Doi = {10.1137/18M1166626},
   Key = {fds349488}
}

@article{fds349467,
   Author = {Li, L and Li, Y and Liu, JG and Liu, Z and Lu, J},
   Title = {A stochastic version of stein variational gradient descent
             for efficient sampling},
   Journal = {Communications in Applied Mathematics and Computational
             Science},
   Volume = {15},
   Number = {1},
   Pages = {37-63},
   Publisher = {Mathematical Sciences Publishers},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.2140/camcos.2020.15.37},
   Abstract = {We propose in this work RBM-SVGD, a stochastic version of
             the Stein variational gradient descent (SVGD) method for
             efficiently sampling from a given probability measure, which
             is thus useful for Bayesian inference. The method is to
             apply the random batch method (RBM) for interacting particle
             systems proposed by Jin et al. to the interacting particle
             systems in SVGD. While keeping the behaviors of SVGD, it
             reduces the computational cost, especially when the
             interacting kernel has long range. We prove that the one
             marginal distribution of the particles generated by this
             method converges to the one marginal of the interacting
             particle systems under Wasserstein-2 distance on fixed time
             interval T0; T U. Numerical examples verify the efficiency
             of this new version of SVGD.},
   Doi = {10.2140/camcos.2020.15.37},
   Key = {fds349467}
}

@article{fds349647,
   Author = {Lu, J and Watson, AB and Weinstein, MI},
   Title = {Dirac operators and domain walls},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {52},
   Number = {2},
   Pages = {1115-1145},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1137/19M127416X},
   Abstract = {We study the eigenvalue problem for a one-dimensional Dirac
             operator with a spatially varying "mass" term. It is
             well-known that when the mass function has the form of a
             kink, or domain wall, transitioning between strictly
             positive and strictly negative asymptotic mass, ±κ∞, at
             ±∞, the Dirac operator has a simple eigenvalue of zero
             energy (geometric multiplicity equal to one) within a gap in
             the continuous spectrum, with corresponding exponentially
             localized zero mode. We consider the eigenvalue problem for
             the one-dimensional Dirac operator with mass function
             defined by "gluing" together n domain wall-type transitions,
             assuming that the distance between transitions, 2δ, is
             sufficiently large, focusing on the illustrative cases n = 2
             and 3. When n = 2 we prove that the Dirac operator has two
             real simple eigenvalues of opposite sign and of order
             e-2|κ∞|δ. The associated eigenfunctions are, up to L2
             error of order e-2|κ∞|δ, linear combinations of shifted
             copies of the single domain wall zero mode. For the case n =
             3, we prove the Dirac operator has two nonzero simple
             eigenvalues as in the two domain wall case and a simple
             eigenvalue at energy zero. The associated eigenfunctions of
             these eigenvalues can again, up to small error, be expressed
             as linear combinations of shifted copies of the single
             domain wall zero mode. When n > 3 no new technical
             difficulty arises and the result is similar. Our methods are
             based on a Lyapunov-Schmidt reduction/ Schur complement
             strategy, which maps the Dirac operator eigenvalue problem
             for eigenstates with near-zero energies to the problem of
             determining the kernel of an n×n matrix reduction, which
             depends nonlinearly on the eigenvalue parameter. The class
             of Dirac operators we consider controls the bifurcation of
             topologically protected "edge states" from Dirac points
             (linear band crossings) for classes of Schrödinger
             operators with domain wall modulated periodic potentials in
             one and two space dimensions. The present results may be
             used to construct a rich class of defect modes in periodic
             structures modulated by multiple domain walls.},
   Doi = {10.1137/19M127416X},
   Key = {fds349647}
}

@article{fds352547,
   Author = {CHEN, Z and LI, Y and LU, J},
   Title = {Tensor ring decomposition: Optimization landscape and
             one-loop convergence of alternating least
             squares},
   Journal = {SIAM Journal on Matrix Analysis and Applications},
   Volume = {41},
   Number = {3},
   Pages = {1416-1442},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1137/19M1270689},
   Abstract = {In this work, we study the tensor ring decomposition and its
             associated numerical algorithms. We establish a sharp
             transition of algorithmic difficulty of the optimization
             problem as the bond dimension increases: On one hand, we
             show the existence of spurious local minima for the
             optimization landscape even when the tensor ring format is
             much overparameterized, i.e., with bond dimension much
             larger than that of the true target tensor. On the other
             hand, when the bond dimension is further increased, we
             establish one-loop convergence for the alternating least
             squares algorithm for the tensor ring decomposition. The
             theoretical results are complemented by numerical
             experiments for both local minima and the one-loop
             convergence for the alternating least squares
             algorithm.},
   Doi = {10.1137/19M1270689},
   Key = {fds352547}
}

@article{fds352784,
   Author = {Chen, K and Li, Q and Lu, J and Wright, SJ},
   Title = {Random sampling and efficient algorithms for multiscale
             pdes},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {42},
   Number = {5},
   Pages = {A2974-A3005},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1137/18M1207430},
   Abstract = {We describe a numerical framework that uses random sampling
             to efficiently capture low-rank local solution spaces of
             multiscale PDE problems arising in domain decomposition. In
             contrast to existing techniques, our method does not rely on
             detailed analytical understanding of specific multiscale
             PDEs, in particular, their asymptotic limits. We present the
             application of the framework on two examples-a linear
             kinetic equation and an elliptic equation with rough media.
             On these two examples, this framework achieves the
             asymptotic preserving property for the kinetic equations and
             numerical homogenization for the elliptic
             equations.},
   Doi = {10.1137/18M1207430},
   Key = {fds352784}
}

@article{fds355479,
   Author = {An, J and Lu, J and Ying, L},
   Title = {Stochastic modified equations for the asynchronous
             stochastic gradient descent},
   Journal = {Information and Inference},
   Volume = {9},
   Number = {4},
   Pages = {851-873},
   Year = {2020},
   Month = {January},
   url = {http://dx.doi.org/10.1093/IMAIAI/IAZ030},
   Abstract = {We propose stochastic modified equations (SMEs) for
             modelling the asynchronous stochastic gradient descent
             (ASGD) algorithms. The resulting SME of Langevin type
             extracts more information about the ASGD dynamics and
             elucidates the relationship between different types of
             stochastic gradient algorithms. We show the convergence of
             ASGD to the SME in the continuous time limit, as well as the
             SME's precise prediction to the trajectories of ASGD with
             various forcing terms. As an application, we propose an
             optimal mini-batching strategy for ASGD via solving the
             optimal control problem of the associated
             SME.},
   Doi = {10.1093/IMAIAI/IAZ030},
   Key = {fds355479}
}

@article{fds356446,
   Author = {Lu, Y and Ma, C and Lu, J and Ying, L},
   Title = {A mean-field analysis of deep resnet and beyond: Towards
             provable optimization via overparameterization from
             depth},
   Journal = {37th International Conference on Machine Learning, ICML
             2020},
   Volume = {PartF168147-9},
   Pages = {6382-6392},
   Year = {2020},
   Month = {January},
   Abstract = {Training deep neural networks with stochastic gradient
             descent (SGD) can often achieve zero training loss on
             real-world tasks although the optimization landscape is
             known to be highly non-convex. To understand the success of
             SGD for training deep neural networks, this work presents a
             meanfield analysis of deep residual networks, based on a
             line of works that interpret the continuum limit of the deep
             residual network as an ordinary differential equation when
             the network capacity tends to infinity. Specifically, we
             propose a new continuum limit of deep residual networks,
             which enjoys a good landscape in the sense that every local
             minimizer is global. This characterization enables us to
             derive the first global convergence result for multilayer
             neural networks in the meanfield regime. Furthermore,
             without assuming the convexity of the loss landscape, our
             proof relies on a zero-loss assumption at the global
             minimizer that can be achieved when the model shares a
             universal approximation property. Key to our result is the
             observation that a deep residual network resembles a shallow
             network ensemble (Veit et al., 2016), i.e. a two-layer
             network. We bound the difference between the shallow network
             and our ResNet model via the adjoint sensitivity method,
             which enables us to apply existing mean-field analyses of
             two-layer networks to deep networks. Furthermore, we propose
             several novel training schemes based on the new continuous
             model, including one training procedure that switches the
             order of the residual blocks and results in strong empirical
             performance on the benchmark datasets.},
   Key = {fds356446}
}

@article{fds357554,
   Author = {Lu, Y and Lu, J},
   Title = {A universal approximation theorem of deep neural networks
             for expressing probability distributions},
   Journal = {Advances in Neural Information Processing
             Systems},
   Volume = {2020-December},
   Year = {2020},
   Month = {January},
   Abstract = {This paper studies the universal approximation property of
             deep neural networks for representing probability
             distributions. Given a target distribution p and a source
             distribution pz both defined on Rd, we prove under some
             assumptions that there exists a deep neural network g : Rd?R
             with ReLU activation such that the push-forward measure
             (?g)#pz of pz under the map ?g is arbitrarily close to the
             target measure p. The closeness are measured by three
             classes of integral probability metrics between probability
             distributions: 1-Wasserstein distance, maximum mean distance
             (MMD) and kernelized Stein discrepancy (KSD). We prove upper
             bounds for the size (width and depth) of the deep neural
             network in terms of the dimension d and the approximation
             error e with respect to the three discrepancies. In
             particular, the size of neural network can grow
             exponentially in d when 1-Wasserstein distance is used as
             the discrepancy, whereas for both MMD and KSD the size of
             neural network only depends on d at most polynomially. Our
             proof relies on convergence estimates of empirical measures
             under aforementioned discrepancies and semi-discrete optimal
             transport.},
   Key = {fds357554}
}

@article{fds361691,
   Author = {Han, J and Li, Y and Lin, L and Lu, J and Zhang, J and Zhang,
             L},
   Title = {Universal approximation of symmetric and anti-symmetric
             functions},
   Year = {2019},
   Month = {December},
   Abstract = {We consider universal approximations of symmetric and
             anti-symmetric functions, which are important for
             applications in quantum physics, as well as other scientific
             and engineering computations. We give constructive
             approximations with explicit bounds on the number of
             parameters with respect to the dimension and the target
             accuracy $\epsilon$. While the approximation still suffers
             from the curse of dimensionality, to the best of our
             knowledge, these are the first results in the literature
             with explicit error bounds for functions with symmetry or
             anti-symmetry constraints.},
   Key = {fds361691}
}

@article{fds346283,
   Author = {Chen, H and Li, Q and Lu, J},
   Title = {A numerical method for coupling the BGK model and Euler
             equations through the linearized Knudsen
             layer},
   Journal = {Journal of Computational Physics},
   Volume = {398},
   Year = {2019},
   Month = {December},
   url = {http://dx.doi.org/10.1016/j.jcp.2019.108893},
   Abstract = {The Bhatnagar-Gross-Krook (BGK) model, a simplification of
             the Boltzmann equation, in the absence of boundary effect,
             converges to the Euler equations when the Knudsen number is
             small. In practice, however, Knudsen layers emerge at the
             physical boundary, or at the interfaces between the two
             regimes. We model the Knudsen layer using a half-space
             kinetic equation, and apply a half-space numerical solver
             [19,20] to quantify the transition between the kinetic to
             the fluid regime. A full domain numerical solver is
             developed with a domain-decomposition approach, where we
             apply the Euler solver and kinetic solver on the appropriate
             subdomains and connect them via the half-space solver. In
             the nonlinear case, linearization is performed upon local
             Maxwellian. Despite the lack of analytical support, the
             numerical evidence nevertheless demonstrate that the
             linearization approach is promising.},
   Doi = {10.1016/j.jcp.2019.108893},
   Key = {fds346283}
}

@article{fds343709,
   Author = {Lu, J and Sogge, CD and Steinerberger, S},
   Title = {Approximating pointwise products of Laplacian
             eigenfunctions},
   Journal = {Journal of Functional Analysis},
   Volume = {277},
   Number = {9},
   Pages = {3271-3282},
   Year = {2019},
   Month = {November},
   url = {http://dx.doi.org/10.1016/j.jfa.2019.05.025},
   Abstract = {We consider Laplacian eigenfunctions on a d-dimensional
             bounded domain M (or a d-dimensional compact manifold M)
             with Dirichlet conditions. These operators give rise to a
             sequence of eigenfunctions (eℓ)ℓ∈N. We study the
             subspace of all pointwise products An=span{ei(x)ej(x):1≤i,j≤n}⊆L2(M).
             Clearly, that vector space has dimension dim(An)=n(n+1)/2.
             We prove that products eiej of eigenfunctions are simple in
             a certain sense: for any ε>0, there exists a
             low-dimensional vector space Bn that almost contains all
             products. More precisely, denoting the orthogonal projection
             ΠBn:L2(M)→Bn, we have ∀1≤i,j≤n‖eiej−ΠBn(eiej)‖L2≤ε
             and the size of the space dim(Bn) is relatively small: for
             every δ>0, dim(Bn)≲M,δε−δn1+δ. We obtain the same
             sort of bounds for products of arbitrary length, as well for
             approximation in H−1 norm. Pointwise products of
             eigenfunctions are low-rank. This has implications, among
             other things, for the validity of fast algorithms in
             electronic structure computations.},
   Doi = {10.1016/j.jfa.2019.05.025},
   Key = {fds343709}
}

@article{fds344664,
   Author = {Cao, Y and Lu, J and Lu, Y},
   Title = {Exponential Decay of Rényi Divergence Under Fokker–Planck
             Equations},
   Journal = {Journal of Statistical Physics},
   Volume = {176},
   Number = {5},
   Pages = {1172-1184},
   Year = {2019},
   Month = {September},
   url = {http://dx.doi.org/10.1007/s10955-019-02339-8},
   Abstract = {We prove the exponential convergence to the equilibrium,
             quantified by Rényi divergence, of the solution of the
             Fokker–Planck equation with drift given by the gradient of
             a strictly convex potential. This extends the classical
             exponential decay result on the relative entropy for the
             same equation.},
   Doi = {10.1007/s10955-019-02339-8},
   Key = {fds344664}
}

@article{fds362601,
   Author = {Cao, Y and Lu, J and Wang, L},
   Title = {On explicit $L^2$-convergence rate estimate for underdamped
             Langevin dynamics},
   Journal = {Arch Rational Mech Anal},
   Volume = {247},
   Pages = {90},
   Year = {2019},
   Month = {August},
   Abstract = {We provide a refined explicit estimate of exponential decay
             rate of underdamped Langevin dynamics in $L^2$ distance,
             based on a framework developed in [1]. To achieve this, we
             first prove a Poincar\'{e}-type inequality with Gibbs
             measure in space and Gaussian measure in momentum. Our
             estimate provides a more explicit and simpler expression of
             decay rate; moreover, when the potential is convex with
             Poincar\'{e} constant $m \ll 1$, our estimate shows the
             decay rate of $O(\sqrt{m})$ after optimizing the choice of
             friction coefficient, which is much faster than $m$ for the
             overdamped Langevi dynamics.},
   Key = {fds362601}
}

@article{fds342763,
   Author = {Wang, Z and Li, Y and Lu, J},
   Title = {Coordinate Descent Full Configuration Interaction.},
   Journal = {Journal of chemical theory and computation},
   Volume = {15},
   Number = {6},
   Pages = {3558-3569},
   Year = {2019},
   Month = {June},
   url = {http://dx.doi.org/10.1021/acs.jctc.9b00138},
   Abstract = {We develop an efficient algorithm, coordinate descent FCI
             (CDFCI), for the electronic structure ground-state
             calculation in the configuration interaction framework.
             CDFCI solves an unconstrained nonconvex optimization
             problem, which is a reformulation of the full configuration
             interaction eigenvalue problem, via an adaptive coordinate
             descent method with a deterministic compression strategy.
             CDFCI captures and updates appreciative determinants with
             different frequencies proportional to their importance. We
             show that CDFCI produces accurate variational energy for
             both static and dynamic correlation by benchmarking the
             binding curve of nitrogen dimer in the cc-pVDZ basis with
             10<sup>-3</sup> mHa accuracy. We also demonstrate the
             efficiency and accuracy of CDFCI for strongly correlated
             chromium dimer in the Ahlrichs VDZ basis and produce
             state-of-the-art variational energy.},
   Doi = {10.1021/acs.jctc.9b00138},
   Key = {fds342763}
}

@article{fds341501,
   Author = {Liu, JG and Lu, J and Margetis, D and Marzuola, JL},
   Title = {Asymmetry in crystal facet dynamics of homoepitaxy by a
             continuum model},
   Journal = {Physica D: Nonlinear Phenomena},
   Volume = {393},
   Pages = {54-67},
   Year = {2019},
   Month = {June},
   url = {http://dx.doi.org/10.1016/j.physd.2019.01.004},
   Abstract = {In the absence of external material deposition, crystal
             surfaces usually relax to become flat by decreasing their
             free energy. We study analytically an asymmetry in the
             relaxation of macroscopic plateaus, facets, of a periodic
             surface corrugation in 1+1 dimensions via a continuum model
             below the roughening transition temperature. The model
             invokes a continuum evolution law expressed by a highly
             degenerate parabolic partial differential equation (PDE) for
             surface diffusion, which is related to the nonlinear
             gradient flow of a convex, singular surface free energy with
             a certain exponential mobility in homoepitaxy. This
             evolution law is motivated both by an atomistic broken-bond
             model and a mesoscale model for crystal steps. By
             constructing an explicit solution to this PDE, we
             demonstrate the lack of symmetry in the evolution of top and
             bottom facets in periodic surface profiles. Our explicit,
             analytical solution is compared to numerical simulations of
             the continuum law via a regularized surface free
             energy.},
   Doi = {10.1016/j.physd.2019.01.004},
   Key = {fds341501}
}

@article{fds347600,
   Author = {Agazzi, A and Lu, J},
   Title = {Temporal-difference learning with nonlinear function
             approximation: lazy training and mean field
             regimes},
   Journal = {PMLR},
   Volume = {145},
   Pages = {37-74},
   Year = {2019},
   Month = {May},
   Abstract = {We discuss the approximation of the value function for
             infinite-horizon discounted Markov Reward Processes (MRP)
             with nonlinear functions trained with the
             Temporal-Difference (TD) learning algorithm. We first
             consider this problem under a certain scaling of the
             approximating function, leading to a regime called lazy
             training. In this regime, the parameters of the model vary
             only slightly during the learning process, a feature that
             has recently been observed in the training of neural
             networks, where the scaling we study arises naturally,
             implicit in the initialization of their parameters. Both in
             the under- and over-parametrized frameworks, we prove
             exponential convergence to local, respectively global
             minimizers of the above algorithm in the lazy training
             regime. We then compare this scaling of the parameters to
             the mean-field regime, where the approximately linear
             behavior of the model is lost. Under this alternative
             scaling we prove that all fixed points of the dynamics in
             parameter space are global minimizers. We finally give
             examples of our convergence results in the case of models
             that diverge if trained with non-lazy TD learning, and in
             the case of neural networks.},
   Key = {fds347600}
}

@article{fds361538,
   Author = {Lu, Y and Lu, J and Nolen, J},
   Title = {Accelerating Langevin Sampling with Birth-death},
   Year = {2019},
   Month = {May},
   Abstract = {A fundamental problem in Bayesian inference and statistical
             machine learning is to efficiently sample from multimodal
             distributions. Due to metastability, multimodal
             distributions are difficult to sample using standard Markov
             chain Monte Carlo methods. We propose a new sampling
             algorithm based on a birth-death mechanism to accelerate the
             mixing of Langevin diffusion. Our algorithm is motivated by
             its mean field partial differential equation (PDE), which is
             a Fokker-Planck equation supplemented by a nonlocal
             birth-death term. This PDE can be viewed as a gradient flow
             of the Kullback-Leibler divergence with respect to the
             Wasserstein-Fisher-Rao metric. We prove that under some
             assumptions the asymptotic convergence rate of the nonlocal
             PDE is independent of the potential barrier, in contrast to
             the exponential dependence in the case of the Langevin
             diffusion. We illustrate the efficiency of the birth-death
             accelerated Langevin method through several analytical
             examples and numerical experiments.},
   Key = {fds361538}
}

@article{fds343500,
   Author = {Cao, Y and Lu, J and Lu, Y},
   Title = {Gradient flow structure and exponential decay of the
             sandwiched Rényi divergence for primitive Lindblad
             equations with GNS-detailed balance},
   Journal = {Journal of Mathematical Physics},
   Volume = {60},
   Number = {5},
   Pages = {052202-052202},
   Publisher = {AIP Publishing},
   Year = {2019},
   Month = {May},
   url = {http://dx.doi.org/10.1063/1.5083065},
   Abstract = {We study the entropy production of the sandwiched Rényi
             divergence under the primitive Lindblad equation with
             Gel'fand-Naimark-Segal-detailed balance. We prove that the
             Lindblad equation can be identified as the gradient flow of
             the sandwiched Rényi divergence of any order α ∈ (0,
             ∞). This extends a previous result by Carlen and Maas [J.
             Funct. Anal. 273(5), 1810-1869 (2017)] for the quantum
             relative entropy (i.e., α = 1). Moreover, we show that the
             sandwiched Rényi divergence of any order α ∈ (0, ∞)
             decays exponentially fast under the time evolution of such a
             Lindblad equation.},
   Doi = {10.1063/1.5083065},
   Key = {fds343500}
}

@article{fds346699,
   Author = {Lin, L and Lu, J and Ying, L},
   Title = {Numerical methods for Kohn-Sham density functional
             theory},
   Journal = {Acta Numerica},
   Volume = {28},
   Pages = {405-539},
   Year = {2019},
   Month = {May},
   url = {http://dx.doi.org/10.1017/S0962492919000047},
   Abstract = {Kohn-Sham density functional theory (DFT) is the most widely
             used electronic structure theory. Despite significant
             progress in the past few decades, the numerical solution of
             Kohn-Sham DFT problems remains challenging, especially for
             large-scale systems. In this paper we review the basics as
             well as state-of-the-art numerical methods, and focus on the
             unique numerical challenges of DFT.},
   Doi = {10.1017/S0962492919000047},
   Key = {fds346699}
}

@article{fds346493,
   Author = {Yu, V and Dawson, W and Garcia, A and Havu, V and Hourahine, B and Huhn, W and Jacquelin, M and Jia, W and Keceli, M and Laasner, R and Li, Y and Lin, L and Lu, J and Roman, J and Vazquez-Mayagoitia, A and Yang, C and Blum,
             V},
   Title = {Large-scale benchmark of electronic structure solvers with
             the ELSI infrastructure},
   Journal = {ABSTRACTS OF PAPERS OF THE AMERICAN CHEMICAL
             SOCIETY},
   Volume = {257},
   Pages = {1 pages},
   Publisher = {AMER CHEMICAL SOC},
   Year = {2019},
   Month = {March},
   Key = {fds346493}
}

@article{fds340897,
   Author = {Lu, J and Vanden-Eijnden, E},
   Title = {Methodological and Computational Aspects of Parallel
             Tempering Methods in the Infinite Swapping
             Limit},
   Journal = {Journal of Statistical Physics},
   Volume = {174},
   Number = {3},
   Pages = {715-733},
   Year = {2019},
   Month = {February},
   url = {http://dx.doi.org/10.1007/s10955-018-2210-y},
   Abstract = {A variant of the parallel tempering method is proposed in
             terms of a stochastic switching process for the coupled
             dynamics of replica configuration and temperature
             permutation. This formulation is shown to facilitate the
             analysis of the convergence properties of parallel tempering
             by large deviation theory, which indicates that the method
             should be operated in the infinite swapping limit to
             maximize sampling efficiency. The effective equation for the
             replica alone that arises in this infinite swapping limit
             simply involves replacing the original potential by a
             mixture potential. The analysis of the geometric properties
             of this potential offers a new perspective on the issues of
             how to choose of temperature ladder, and why many
             temperatures should typically be introduced to boost the
             sampling efficiency. It is also shown how to simulate the
             effective equation in this many temperature regime using
             multiscale integrators. Finally, similar ideas are also used
             to discuss extensions of the infinite swapping limits to the
             technique of simulated tempering.},
   Doi = {10.1007/s10955-018-2210-y},
   Key = {fds340897}
}

@article{fds361345,
   Author = {Holst, M and Hu, H and Lu, J and Marzuola, JL and Song, D and Weare,
             J},
   Title = {Symmetry Breaking in Density Functional Theory due to Dirac
             Exchange for a Hydrogen Molecule},
   Year = {2019},
   Month = {February},
   Abstract = {We study symmetry breaking in the mean field solutions to
             the 2 electron hydrogen molecule within Kohn Sham (KS) local
             spin density function theory with Dirac exchange (the XLDA
             model). This simplified model shows behavior related to that
             of the (KS) spin density functional theory (SDFT)
             predictions in condensed and molecular systems. The Kohn
             Sham solutions to the constrained SDFT variation problem
             undergo spontaneous symmetry breaking as the relative
             strength of the non-convex exchange term increases. This
             results in the change of the molecular ground state from a
             paramagnetic state to an antiferromagnetic ground states and
             a stationary symmetric delocalized 1st excited state. We
             further characterize the limiting behavior of the minimizer
             when the strength of the exchange term goes to infinity.
             This leads to further bifurcations and highly localized
             states with varying character. The stability of the various
             solution classes is demonstrated by Hessian analysis. Finite
             element numerical results provide support for the formal
             conjectures.},
   Key = {fds361345}
}

@article{fds341434,
   Author = {Li, Y and Lu, J},
   Title = {Bold diagrammatic Monte Carlo in the lens of stochastic
             iterative methods},
   Journal = {Transactions of Mathematics and Its Applications},
   Volume = {3},
   Number = {1},
   Pages = {1-17},
   Publisher = {Oxford University Press (OUP)},
   Year = {2019},
   Month = {February},
   url = {http://dx.doi.org/10.1093/imatrm/tnz001},
   Abstract = {<jats:title>Abstract</jats:title> <jats:p>This work aims at
             understanding of bold diagrammatic Monte Carlo (BDMC)
             methods for stochastic summation of Feynman diagrams from
             the angle of stochastic iterative methods. The convergence
             enhancement trick of the BDMC is investigated from the
             analysis of condition number and convergence of the
             stochastic iterative methods. Numerical experiments are
             carried out for model systems to compare the BDMC with
             related stochastic iterative approaches.</jats:p>},
   Doi = {10.1093/imatrm/tnz001},
   Key = {fds341434}
}

@article{fds341334,
   Author = {Martinsson, A and Lu, J and Leimkuhler, B and Vanden-Eijnden,
             E},
   Title = {The simulated tempering method in the infinite switch limit
             with adaptive weight learning},
   Journal = {Journal of Statistical Mechanics: Theory and
             Experiment},
   Volume = {2019},
   Number = {1},
   Pages = {013207-013207},
   Publisher = {IOP Publishing},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1088/1742-5468/aaf323},
   Abstract = {We investigate the theoretical foundations of the simulated
             tempering (ST) method and use our findings to design an
             efficient accelerated sampling algorithm. Employing a large
             deviation argument first used for replica exchange molecular
             dynamics (Plattner et al 2011 J. Chem. Phys. 135 134111), we
             demonstrate that the most efficient approach to simulated
             tempering is to vary the temperature infinitely rapidly. In
             this limit, we can replace the equations of motion for the
             temperature and physical variables by averaged equations for
             the latter alone, with the forces rescaled according to a
             position-dependent function defined in terms of temperature
             weights. The averaged equations are similar to those used in
             Gao's integrated-over-temperature method, except that we
             show that it is better to use a continuous rather than a
             discrete set of temperatures. We give a theoretical argument
             for the choice of the temperature weights as the reciprocal
             partition function, thereby relating simulated tempering to
             Wang-Landau sampling. Finally, we describe a self-consistent
             algorithm for simultaneously sampling the canonical ensemble
             and learning the weights during simulation. This infinite
             switch simulated tempering (ISST) algorithm is tested on
             three examples of increasing complexity: a system of
             harmonic oscillators; a continuous variant of the
             Curie-Weiss model, where ISST is shown to perform better
             than standard ST and to accurately capture the second-order
             phase transition observed in this model; and alanine-12 in
             vacuum, where ISST also compares favorably with standard ST
             in its ability to calculate the free energy profiles of the
             root mean square deviation (RMSD) and radius of gyration of
             the molecule in the 300-500 K temperature
             range.},
   Doi = {10.1088/1742-5468/aaf323},
   Key = {fds341334}
}

@article{fds340898,
   Author = {Huang, H and Liu, JG and Lu, J},
   Title = {Learning interacting particle systems: Diffusion parameter
             estimation for aggregation equations},
   Journal = {Mathematical Models and Methods in Applied
             Sciences},
   Volume = {29},
   Number = {1},
   Pages = {1-29},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1142/S0218202519500015},
   Abstract = {In this paper, we study the parameter estimation of
             interacting particle systems subject to the Newtonian
             aggregation and Brownian diffusion. Specifically, we
             construct an estimator with partial observed data to
             approximate the diffusion parameter , and the estimation
             error is achieved. Furthermore, we extend this result to
             general aggregation equations with a bounded Lipschitz
             interaction field.},
   Doi = {10.1142/S0218202519500015},
   Key = {fds340898}
}

@article{fds340591,
   Author = {Gauckler, L and Lu, J and Marzuola, JL and Rousset, F and Schratz,
             K},
   Title = {Trigonometric integrators for quasilinear wave
             equations},
   Journal = {Mathematics of Computation},
   Volume = {88},
   Number = {316},
   Pages = {717-749},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3339},
   Abstract = {Trigonometric time integrators are introduced as a class of
             explicit numerical methods for quasilinear wave equations.
             Second-order convergence for the semidiscretization in time
             with these integrators is shown for a sufficiently regular
             exact solution. The time integrators are also combined with
             a Fourier spectral method into a fully discrete scheme, for
             which error bounds are provided without requiring any
             CFL-type coupling of the discretization parameters. The
             proofs of the error bounds are based on energy techniques
             and on the semiclassical Gårding inequality.},
   Doi = {10.1090/mcom/3339},
   Key = {fds340591}
}

@article{fds348057,
   Author = {Khoo, Y and Lu, J and Ying, L},
   Title = {Solving for high-dimensional committor functions using
             artificial neural networks},
   Journal = {Research in Mathematical Sciences},
   Volume = {6},
   Number = {1},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1007/s40687-018-0160-2},
   Abstract = {In this note we propose a method based on artificial neural
             network to study the transition between states governed by
             stochastic processes. In particular, we aim for numerical
             schemes for the committor function, the central object of
             transition path theory, which satisfies a high-dimensional
             Fokker–Planck equation. By working with the variational
             formulation of such partial differential equation and
             parameterizing the committor function in terms of a neural
             network, approximations can be obtained via optimizing the
             neural network weights using stochastic algorithms. The
             numerical examples show that moderate accuracy can be
             achieved for high-dimensional problems.},
   Doi = {10.1007/s40687-018-0160-2},
   Key = {fds348057}
}

@article{fds345876,
   Author = {Yingzhou, LI and Jianfeng, LU and Wang, AZHE},
   Title = {Coordinatewise descent methods for leading eigenvalue
             problem},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {41},
   Number = {4},
   Pages = {A2681-A2716},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1137/18M1202505},
   Abstract = {Leading eigenvalue problems for large scale matrices arise
             in many applications. Coordinatewise descent methods are
             considered in this work for such problems based on a
             reformulation of the leading eigenvalue problem as a
             nonconvex optimization problem. The convergence of several
             coordinatewise methods is analyzed and compared. Numerical
             examples of applications to quantum many-body problems
             demonstrate the efficiency and provide benchmarks of the
             proposed coordinatewise descent methods.},
   Doi = {10.1137/18M1202505},
   Key = {fds345876}
}

@article{fds352987,
   Author = {Cao, Y and Lu, J},
   Title = {Tensorization of the strong data processing inequality for
             quantum chi-square divergences},
   Journal = {Quantum},
   Volume = {3},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.22331/q-2019-10-28-199},
   Abstract = {It is well-known that any quantum channel E satisfies the
             data processing inequality (DPI), with respect to various
             divergences, e.g., quantum χ2κ divergences and quantum
             relative entropy. More specifically, the data processing
             inequality states that the divergence between two arbitrary
             quantum states ρ and σ does not increase under the action
             of any quantum channel E. For a fixed channel E and a state
             σ, the divergence between output states E(ρ) and E(σ)
             might be strictly smaller than the divergence between input
             states ρ and σ, which is characterized by the strong data
             processing inequality (SDPI). Among various input states ρ,
             the largest value of the rate of contraction is known as the
             SDPI constant. An important and widely studied property for
             classical channels is that SDPI constants tensorize. In this
             paper, we extend the tensorization property to the quantum
             regime: we establish the tensorization of SDPIs for the
             quantum χ2κ1/2 divergence for arbitrary quantum channels
             and also for a family of χ2κ divergences (with κ ≥
             κ1/2) for arbitrary quantum-classical channels.},
   Doi = {10.22331/q-2019-10-28-199},
   Key = {fds352987}
}

@article{fds362642,
   Author = {Zhu, W and Qiu, Q and Wang, B and Lu, J and Sapiro, G and Daubechies,
             I},
   Title = {Stop Memorizing: A Data-Dependent Regularization Framework
             for Intrinsic Pattern Learning},
   Journal = {SIAM Journal on Mathematics of Data Science},
   Volume = {1},
   Number = {3},
   Pages = {476-496},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1137/19m1236886},
   Doi = {10.1137/19m1236886},
   Key = {fds362642}
}

@article{fds342550,
   Author = {Nolen, JH and Lu, J and Lu, Y},
   Title = {Scaling limit of the Stein variational gradient descent: the
             mean field regime},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {51},
   Number = {2},
   Pages = {648-671},
   Publisher = {Society for Industrial and Applied Mathematics},
   Year = {2019},
   url = {http://dx.doi.org/10.1137/18M1187611},
   Abstract = {We study an interacting particle system in Rd motivated by
             Stein variational gradient descent [Q. Liu and D. Wang,
             Proceedings of NIPS, 2016], a deterministic algorithm for
             approximating a given probability density with unknown
             normalization based on particles. We prove that in the large
             particle limit the empirical measure of the particle system
             converges to a solution of a nonlocal and nonlinear PDE. We
             also prove the global existence, uniqueness, and regularity
             of the solution to the limiting PDE. Finally, we prove that
             the solution to the PDE converges to the unique invariant
             solution in a long time limit.},
   Doi = {10.1137/18M1187611},
   Key = {fds342550}
}

@article{fds365310,
   Author = {Lu, Y and Lu, J and Nolen, J},
   Title = {Accelerating Langevin Sampling with Birth-death},
   Year = {2019},
   Key = {fds365310}
}

@article{fds340245,
   Author = {Chen, H and Lu, J and Ortner, C},
   Title = {Thermodynamic Limit of Crystal Defects with Finite
             Temperature Tight Binding},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {230},
   Number = {2},
   Pages = {701-733},
   Publisher = {Springer Nature America, Inc},
   Year = {2018},
   Month = {November},
   url = {http://dx.doi.org/10.1007/s00205-018-1256-y},
   Abstract = {We consider a tight binding model for localised crystalline
             defects with electrons in the canonical ensemble (finite
             Fermi temperature) and nuclei positions relaxed according to
             the Born–Oppenheimer approximation. We prove that the
             limit model as the computational domain size grows to
             infinity is formulated in the grand-canonical ensemble for
             the electrons. The Fermi-level for the limit model is fixed
             at a homogeneous crystal level, independent of the defect or
             electron number in the sequence of finite-domain
             approximations. We quantify the rates of convergence for the
             nuclei configuration and for the Fermi-level.},
   Doi = {10.1007/s00205-018-1256-y},
   Key = {fds340245}
}

@article{fds337607,
   Author = {Li, X and Liu, J and Lu, J and Zhou, X},
   Title = {Moderate deviation for random elliptic PDE with small
             noise},
   Journal = {Annals of Applied Probability},
   Volume = {28},
   Number = {5},
   Pages = {2781-2813},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2018},
   Month = {October},
   url = {http://dx.doi.org/10.1214/17-AAP1373},
   Abstract = {Partial differential equations with random inputs have
             become popular models to characterize physical systems with
             uncertainty coming from imprecise measurement and intrinsic
             randomness. In this paper, we perform asymptotic rare-event
             analysis for such elliptic PDEs with random inputs. In
             particular, we consider the asymptotic regime that the noise
             level converges to zero suggesting that the system
             uncertainty is low, but does exist. We develop sharp
             approximations of the probability of a large class of rare
             events.},
   Doi = {10.1214/17-AAP1373},
   Key = {fds337607}
}

@article{fds361713,
   Author = {Lu, J and Steinerberger, S},
   Title = {On Pointwise Products of Elliptic Eigenfunctions},
   Year = {2018},
   Month = {October},
   Abstract = {We consider eigenfunctions of Schr\"odinger operators on a
             $d-$dimensional bounded domain $\Omega$ (or a
             $d-$dimensional compact manifold $\Omega$) with Dirichlet
             conditions. These operators give rise to a sequence of
             eigenfunctions $(\phi_n)_{n \in \mathbb{N}}$. We study the
             subspace of all pointwise products $$ A_n = \mbox{span}
             \left\{ \phi_i(x) \phi_j(x): 1 \leq i,j \leq n\right\}
             \subseteq L^2(\Omega).$$ Clearly, that vector space has
             dimension $\mbox{dim}(A_n) = n(n+1)/2$. We prove that
             products $\phi_i \phi_j$ of eigenfunctions are simple in a
             certain sense: for any $\varepsilon > 0$, there exists a
             low-dimensional vector space $B_n$ that almost contains all
             products. More precisely, denoting the orthogonal projection
             $\Pi_{B_n}:L^2(\Omega) \rightarrow B_n$, we have $$
             \forall~1 \leq i,j \leq n~ \qquad \|\phi_i\phi_j -
             \Pi_{B_n}( \phi_i \phi_j) \|_{L^2} \leq \varepsilon$$ and
             the size of the space $\mbox{dim}(B_n)$ is relatively small
             $$ \mbox{dim}(B_n) \lesssim \left( \frac{1}{\varepsilon}
             \max_{1 \leq i \leq n} \|\phi_i\|_{L^{\infty}} \right)^d
             n.$$ In the generic delocalized setting, this bound grows
             linearly up to logarithmic factors: pointwise products of
             eigenfunctions are low-rank. This has implications, among
             other things, for the validity of fast algorithms in
             electronic structure computations.},
   Key = {fds361713}
}

@article{fds348058,
   Author = {Lu, J and Steinerberger, S},
   Title = {Detecting localized eigenstates of linear
             operators},
   Journal = {Research in Mathematical Sciences},
   Volume = {5},
   Number = {3},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2018},
   Month = {September},
   url = {http://dx.doi.org/10.1007/s40687-018-0152-2},
   Abstract = {We describe a way of detecting the location of localized
             eigenvectors of the eigenvalue problem Ax = λx for
             eigenvalues λ with |λ| comparatively large. We define the
             family of functions fα: {1, 2, …,n} → R fα (k) =
             log(‖Aα ek ‖ℓ2), where α ≥ 0 is a parameter and ek
             = (0, 0, …, 0, 1, 0, …, 0) is the kth standard basis
             vector. We prove that eigenvectors associated with
             eigenvalues with large absolute value localize around local
             maxima of fα: the metastable states in the power iteration
             method (slowing down its convergence) can be used to predict
             localization. We present a fast randomized algorithm and
             discuss different examples: a random band matrix,
             discretizations of the local operator −Δ + V, and the
             nonlocal operator (−Δ)3/4 + V.},
   Doi = {10.1007/s40687-018-0152-2},
   Key = {fds348058}
}

@article{fds338041,
   Author = {Barthel, T and Lu, J},
   Title = {Fundamental Limitations for Measurements in Quantum
             Many-Body Systems},
   Journal = {Physical Review Letters},
   Volume = {121},
   Number = {8},
   Pages = {080406},
   Year = {2018},
   Month = {August},
   url = {http://dx.doi.org/10.1103/PhysRevLett.121.080406},
   Abstract = {Dynamical measurement schemes are an important tool for the
             investigation of quantum many-body systems, especially in
             the age of quantum simulation. Here, we address the question
             whether generic measurements can be implemented efficiently
             if we have access to a certain set of experimentally
             realizable measurements and can extend it through time
             evolution. For the latter, two scenarios are considered: (a)
             evolution according to unitary circuits and (b) evolution
             due to Hamiltonians that we can control in a time-dependent
             fashion. We find that the time needed to realize a certain
             measurement to a predefined accuracy scales in general
             exponentially with the system size - posing a fundamental
             limitation. The argument is based on the construction of
             μ-packings for manifolds of observables with identical
             spectra and a comparison of their cardinalities to those of
             μ-coverings for quantum circuits and unitary time-evolution
             operators. The former is related to the study of Grassmann
             manifolds.},
   Doi = {10.1103/PhysRevLett.121.080406},
   Key = {fds338041}
}

@article{fds332859,
   Author = {Huang, Y and Lu, J and Ming, P},
   Title = {A Concurrent Global–Local Numerical Method for Multiscale
             PDEs},
   Journal = {Journal of Scientific Computing},
   Volume = {76},
   Number = {2},
   Pages = {1188-1215},
   Publisher = {Springer Nature},
   Year = {2018},
   Month = {August},
   url = {http://dx.doi.org/10.1007/s10915-018-0662-5},
   Abstract = {We present a new hybrid numerical method for multiscale
             partial differential equations, which simultaneously
             captures the global macroscopic information and resolves the
             local microscopic events over regions of relatively small
             size. The method couples concurrently the microscopic
             coefficients in the region of interest with the homogenized
             coefficients elsewhere. The cost of the method is comparable
             to the heterogeneous multiscale method, while being able to
             recover microscopic information of the solution. The
             convergence of the method is proved for problems with
             bounded and measurable coefficients, while the rate of
             convergence is established for problems with rapidly
             oscillating periodic or almost-periodic coefficients.
             Numerical results are reported to show the efficiency and
             accuracy of the proposed method.},
   Doi = {10.1007/s10915-018-0662-5},
   Key = {fds332859}
}

@article{fds337608,
   Author = {You, Z and Li, L and Lu, J and Ge, H},
   Title = {Integrated tempering enhanced sampling method as the
             infinite switching limit of simulated tempering.},
   Journal = {The Journal of chemical physics},
   Volume = {149},
   Number = {8},
   Pages = {084114},
   Year = {2018},
   Month = {August},
   url = {http://dx.doi.org/10.1063/1.5045369},
   Abstract = {A fast and accurate sampling method is in high demand, in
             order to bridge the large gaps between molecular dynamic
             simulations and experimental observations. Recently, an
             integrated tempering enhanced sampling (ITS) method has been
             proposed and successfully applied to various biophysical
             examples, significantly accelerating conformational
             sampling. The mathematical validation for its effectiveness
             has not been elucidated yet. Here we show that the
             integrated tempering enhanced sampling method can be viewed
             as a reformulation of the infinite switching limit of the
             simulated tempering method over a mixed potential. Moreover,
             we demonstrate that the efficiency of simulated tempering
             molecular dynamics improves as the frequency of switching
             between the temperatures is increased, based on the large
             deviation principle of empirical distributions. Our theory
             provides the theoretical justification of the advantage of
             ITS. Finally, we illustrate the utility of the infinite
             switching simulated tempering method through several
             numerical examples.},
   Doi = {10.1063/1.5045369},
   Key = {fds337608}
}

@article{fds333284,
   Author = {Lin, L and Lu, J and Vanden-Eijnden, E},
   Title = {A Mathematical Theory of Optimal Milestoning (with a Detour
             via Exact Milestoning)},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {71},
   Number = {6},
   Pages = {1149-1177},
   Publisher = {WILEY},
   Year = {2018},
   Month = {June},
   url = {http://dx.doi.org/10.1002/cpa.21725},
   Abstract = {Milestoning is a computational procedure that reduces the
             dynamics of complex systems to memoryless jumps between
             intermediates, or milestones, and only retains some
             information about the probability of these jumps and the
             time lags between them. Here we analyze a variant of this
             procedure, termed optimal milestoning, which relies on a
             specific choice of milestones to capture exactly some
             kinetic features of the original dynamical system. In
             particular, we prove that optimal milestoning permits the
             exact calculation of the mean first passage times (MFPT)
             between any two milestones. In so doing, we also analyze
             another variant of the method, called exact milestoning,
             which also permits the exact calculation of certain MFPTs,
             but at the price of retaining more information about the
             original system's dynamics. Finally, we discuss importance
             sampling strategies based on optimal and exact milestoning
             that can be used to bypass the simulation of the original
             system when estimating the statistical quantities used in
             these methods.© 2017 Wiley Periodicals,
             Inc.},
   Doi = {10.1002/cpa.21725},
   Key = {fds333284}
}

@article{fds335541,
   Author = {Zhu, W and Qiu, Q and Wang, B and Lu, J and Sapiro, G and Daubechies,
             I},
   Title = {Stop memorizing: A data-dependent regularization framework
             for intrinsic pattern learning},
   Volume = {1},
   Number = {3},
   Pages = {476-496},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {May},
   url = {http://dx.doi.org/10.1137/19m1236886},
   Abstract = {Deep neural networks (DNNs) typically have enough capacity
             to fit random data by brute force even when conventional
             data-dependent regularizations focusing on the geometry of
             the features are imposed. We find out that the reason for
             this is the inconsistency between the enforced geometry and
             the standard softmax cross entropy loss. To resolve this, we
             propose a new framework for data-dependent DNN
             regularization, the Geometrically-Regularized-Self-Validating
             neural Networks (GRSVNet). During training, the geometry
             enforced on one batch of features is simultaneously
             validated on a separate batch using a validation loss
             consistent with the geometry. We study a particular case of
             GRSVNet, the Orthogonal-Low-rank Embedding (OLE)-GRSVNet,
             which is capable of producing highly discriminative features
             residing in orthogonal low-rank subspaces. Numerical
             experiments show that OLE-GRSVNet outperforms DNNs with
             conventional regularization when trained on real data. More
             importantly, unlike conventional DNNs, OLE-GRSVNet refuses
             to memorize random data or random labels, suggesting it only
             learns intrinsic patterns by reducing the memorizing
             capacity of the baseline DNN.},
   Doi = {10.1137/19m1236886},
   Key = {fds335541}
}

@article{fds340186,
   Author = {Yu, V and Huhn, W and Lin, L and Lu, J and Vazquez-Mayagoitia, A and Yang,
             C and Blum, V},
   Title = {ELSI: A unified software interface for Kohn-Sham electronic
             structure solvers},
   Journal = {ABSTRACTS OF PAPERS OF THE AMERICAN CHEMICAL
             SOCIETY},
   Volume = {255},
   Pages = {1 pages},
   Publisher = {AMER CHEMICAL SOC},
   Year = {2018},
   Month = {March},
   Key = {fds340186}
}

@article{fds361346,
   Author = {Cai, Z and Lu, J and Stubbs, K},
   Title = {On discrete Wigner transforms},
   Year = {2018},
   Month = {February},
   Abstract = {In this work, we derive a discrete analog of the Wigner
             transform over the space $(\mathbb{C}^p)^{\otimes N}$ for
             any prime $p$ and any positive integer $N$. We show that the
             Wigner transform over this space can be constructed as the
             inverse Fourier transform of the standard Pauli matrices for
             $p=2$ or more generally of the Heisenberg-Weyl group
             elements for $p > 2$. We connect our work to a previous
             construction by Wootters of a discrete Wigner transform by
             showing that for all $p$, Wootters' construction corresponds
             to taking the inverse symplectic Fourier transform instead
             of the inverse Fourier transform. Finally, we discuss some
             implications of these results for the numerical simulation
             of many-body quantum spin systems.},
   Key = {fds361346}
}

@article{fds332860,
   Author = {Lu, J and Zhou, Z},
   Title = {Accelerated sampling by infinite swapping of path integral
             molecular dynamics with surface hopping.},
   Journal = {The Journal of chemical physics},
   Volume = {148},
   Number = {6},
   Pages = {064110},
   Year = {2018},
   Month = {February},
   url = {http://dx.doi.org/10.1063/1.5005024},
   Abstract = {To accelerate the thermal equilibrium sampling of
             multi-level quantum systems, the infinite swapping limit of
             a recently proposed multi-level ring polymer representation
             is investigated. In the infinite swapping limit, the ring
             polymer evolves according to an averaged Hamiltonian with
             respect to all possible surface index configurations of the
             ring polymer and thus connects the surface hopping approach
             to the mean-field path-integral molecular dynamics. A
             multiscale integrator for the infinite swapping limit is
             also proposed to enable efficient sampling based on the
             limiting dynamics. Numerical results demonstrate the huge
             improvement of sampling efficiency of the infinite swapping
             compared with the direct simulation of path-integral
             molecular dynamics with surface hopping.},
   Doi = {10.1063/1.5005024},
   Key = {fds332860}
}

@article{fds336984,
   Author = {Cai, Z and Lu, J},
   Title = {A quantum kinetic monte carlo method for quantum many-body
             spin dynamics},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {40},
   Number = {3},
   Pages = {B706-B722},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1145446},
   Abstract = {We propose a general framework of a quantum kinetic Monte
             Carlo algorithm, based on a stochastic representation of a
             series expansion of the quantum evolution. Two approaches
             have been developed in the context of quantum many-body spin
             dynamics, using different decomposition of the Hamiltonian.
             The effectiveness of the methods is tested for many-body
             spin systems up to 40 spins.},
   Doi = {10.1137/17M1145446},
   Key = {fds336984}
}

@article{fds339290,
   Author = {Yang, H and lu, J},
   Title = {Phase Space Sketching for Crystal Image Analysis based on
             Synchrosqueezed Transforms},
   Volume = {11},
   Number = {3},
   Pages = {1954-1978},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1129441},
   Abstract = {Recent developments of imaging techniques enable researchers
             to visualize materials at atomic resolution to better
             understand the microscopic structures of materials. This
             paper aims at automatic and quantitative characterization of
             potentially complicated microscopic crystal images,
             providing feedback to tweak theories and improve synthesis
             in materials science. As such, an efficient phase-space
             sketching method is proposed to encode microscopic crystal
             images in a translation, rotation, illumination, and scale
             invariant representation, which is also stable with respect
             to small deformations. Based on the phase-space sketching,
             we generalize our previous analysis framework for crystal
             images with simple structures to those with complicated
             geometry.},
   Doi = {10.1137/17M1129441},
   Key = {fds339290}
}

@article{fds339744,
   Author = {Delgadillo, R and Lu, J and Yang, X},
   Title = {Frozen Gaussian approximation for high frequency wave
             propagation in periodic media},
   Journal = {Asymptotic Analysis},
   Volume = {110},
   Number = {3-4},
   Pages = {113-135},
   Publisher = {IOS Press},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.3233/ASY-181479},
   Abstract = {Propagation of high-frequency wave in periodic media is a
             challenging problem due to the existence of multiscale
             characterized by short wavelength, small lattice constant
             and large physical domain size. Conventional computational
             methods lead to extremely expensive costs, especially in
             high dimensions. In this paper, based on Bloch decomposition
             and asymptotic analysis in the phase space, we derive the
             frozen Gaussian approximation for high-frequency wave
             propagation in periodic media and establish its converge to
             the true solution. The formulation leads to efficient
             numerical algorithms, which are presented in a companion
             paper [SIAM J. Sci. Comput. 38 (2016), A2440-A2463].},
   Doi = {10.3233/ASY-181479},
   Key = {fds339744}
}

@article{fds329344,
   Author = {Yu, VWZ and Corsetti, F and García, A and Huhn, WP and Jacquelin, M and Jia, W and Lange, B and Lin, L and Lu, J and Mi, W and Seifitokaldani, A and Vázquez-Mayagoitia, Á and Yang, C and Yang, H and Blum,
             V},
   Title = {ELSI: A unified software interface for Kohn–Sham
             electronic structure solvers},
   Journal = {Computer Physics Communications},
   Volume = {222},
   Pages = {267-285},
   Publisher = {Elsevier BV},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1016/j.cpc.2017.09.007},
   Abstract = {Solving the electronic structure from a generalized or
             standard eigenproblem is often the bottleneck in large scale
             calculations based on Kohn–Sham density-functional theory.
             This problem must be addressed by essentially all current
             electronic structure codes, based on similar matrix
             expressions, and by high-performance computation. We here
             present a unified software interface, ELSI, to access
             different strategies that address the Kohn–Sham eigenvalue
             problem. Currently supported algorithms include the dense
             generalized eigensolver library ELPA, the orbital
             minimization method implemented in libOMM, and the pole
             expansion and selected inversion (PEXSI) approach with lower
             computational complexity for semilocal density functionals.
             The ELSI interface aims to simplify the implementation and
             optimal use of the different strategies, by offering (a) a
             unified software framework designed for the electronic
             structure solvers in Kohn–Sham density-functional theory;
             (b) reasonable default parameters for a chosen solver; (c)
             automatic conversion between input and internal working
             matrix formats, and in the future (d) recommendation of the
             optimal solver depending on the specific problem.
             Comparative benchmarks are shown for system sizes up to
             11,520 atoms (172,800 basis functions) on distributed memory
             supercomputing architectures. Program summary Program title:
             ELSI Interface Program Files doi: http://dx.doi.org/10.17632/y8vzhzdm62.1
             Licensing provisions: BSD 3-clause Programming language:
             Fortran 2003, with interface to C/C++ External
             routines/libraries: MPI, BLAS, LAPACK, ScaLAPACK, ELPA,
             libOMM, PEXSI, ParMETIS, SuperLU_DIST Nature of problem:
             Solving the electronic structure from a generalized or
             standard eigenvalue problem in calculations based on
             Kohn–Sham density functional theory (KS-DFT). Solution
             method: To connect the KS-DFT codes and the KS electronic
             structure solvers, ELSI provides a unified software
             interface with reasonable default parameters, hierarchical
             control over the interface and the solvers, and automatic
             conversions between input and internal working matrix
             formats. Supported solvers are: ELPA (dense generalized
             eigensolver), libOMM (orbital minimization method), and
             PEXSI (pole expansion and selected inversion method).
             Restrictions: The ELSI interface requires complete
             information of the Hamiltonian matrix.},
   Doi = {10.1016/j.cpc.2017.09.007},
   Key = {fds329344}
}

@article{fds337014,
   Author = {Lu, J and Zhou, Z},
   Title = {Frozen gaussian approximation with surface hopping for mixed
             quantum-classical dynamics: A mathematical justification of
             fewest switches surface hopping algorithms},
   Journal = {Mathematics of Computation},
   Volume = {87},
   Number = {313},
   Pages = {2189-2232},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3310},
   Abstract = {We develop a surface hopping algorithm based on frozen
             Gaussian approximation for semiclassical matrix Schrödinger
             equations, in the spirit of Tully's fewest switches surface
             hopping method. The algorithm is asymptotically derived from
             the Schrödinger equation with rigorous approximation error
             analysis. The resulting algorithm can be viewed as a path
             integral stochastic representation of the semiclassical
             matrix Schrödinger equations. Our results provide
             mathematical understanding to and shed new light on the
             important class of surface hopping methods in theoretical
             and computational chemistry.},
   Doi = {10.1090/mcom/3310},
   Key = {fds337014}
}

@article{fds335540,
   Author = {Du, Q and Li, XH and Lu, J and Tian, X},
   Title = {A quasi-nonlocal coupling method for nonlocal and local
             diffusion models},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {56},
   Number = {3},
   Pages = {1386-1404},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1124012},
   Abstract = {In this paper, we extend the idea of “geometric
             reconstruction” to couple a nonlocal diffusion model
             directly with the classical local diffusion in one
             dimensional space. This new coupling framework removes
             interfacial inconsistency, ensures the flux balance, and
             satisfies energy conservation as well as the maximum
             principle, whereas none of existing coupling methods for
             nonlocal-to-local coupling satisfies all of these
             properties. We establish the well-posedness and provide the
             stability analysis of the coupling method. We investigate
             the difference to the local limiting problem in terms of the
             nonlocal interaction range. Furthermore, we propose a first
             order finite difference numerical discretization and perform
             several numerical tests to confirm the theoretical findings.
             In particular, we show that the resulting numerical result
             is free of artifacts near the boundary of the domain where a
             classical local boundary condition is used, together with a
             coupled fully nonlocal model in the interior of the
             domain.},
   Doi = {10.1137/17M1124012},
   Key = {fds335540}
}

@article{fds332861,
   Author = {Dai, S and Li, B and Lu, J},
   Title = {Convergence of Phase-Field Free Energy and Boundary Force
             for Molecular Solvation},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {227},
   Number = {1},
   Pages = {105-147},
   Publisher = {Springer Nature},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1007/s00205-017-1158-4},
   Abstract = {We study a phase-field variational model for the solvation
             of charged molecules with an implicit solvent. The solvation
             free-energy functional of all phase fields consists of the
             surface energy, solute excluded volume and solute-solvent
             van der Waals dispersion energy, and electrostatic free
             energy. The surface energy is defined by the van der
             Waals–Cahn–Hilliard functional with squared gradient and
             a double-well potential. The electrostatic part of free
             energy is defined through the electrostatic potential
             governed by the Poisson–Boltzmann equation in which the
             dielectric coefficient is defined through the underlying
             phase field. We prove the continuity of the
             electrostatics—its potential, free energy, and dielectric
             boundary force—with respect to the perturbation of the
             dielectric boundary. We also prove the Γ -convergence of
             the phase-field free-energy functionals to their
             sharp-interface limit, and the equivalence of the
             convergence of total free energies to that of all individual
             parts of free energy. We finally prove the convergence of
             phase-field forces to their sharp-interface limit. Such
             forces are defined as the negative first variations of the
             free-energy functional; and arise from stress tensors. In
             particular, we obtain the force convergence for the van der
             Waals–Cahn–Hilliard functionals with minimal
             assumptions.},
   Doi = {10.1007/s00205-017-1158-4},
   Key = {fds332861}
}

@article{fds339637,
   Author = {Cai, Z and Lu, J},
   Title = {A surface hopping Gaussian beam method for high-dimensional
             transport systems},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {40},
   Number = {5},
   Pages = {B1277-B1301},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1121299},
   Abstract = {We consider a set of linear hyperbolic equations coupled by
             a linear source term and introduce a surface hopping
             Gaussian beam method as its numerical solver. The Gaussian
             beam part is basically classic, while the surface hopping
             part is derived from the equations. The whole algorithm
             shows high efficiency and good parallelizability. An
             application on the quantum-classical Liouville equations is
             presented to show its potential use in practice.},
   Doi = {10.1137/17M1121299},
   Key = {fds339637}
}

@article{fds340383,
   Author = {Lai, R and Lu, J},
   Title = {Point cloud discretization of Fokker-planck operators for
             committor functions},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {16},
   Number = {2},
   Pages = {710-726},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1123018},
   Abstract = {The committor functions provide useful information to the
             understanding of transitions of a stochastic system between
             disjoint regions in phase space. In this work, we develop a
             point cloud discretization for Fokker-Planck operators to
             numerically calculate the committor function, with the
             assumption that the transition occurs on an intrinsically
             low dimensional manifold in the ambient potentially high
             dimensional configurational space of the stochastic system.
             Numerical examples on model systems validate the
             effectiveness of the proposed method.},
   Doi = {10.1137/17M1123018},
   Key = {fds340383}
}

@article{fds340592,
   Author = {Lu, J and Spiliopoulos, K},
   Title = {Analysis of multiscale integrators for multiple attractors
             and irreversible langevin samplers},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {16},
   Number = {4},
   Pages = {1859-1883},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1083748},
   Abstract = {We study multiscale integrator numerical schemes for a class
             of stiff stochastic differential equations (SDEs). We
             consider multiscale SDEs with potentially multiple
             attractors that behave as diffusions on graphs as the
             stiffness parameter goes to its limit. Classical numerical
             discretization schemes, such as the Euler-Maruyama scheme,
             become unstable as the stiffness parameter converges to its
             limit and appropriate multiscale integrators can correct for
             this. We rigorously establish the convergence of the
             numerical method to the related diffusion on graph,
             identifying the appropriate choice of discretization
             parameters. Theoretical results are supplemented by
             numerical studies on the problem of the recently developing
             area of introducing irreversibility in Langevin samplers in
             order to accelerate convergence to equilibrium.},
   Doi = {10.1137/16M1083748},
   Key = {fds340592}
}

@article{fds340593,
   Author = {Fang, D and Lu, J},
   Title = {A diabatic surface hopping algorithm based on time dependent
             perturbation theory and semiclassical analysis},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {16},
   Number = {4},
   Pages = {1603-1622},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1137/17M1138042},
   Abstract = {Surface hopping algorithms are popular tools to study
             dynamics of the quantumclassical mixed systems. In this
             paper, we propose a surface hopping algorithm in diabatic
             representations, based on time dependent perturbation theory
             and semiclassical analysis. The algorithm can be viewed as a
             Monte Carlo sampling algorithm on the semiclassical path
             space for a piecewise deterministic path with stochastic
             jumps between the energy surfaces. The algorithm is
             validated numerically and it shows good performance in both
             weak coupling and avoided crossing regimes.},
   Doi = {10.1137/17M1138042},
   Key = {fds340593}
}

@article{fds337144,
   Author = {Cao, Y and Lu, J},
   Title = {Stochastic dynamical low-rank approximation
             method},
   Journal = {Journal of Computational Physics},
   Volume = {372},
   Pages = {564-586},
   Publisher = {Elsevier BV},
   Year = {2018},
   url = {http://dx.doi.org/10.1016/j.jcp.2018.06.058},
   Abstract = {In this paper, we extend the dynamical low-rank
             approximation method to the space of finite signed measures.
             Under this framework, we derive stochastic low-rank dynamics
             for stochastic differential equations (SDEs) coming from
             classical stochastic dynamics or unraveling of Lindblad
             quantum master equations. We justify the proposed method by
             error analysis and also numerical examples for applications
             in solving high-dimensional SDE, stochastic Burgers'
             equation, and high-dimensional Lindblad equation.},
   Doi = {10.1016/j.jcp.2018.06.058},
   Key = {fds337144}
}

@article{fds367495,
   Author = {Li, X and Lin, L and Lu, J},
   Title = {PEXSI-$\Sigma$: a Green’s function embedding method for
             Kohn–Sham density functional theory},
   Journal = {Annals of Mathematical Sciences and Applications},
   Volume = {3},
   Number = {2},
   Pages = {441-472},
   Publisher = {International Press of Boston},
   Year = {2018},
   url = {http://dx.doi.org/10.4310/amsa.2018.v3.n2.a3},
   Doi = {10.4310/amsa.2018.v3.n2.a3},
   Key = {fds367495}
}

@article{fds329343,
   Author = {Lu, J and Thicke, K},
   Title = {Cubic scaling algorithms for RPA correlation using
             interpolative separable density fitting},
   Journal = {Journal of Computational Physics},
   Volume = {351},
   Pages = {187-202},
   Publisher = {Elsevier BV},
   Year = {2017},
   Month = {December},
   url = {http://dx.doi.org/10.1016/j.jcp.2017.09.012},
   Abstract = {We present a new cubic scaling algorithm for the calculation
             of the RPA correlation energy. Our scheme splits up the
             dependence between the occupied and virtual orbitals in χ0
             by use of Cauchy's integral formula. This introduces an
             additional integral to be carried out, for which we provide
             a geometrically convergent quadrature rule. Our scheme also
             uses the newly developed Interpolative Separable Density
             Fitting algorithm to further reduce the computational cost
             in a way analogous to that of the Resolution of Identity
             method.},
   Doi = {10.1016/j.jcp.2017.09.012},
   Key = {fds329343}
}

@article{fds332172,
   Author = {Cao, Y and Lu, J},
   Title = {Lindblad equation and its semiclassical limit of the
             Anderson-Holstein model},
   Journal = {Journal of Mathematical Physics},
   Volume = {58},
   Number = {12},
   Pages = {122105-122105},
   Publisher = {AIP Publishing},
   Year = {2017},
   Month = {December},
   url = {http://dx.doi.org/10.1063/1.4993431},
   Abstract = {For multi-level open quantum systems, the interaction
             between different levels could pose a challenge to
             understand the quantum system both analytically and
             numerically. In this work, we study the approximation of the
             dynamics of the Anderson-Holstein model, as a model of the
             multi-level open quantum system, by Redfield and Lindblad
             equations. Both equations have a desirable property that if
             the density operators for different levels are diagonal
             initially, they remain to be diagonal for any time. Thanks
             to this nice property, the semiclassical limit of both
             Redfield and Lindblad equations could be derived explicitly;
             the resulting classical master equations share similar
             structures of transport and hopping terms. The Redfield and
             Lindblad equations are also compared from the angle of time
             dependent perturbation theory.},
   Doi = {10.1063/1.4993431},
   Key = {fds332172}
}

@article{fds361457,
   Author = {Lu, J and Steinerberger, S},
   Title = {Riesz Energy on the Torus: Regularity of
             Minimizers},
   Year = {2017},
   Month = {October},
   Abstract = {We study sets of $N$ points on the $d-$dimensional torus
             $\mathbb{T}^d$ minimizing interaction functionals of the
             type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \]
             The main result states that for a class of functions $f$
             that behave like Riesz energies $f(x) \sim \|x\|^{-s}$ for
             $0< s < d$, the minimizing configuration of points has
             optimal regularity w.r.t. a Fourier-analytic regularity
             measure that arises in the study of irregularities of
             distribution. A particular consequence is that they are
             optimal quadrature points in the space of trigonometric
             polynomials up to a certain degree. The proof extends to
             other settings and also covers less singular functions such
             as $f(x) = \exp\bigl(- N^{\frac{2}{d}} \|x\|^2
             \bigr)$.},
   Key = {fds361457}
}

@article{fds328894,
   Author = {Li, L and Liu, JG and Lu, J},
   Title = {Fractional Stochastic Differential Equations Satisfying
             Fluctuation-Dissipation Theorem},
   Journal = {Journal of Statistical Physics},
   Volume = {169},
   Number = {2},
   Pages = {316-339},
   Publisher = {Springer Nature America, Inc},
   Year = {2017},
   Month = {October},
   url = {http://dx.doi.org/10.1007/s10955-017-1866-z},
   Abstract = {We propose in this work a fractional stochastic differential
             equation (FSDE) model consistent with the over-damped limit
             of the generalized Langevin equation model. As a result of
             the ‘fluctuation-dissipation theorem’, the differential
             equations driven by fractional Brownian noise to model
             memory effects should be paired with Caputo derivatives, and
             this FSDE model should be understood in an integral form. We
             establish the existence of strong solutions for such
             equations and discuss the ergodicity and convergence to
             Gibbs measure. In the linear forcing regime, we show
             rigorously the algebraic convergence to Gibbs measure when
             the ‘fluctuation-dissipation theorem’ is satisfied, and
             this verifies that satisfying ‘fluctuation-dissipation
             theorem’ indeed leads to the correct physical behavior. We
             further discuss possible approaches to analyze the
             ergodicity and convergence to Gibbs measure in the nonlinear
             forcing regime, while leave the rigorous analysis for future
             works. The FSDE model proposed is suitable for systems in
             contact with heat bath with power-law kernel and
             subdiffusion behaviors.},
   Doi = {10.1007/s10955-017-1866-z},
   Key = {fds328894}
}

@article{fds333283,
   Author = {Li, Q and Lu, J and Sun, W},
   Title = {A convergent method for linear half-space kinetic
             equations},
   Journal = {ESAIM: Mathematical Modelling and Numerical
             Analysis},
   Volume = {51},
   Number = {5},
   Pages = {1583-1615},
   Publisher = {E D P SCIENCES},
   Year = {2017},
   Month = {September},
   url = {http://dx.doi.org/10.1051/m2an/2016076},
   Abstract = {We give a unified proof for the well-posedness of a class of
             linear half-space equations with general incoming data and
             construct a Galerkin method to numerically resolve this type
             of equations in a systematic way. Our main strategy in both
             analysis and numerics includes three steps: Adding damping
             terms to the original half-space equation, using an inf-sup
             argument and even-odd decomposition to establish the
             well-posedness of the damped equation, and then recovering
             solutions to the original half-space equation. The proposed
             numerical methods for the damped equation is shown to be
             quasi-optimal and the numerical error of approximations to
             the original equation is controlled by that of the damped
             equation. This efficient solution to the half-space problem
             is useful for kinetic-fluid coupling simulations.},
   Doi = {10.1051/m2an/2016076},
   Key = {fds333283}
}

@article{fds328895,
   Author = {Lu, J and Steinerberger, S},
   Title = {A variation on the Donsker-Varadhan inequality for the
             principal eigenvalue.},
   Journal = {Proceedings. Mathematical, physical, and engineering
             sciences},
   Volume = {473},
   Number = {2204},
   Pages = {20160877},
   Year = {2017},
   Month = {August},
   url = {http://dx.doi.org/10.1098/rspa.2016.0877},
   Abstract = {The purpose of this short paper is to give a variation on
             the classical Donsker-Varadhan inequality, which bounds the
             first eigenvalue of a second-order elliptic operator on a
             bounded domain <i>Ω</i> by the largest mean first exit time
             of the associated drift-diffusion process via [Formula: see
             text]Instead of looking at the mean of the first exit time,
             we study quantiles: let [Formula: see text] be the smallest
             time <i>t</i> such that the likelihood of exiting within
             that time is <i>p</i>, then [Formula: see text]Moreover, as
             [Formula: see text], this lower bound converges to
             λ<sub>1</sub>.},
   Doi = {10.1098/rspa.2016.0877},
   Key = {fds328895}
}

@article{fds325888,
   Author = {Lu, JL and Yang, HY},
   Title = {A Cubic Scaling Algorithm for Excited States Calculations in
             Particle-Particle Random Phase Approximation},
   Volume = {340},
   Pages = {297-308},
   Publisher = {Elsevier BV},
   Year = {2017},
   Month = {July},
   url = {http://dx.doi.org/10.1016/j.jcp.2017.03.055},
   Abstract = {The particle–particle random phase approximation (pp-RPA)
             has been shown to be capable of describing double, Rydberg,
             and charge transfer excitations, for which the conventional
             time-dependent density functional theory (TDDFT) might not
             be suitable. It is thus desirable to reduce the
             computational cost of pp-RPA so that it can be efficiently
             applied to larger molecules and even solids. This paper
             introduces an O(N3) algorithm, where N is the number of
             orbitals, based on an interpolative separable density
             fitting technique and the Jacobi–Davidson eigensolver to
             calculate a few low-lying excitations in the pp-RPA
             framework. The size of the pp-RPA matrix can also be reduced
             by keeping only a small portion of orbitals with orbital
             energy close to the Fermi energy. This reduced system leads
             to a smaller prefactor of the cubic scaling algorithm, while
             keeping the accuracy for the low-lying excitation
             energies.},
   Doi = {10.1016/j.jcp.2017.03.055},
   Key = {fds325888}
}

@article{fds326080,
   Author = {Gao, Y and Liu, JG and Lu, J},
   Title = {Continuum Limit of a Mesoscopic Model with Elasticity of
             Step Motion on Vicinal Surfaces},
   Journal = {Journal of Nonlinear Science},
   Volume = {27},
   Number = {3},
   Pages = {873-926},
   Publisher = {Springer Nature},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.1007/s00332-016-9354-1},
   Abstract = {This work considers the rigorous derivation of continuum
             models of step motion starting from a mesoscopic
             Burton–Cabrera–Frank-type model following the Xiang’s
             work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We
             prove that as the lattice parameter goes to zero, for a
             finite time interval, a modified discrete model converges to
             the strong solution of the limiting PDE with first-order
             convergence rate.},
   Doi = {10.1007/s00332-016-9354-1},
   Key = {fds326080}
}

@article{fds326484,
   Author = {Li, C and Lu, J and Yang, W},
   Title = {On extending Kohn-Sham density functionals to systems with
             fractional number of electrons.},
   Journal = {The Journal of chemical physics},
   Volume = {146},
   Number = {21},
   Pages = {214109},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.1063/1.4982951},
   Abstract = {We analyze four ways of formulating the Kohn-Sham (KS)
             density functionals with a fractional number of electrons,
             through extending the constrained search space from the
             Kohn-Sham and the generalized Kohn-Sham (GKS)
             non-interacting v-representable density domain for integer
             systems to four different sets of densities for fractional
             systems. In particular, these density sets are (I) ensemble
             interacting N-representable densities, (II) ensemble
             non-interacting N-representable densities, (III)
             non-interacting densities by the Janak construction, and
             (IV) non-interacting densities whose composing orbitals
             satisfy the Aufbau occupation principle. By proving the
             equivalence of the underlying first order reduced density
             matrices associated with these densities, we show that sets
             (I), (II), and (III) are equivalent, and all reduce to the
             Janak construction. Moreover, for functionals with the
             ensemble v-representable assumption at the minimizer, (III)
             reduces to (IV) and thus justifies the previous use of the
             Aufbau protocol within the (G)KS framework in the study of
             the ground state of fractional electron systems, as defined
             in the grand canonical ensemble at zero temperature. By
             further analyzing the Aufbau solution for different density
             functional approximations (DFAs) in the (G)KS scheme, we
             rigorously prove that there can be one and only one
             fractional occupation for the Hartree Fock functional, while
             there can be multiple fractional occupations for general
             DFAs in the presence of degeneracy. This has been confirmed
             by numerical calculations using the local density
             approximation as a representative of general DFAs. This work
             thus clarifies important issues on density functional theory
             calculations for fractional electron systems.},
   Doi = {10.1063/1.4982951},
   Key = {fds326484}
}

@article{fds324707,
   Author = {Lu, J and Thicke, K},
   Title = {Orbital minimization method with ℓ1
             regularization},
   Journal = {Journal of Computational Physics},
   Volume = {336},
   Pages = {87-103},
   Publisher = {Elsevier BV},
   Year = {2017},
   Month = {May},
   url = {http://dx.doi.org/10.1016/j.jcp.2017.02.005},
   Abstract = {We consider a modification of the orbital minimization
             method (OMM) energy functional which contains an ℓ1
             penalty term in order to find a sparse representation of the
             low-lying eigenspace of self-adjoint operators. We analyze
             the local minima of the modified functional as well as the
             convergence of the modified functional to the original
             functional. Algorithms combining soft thresholding with
             gradient descent are proposed for minimizing this new
             functional. Numerical tests validate our approach. In
             addition, we also prove the unanticipated and remarkable
             property that every local minimum of the OMM functional
             without the ℓ1 term is also a global minimum.},
   Doi = {10.1016/j.jcp.2017.02.005},
   Key = {fds324707}
}

@article{fds339405,
   Author = {Huang, Y and Lu, J and Ming, P},
   Title = {A Hybrid Global-local Numerical Method for Multiscale
             PDEs},
   Year = {2017},
   Month = {April},
   Abstract = {We present a new hybrid numerical method for multiscale
             partial differential equations, which simultaneously
             captures both the global macroscopic information and
             resolves the local microscopic events. The convergence of
             the proposed method is proved for problems with bounded and
             measurable coefficient, while the rate of convergence is
             established for problems with rapidly oscillating periodic
             or almost-periodic coefficients. Numerical results are
             reported to show the efficiency and accuracy of the proposed
             method.},
   Key = {fds339405}
}

@article{fds339404,
   Author = {Li, L and Liu, J-G and Lu, J},
   Title = {Fractional stochastic differential equations satisfying
             fluctuation-dissipation theorem},
   Year = {2017},
   Month = {April},
   Abstract = {We consider in this work stochastic differential equation
             (SDE) model for particles in contact with a heat bath when
             the memory effects are non-negligible. As a result of the
             fluctuation-dissipation theorem, the differential equations
             driven by fractional Brownian noise to model memory effects
             should be paired with Caputo derivatives and based on this
             we consider fractional stochastic differential equations
             (FSDEs), which should be understood in an integral form. We
             establish the existence of strong solutions for such
             equations. In the linear forcing regime, we compute the
             solutions explicitly and analyze the asymptotic behavior,
             through which we verify that satisfying fluctuation-dissipation
             indeed leads to the correct physical behavior. We further
             discuss possible extensions to nonlinear forcing regime,
             while leave the rigorous analysis for future
             works.},
   Key = {fds339404}
}

@article{fds326081,
   Author = {Lu, J and Zhou, Z},
   Title = {Path integral molecular dynamics with surface hopping for
             thermal equilibrium sampling of nonadiabatic
             systems.},
   Journal = {The Journal of chemical physics},
   Volume = {146},
   Number = {15},
   Pages = {154110},
   Year = {2017},
   Month = {April},
   url = {http://dx.doi.org/10.1063/1.4981021},
   Abstract = {In this work, a novel ring polymer representation for a
             multi-level quantum system is proposed for thermal average
             calculations. The proposed representation keeps the
             discreteness of the electronic states: besides position and
             momentum, each bead in the ring polymer is also
             characterized by a surface index indicating the electronic
             energy surface. A path integral molecular dynamics with
             surface hopping (PIMD-SH) dynamics is also developed to
             sample the equilibrium distribution of the ring polymer
             configurational space. The PIMD-SH sampling method is
             validated theoretically and by numerical
             examples.},
   Doi = {10.1063/1.4981021},
   Key = {fds326081}
}

@article{fds325889,
   Author = {Watson, AB and Lu, J and Weinstein, MI},
   Title = {Wavepackets in inhomogeneous periodic media: Effective
             particle-field dynamics and Berry curvature},
   Journal = {Journal of Mathematical Physics},
   Volume = {58},
   Number = {2},
   Pages = {021503-021503},
   Publisher = {AIP Publishing},
   Year = {2017},
   Month = {February},
   url = {http://dx.doi.org/10.1063/1.4976200},
   Abstract = {We consider a model of an electron in a crystal moving under
             the influence of an external electric field: Schrödinger's
             equation with a potential which is the sum of a periodic
             function and a general smooth function. We identify two
             dimensionless parameters: (re-scaled) Planck's constant and
             the ratio of the lattice spacing to the scale of variation
             of the external potential. We consider the special case
             where both parameters are equal and denote this parameter
             ∈. In the limit ∈ ↓ 0, we prove the existence of
             solutions known as semiclassical wavepackets which are
             asymptotic up to "Ehrenfest time" t ln 1/∈. To leading
             order, the center of mass and average quasimomentum of these
             solutions evolve along trajectories generated by the
             classical Hamiltonian given by the sum of the Bloch band
             energy and the external potential. We then derive all
             corrections to the evolution of these observables
             proportional to ∈. The corrections depend on the
             gauge-invariant Berry curvature of the Bloch band and a
             coupling to the evolution of the wave-packet envelope, which
             satisfies Schrödinger's equation with a time-dependent
             harmonic oscillator Hamiltonian. This infinite dimensional
             coupled "particle-field" system may be derived from an
             "extended" ∈-dependent Hamiltonian. It is known that such
             coupling of observables (discrete particle-like degrees of
             freedom) to the wave-envelope (continuum field-like degrees
             of freedom) can have a significant impact on the overall
             dynamics.},
   Doi = {10.1063/1.4976200},
   Key = {fds325889}
}

@article{fds320926,
   Author = {Niu, X and Luo, T and Lu, J and Xiang, Y},
   Title = {Dislocation climb models from atomistic scheme to
             dislocation dynamics},
   Journal = {Journal of the Mechanics and Physics of Solids},
   Volume = {99},
   Pages = {242-258},
   Publisher = {Elsevier BV},
   Year = {2017},
   Month = {February},
   url = {http://dx.doi.org/10.1016/j.jmps.2016.11.012},
   Abstract = {We develop a mesoscopic dislocation dynamics model for
             vacancy-assisted dislocation climb by upscalings from a
             stochastic model on the atomistic scale. Our models
             incorporate microscopic mechanisms of (i) bulk diffusion of
             vacancies, (ii) vacancy exchange dynamics between bulk and
             dislocation core, (iii) vacancy pipe diffusion along the
             dislocation core, and (iv) vacancy attachment-detachment
             kinetics at jogs leading to the motion of jogs. Our
             mesoscopic model consists of the vacancy bulk diffusion
             equation and a dislocation climb velocity formula. The
             effects of these microscopic mechanisms are incorporated by
             a Robin boundary condition near the dislocations for the
             bulk diffusion equation and a new contribution in the
             dislocation climb velocity due to vacancy pipe diffusion
             driven by the stress variation along the dislocation. Our
             climb formulation is able to quantitatively describe the
             translation of prismatic loops at low temperatures when the
             bulk diffusion is negligible. Using this new formulation, we
             derive analytical formulas for the climb velocity of a
             straight edge dislocation and a prismatic circular loop. Our
             dislocation climb formulation can be implemented in
             dislocation dynamics simulations to incorporate all the
             above four microscopic mechanisms of dislocation
             climb.},
   Doi = {10.1016/j.jmps.2016.11.012},
   Key = {fds320926}
}

@article{fds330519,
   Author = {Li, XH and Lu, J},
   Title = {Quasi-nonlocal coupling of nonlocal diffusions},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {55},
   Number = {5},
   Pages = {2394-2415},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1086443},
   Abstract = {We developed a new self-adjoint, consistent, and stable
             coupling strategy for nonlocal diffusion models, inspired by
             the quasi-nonlocal atomistic-to-continuum method for
             crystalline solids. The proposed coupling model is coercive
             with respect to the energy norms induced by the nonlocal
             diffusion kernels as well as the L2 norm, and it satisfies
             the maximum principle. A finite difference approximation is
             used to discretize the coupled system, which inherits the
             property from the continuous formulation. Furthermore, we
             design a numerical example that shows the discrepancy
             between the fully nonlocal and fully local diffusions,
             whereas the result of the coupled diffusion agrees with that
             of the fully nonlocal diffusion.},
   Doi = {10.1137/16M1086443},
   Key = {fds330519}
}

@article{fds325890,
   Author = {Lu, J and Yang, H},
   Title = {Preconditioning Orbital Minimization Method for Planewave
             Discretization},
   Journal = {Multiscale Modeling & Simulation},
   Volume = {15},
   Number = {1},
   Pages = {254-273},
   Publisher = {Society for Industrial and Applied Mathematics},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1068670},
   Abstract = {We present an efficient preconditioner for the orbital
             minimization method when the Hamiltonian is discretized
             using planewaves (i.e., pseudospectral method). This novel
             preconditioner is based on an approximate Fermi operator
             projection by pole expansion, combined with the sparsifying
             preconditioner to efficiently evaluate the pole expansion
             for a wide range of Hamiltonian operators. Numerical results
             validate the performance of the new preconditioner for the
             orbital minimization method, in particular, the iteration
             number is reduced to O(1) and often only a few iterations
             are enough for convergence.},
   Doi = {10.1137/16M1068670},
   Key = {fds325890}
}

@article{fds323661,
   Author = {Li, Q and Lu, J and Sun, W},
   Title = {Validity and Regularization of Classical Half-Space
             Equations},
   Journal = {Journal of Statistical Physics},
   Volume = {166},
   Number = {2},
   Pages = {398-433},
   Publisher = {Springer Nature},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1007/s10955-016-1688-4},
   Abstract = {Recent result (Wu and Guo in Commun Math Phys
             336(3):1473–1553, 2015) has shown that over the 2D unit
             disk, the classical half-space equation (CHS) for the
             neutron transport does not capture the correct boundary
             layer behaviour as long believed. In this paper we develop a
             regularization technique for CHS to any arbitrary order and
             use its first-order regularization to show that in the case
             of the 2D unit disk, although CHS misrepresents the boundary
             layer behaviour, it does give the correct boundary condition
             for the interior macroscopic (Laplace) equation. Therefore
             CHS is still a valid equation to recover the correct
             boundary condition for the interior Laplace equation over
             the 2D unit disk.},
   Doi = {10.1007/s10955-016-1688-4},
   Key = {fds323661}
}

@article{fds332173,
   Author = {Li, Q and Lu, J},
   Title = {An asymptotic preserving method for transport equations with
             oscillatory scattering coefficients},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {15},
   Number = {4},
   Pages = {1694-1718},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M109212X},
   Abstract = {We design a numerical scheme for transport equations with
             oscillatory periodic scattering coefficients. The scheme is
             asymptotic preserving in the diffusion limit as the Knudsen
             number goes to zero. It also captures the homogenization
             limit as the length scale of the scattering coefficient goes
             to zero. The proposed method is based on the construction of
             multiscale finite element basis and a Galerkin projection
             based on the even-odd decomposition. The method is analyzed
             in the asymptotic regime, as well as validated
             numerically.},
   Doi = {10.1137/16M109212X},
   Key = {fds332173}
}

@article{fds327371,
   Author = {Gao, Y and Liu, JG and Lu, J},
   Title = {Weak solution of a continuum model for vicinal surface in
             the attachment-detachment-limited regime},
   Journal = {SIAM Journal on Mathematical Analysis},
   Volume = {49},
   Number = {3},
   Pages = {1705-1731},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1137/16M1094543},
   Abstract = {We study in this work a continuum model derived from a
             one-dimensional attachmentdetachment-limited type step flow
             on a vicinal surface, ut = -u2(u3)hhhh, where u, considered
             as a function of step height h, is the step slope of the
             surface. We formulate a notion of a weak solution to this
             continuum model and prove the existence of a global weak
             solution, which is positive almost everywhere. We also study
             the long time behavior of the weak solution and prove it
             converges to a constant solution as time goes to infinity.
             The space-time Hölder continuity of the weak solution is
             also discussed as a byproduct.},
   Doi = {10.1137/16M1094543},
   Key = {fds327371}
}

@article{fds325467,
   Author = {Cornelis, B and Yang, H and Goodfriend, A and Ocon, N and Lu, J and Daubechies, I},
   Title = {Removal of Canvas Patterns in Digital Acquisitions of
             Paintings},
   Journal = {IEEE Transactions on Image Processing},
   Volume = {26},
   Number = {1},
   Pages = {160-171},
   Publisher = {Institute of Electrical and Electronics Engineers
             (IEEE)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1109/tip.2016.2621413},
   Abstract = {We address the removal of canvas artifacts from
             high-resolution digital photographs and X-ray images of
             paintings on canvas. Both imaging modalities are common
             investigative tools in art history and art conservation.
             Canvas artifacts manifest themselves very differently
             according to the acquisition modality; they can hamper the
             visual reading of the painting by art experts, for instance,
             in preparing a restoration campaign. Computer-aided canvas
             removal is desirable for restorers when the painting on
             canvas they are preparing to restore has acquired over the
             years a much more salient texture. We propose a new
             algorithm that combines a cartoon-texture decomposition
             method with adaptive multiscale thresholding in the
             frequency domain to isolate and suppress the canvas
             components. To illustrate the strength of the proposed
             method, we provide various examples, for acquisitions in
             both imaging modalities, for paintings with different types
             of canvas and from different periods. The proposed algorithm
             outperforms previous methods proposed for visual photographs
             such as morphological component analysis and Wiener
             filtering and it also works for the digital removal of
             canvas artifacts in X-ray images.},
   Doi = {10.1109/tip.2016.2621413},
   Key = {fds325467}
}

@article{fds325891,
   Author = {Li, Q and Lu, J and Sun, W},
   Title = {Half-space kinetic equations with general boundary
             conditions},
   Journal = {Mathematics of Computation},
   Volume = {86},
   Number = {305},
   Pages = {1269-1301},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2017},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3155},
   Abstract = {We study half-space linear kinetic equations with general
             boundary conditions that consist of both given incoming data
             and various types of reflections, extending our previous
             work on half-space equations with incoming boundary
             conditions. As in our previous work, the main technique is a
             damping adding-removing procedure. We establish the
             well-posedness of linear (or linearized) half-space
             equations with general boundary conditions and
             quasioptimality of the numerical scheme. The numerical
             method is validated by examples including a two-species
             transport equation, a multi-frequency transport equation,
             and the linearized BGK equation in 2D velocity
             space.},
   Doi = {10.1090/mcom/3155},
   Key = {fds325891}
}

@article{fds321515,
   Author = {Mendl, CB and Lu, J and Lukkarinen, J},
   Title = {Thermalization of oscillator chains with onsite
             anharmonicity and comparison with kinetic
             theory.},
   Journal = {Physical review. E},
   Volume = {94},
   Number = {6-1},
   Pages = {062104},
   Year = {2016},
   Month = {December},
   url = {http://dx.doi.org/10.1103/physreve.94.062104},
   Abstract = {We perform microscopic molecular dynamics simulations of
             particle chains with an onsite anharmonicity to study
             relaxation of spatially homogeneous states to equilibrium,
             and directly compare the simulations with the corresponding
             Boltzmann-Peierls kinetic theory. The Wigner function serves
             as a common interface between the microscopic and kinetic
             level. We demonstrate quantitative agreement after an
             initial transient time interval. In particular, besides
             energy conservation, we observe the additional
             quasiconservation of the phonon density, defined via an
             ensemble average of the related microscopic field variables
             and exactly conserved by the kinetic equations. On
             superkinetic time scales, density quasiconservation is lost
             while energy remains conserved, and we find evidence for
             eventual relaxation of the density to its canonical ensemble
             value. However, the precise mechanism remains unknown and is
             not captured by the Boltzmann-Peierls equations.},
   Doi = {10.1103/physreve.94.062104},
   Key = {fds321515}
}

@article{fds320186,
   Author = {Yu, T-Q and Lu, J and Abrams, CF and Vanden-Eijnden,
             E},
   Title = {Multiscale implementation of infinite-swap replica exchange
             molecular dynamics.},
   Journal = {Proceedings of the National Academy of Sciences of the
             United States of America},
   Volume = {113},
   Number = {42},
   Pages = {11744-11749},
   Year = {2016},
   Month = {October},
   url = {http://dx.doi.org/10.1073/pnas.1605089113},
   Abstract = {Replica exchange molecular dynamics (REMD) is a popular
             method to accelerate conformational sampling of complex
             molecular systems. The idea is to run several replicas of
             the system in parallel at different temperatures that are
             swapped periodically. These swaps are typically attempted
             every few MD steps and accepted or rejected according to a
             Metropolis-Hastings criterion. This guarantees that the
             joint distribution of the composite system of replicas is
             the normalized sum of the symmetrized product of the
             canonical distributions of these replicas at the different
             temperatures. Here we propose a different implementation of
             REMD in which (i) the swaps obey a continuous-time Markov
             jump process implemented via Gillespie's stochastic
             simulation algorithm (SSA), which also samples exactly the
             aforementioned joint distribution and has the advantage of
             being rejection free, and (ii) this REMD-SSA is combined
             with the heterogeneous multiscale method to accelerate the
             rate of the swaps and reach the so-called infinite-swap
             limit that is known to optimize sampling efficiency. The
             method is easy to implement and can be trivially
             parallelized. Here we illustrate its accuracy and efficiency
             on the examples of alanine dipeptide in vacuum and
             C-terminal β-hairpin of protein G in explicit solvent. In
             this latter example, our results indicate that the landscape
             of the protein is a triple funnel with two folded structures
             and one misfolded structure that are stabilized by
             H-bonds.},
   Doi = {10.1073/pnas.1605089113},
   Key = {fds320186}
}

@article{fds320187,
   Author = {Lu, J and Zhou, Z},
   Title = {Improved sampling and validation of frozen Gaussian
             approximation with surface hopping algorithm for
             nonadiabatic dynamics.},
   Journal = {The Journal of chemical physics},
   Volume = {145},
   Number = {12},
   Pages = {124109},
   Year = {2016},
   Month = {September},
   url = {http://dx.doi.org/10.1063/1.4963107},
   Abstract = {In the spirit of the fewest switches surface hopping, the
             frozen Gaussian approximation with surface hopping (FGA-SH)
             method samples a path integral representation of the
             non-adiabatic dynamics in the semiclassical regime. An
             improved sampling scheme is developed in this work for
             FGA-SH based on birth and death branching processes. The
             algorithm is validated for the standard test examples of
             non-adiabatic dynamics.},
   Doi = {10.1063/1.4963107},
   Key = {fds320187}
}

@article{fds318293,
   Author = {Li, X and Lu, J},
   Title = {Traction boundary conditions for molecular static
             simulations},
   Journal = {Computer Methods in Applied Mechanics and
             Engineering},
   Volume = {308},
   Pages = {310-329},
   Publisher = {Elsevier BV},
   Year = {2016},
   Month = {August},
   url = {http://dx.doi.org/10.1016/j.cma.2016.05.002},
   Abstract = {This paper presents a consistent approach to prescribe
             traction boundary conditions in atomistic models. Due to the
             typical multiple-neighbor interactions, finding an
             appropriate boundary condition that models a desired
             traction is a non-trivial task. We first present a
             one-dimensional example, which demonstrates how such
             boundary conditions can be formulated. We further analyze
             the stability, and derive its continuum limit. We also show
             how the boundary conditions can be extended to higher
             dimensions with an application to a dislocation dipole
             problem under shear stress.},
   Doi = {10.1016/j.cma.2016.05.002},
   Key = {fds318293}
}

@article{fds318294,
   Author = {Lin, L and Lu, J},
   Title = {Decay estimates of discretized Green’s functions for
             Schrödinger type operators},
   Journal = {Science China Mathematics},
   Volume = {59},
   Number = {8},
   Pages = {1561-1578},
   Publisher = {Springer Nature},
   Year = {2016},
   Month = {August},
   url = {http://dx.doi.org/10.1007/s11425-016-0311-4},
   Abstract = {For a sparse non-singular matrix A, generally A−1 is a
             dense matrix. However, for a class of matrices, A−1 can be
             a matrix with off-diagonal decay properties, i.e., |Aij−1|
             decays fast to 0 with respect to the increase of a properly
             defined distance between i and j. Here we consider the
             off-diagonal decay properties of discretized Green’s
             functions for Schrödinger type operators. We provide decay
             estimates for discretized Green’s functions obtained from
             the finite difference discretization, and from a variant of
             the pseudo-spectral discretization. The asymptotic decay
             rate in our estimate is independent of the domain size and
             of the discretization parameter. We verify the decay
             estimate with numerical results for one-dimensional
             Schrödinger type operators.},
   Doi = {10.1007/s11425-016-0311-4},
   Key = {fds318294}
}

@article{fds318295,
   Author = {Lai, R and Lu, J},
   Title = {Localized density matrix minimization and linear-scaling
             algorithms},
   Journal = {Journal of Computational Physics},
   Volume = {315},
   Pages = {194-210},
   Publisher = {Elsevier BV},
   Year = {2016},
   Month = {June},
   url = {http://dx.doi.org/10.1016/j.jcp.2016.02.076},
   Abstract = {We propose a convex variational approach to compute
             localized density matrices for both zero temperature and
             finite temperature cases, by adding an entry-wise ℓ1
             regularization to the free energy of the quantum system.
             Based on the fact that the density matrix decays
             exponentially away from the diagonal for insulating systems
             or systems at finite temperature, the proposed ℓ1
             regularized variational method provides an effective way to
             approximate the original quantum system. We provide
             theoretical analysis of the approximation behavior and also
             design convergence guaranteed numerical algorithms based on
             Bregman iteration. More importantly, the ℓ1 regularized
             system naturally leads to localized density matrices with
             banded structure, which enables us to develop approximating
             algorithms to find the localized density matrices with
             computation cost linearly dependent on the problem
             size.},
   Doi = {10.1016/j.jcp.2016.02.076},
   Key = {fds318295}
}

@article{fds318296,
   Author = {Lu, J and Ying, L},
   Title = {Sparsifying preconditioner for soliton calculations},
   Journal = {Journal of Computational Physics},
   Volume = {315},
   Pages = {458-466},
   Publisher = {Elsevier BV},
   Year = {2016},
   Month = {June},
   url = {http://dx.doi.org/10.1016/j.jcp.2016.03.061},
   Abstract = {We develop a robust and efficient method for soliton
             calculations for nonlinear Schrödinger equations. The
             method is based on the recently developed sparsifying
             preconditioner combined with Newton's iterative method. The
             performance of the method is demonstrated by numerical
             examples of gap solitons in the context of nonlinear
             optics.},
   Doi = {10.1016/j.jcp.2016.03.061},
   Key = {fds318296}
}

@article{fds361458,
   Author = {Li, X and Lin, L and Lu, J},
   Title = {PEXSI-$Σ$: A Green's function embedding method for
             Kohn-Sham density functional theory},
   Year = {2016},
   Month = {June},
   Abstract = {In this paper, we propose a new Green's function embedding
             method called PEXSI-$\Sigma$ for describing complex systems
             within the Kohn-Sham density functional theory (KSDFT)
             framework, after revisiting the physics literature of
             Green's function embedding methods from a numerical linear
             algebra perspective. The PEXSI-$\Sigma$ method approximates
             the density matrix using a set of nearly optimally chosen
             Green's functions evaluated at complex frequencies. For each
             Green's function, the complex boundary conditions are
             described by a self energy matrix $\Sigma$ constructed from
             a physical reference Green's function, which can be computed
             relatively easily. In the linear regime, such treatment of
             the boundary condition can be numerically exact. The support
             of the $\Sigma$ matrix is restricted to degrees of freedom
             near the boundary of computational domain, and can be
             interpreted as a frequency dependent surface potential. This
             makes it possible to perform KSDFT calculations with
             $\mathcal{O}(N^2)$ computational complexity, where $N$ is
             the number of atoms within the computational domain. Green's
             function embedding methods are also naturally compatible
             with atomistic Green's function methods for relaxing the
             atomic configuration outside the computational domain. As a
             proof of concept, we demonstrate the accuracy of the
             PEXSI-$\Sigma$ method for graphene with divacancy and
             dislocation dipole type of defects using the DFTB+ software
             package.},
   Key = {fds361458}
}

@article{fds316401,
   Author = {Lu, J and Wirth, B and Yang, H},
   Title = {Combining 2D synchrosqueezed wave packet transform with
             optimization for crystal image analysis},
   Journal = {Journal of the Mechanics and Physics of Solids},
   Volume = {89},
   Pages = {194-210},
   Publisher = {Elsevier},
   Year = {2016},
   Month = {April},
   ISSN = {0022-5096},
   url = {http://hdl.handle.net/10161/11296 Duke open
             access},
   Abstract = {We develop a variational optimization method for crystal
             analysis in atomic resolution images, which uses information
             from a 2D synchrosqueezed transform (SST) as input. The
             synchrosqueezed transform is applied to extract initial
             information from atomic crystal images: crystal defects,
             rotations and the gradient of elastic deformation. The
             deformation gradient estimate is then improved outside the
             identified defect region via a variational approach, to
             obtain more robust results agreeing better with the physical
             constraints. The variational model is optimized by a
             nonlinear projected conjugate gradient method. Both examples
             of images from computer simulations and imaging experiments
             are analyzed, with results demonstrating the effectiveness
             of the proposed method.},
   Doi = {10.1016/j.jmps.2016.01.002},
   Key = {fds316401}
}

@article{fds320188,
   Author = {Delgadillo, R and Lu, J and Yang, X},
   Title = {Gauge-invariant frozen Gaussian approximation method for the
             schrödinger equation with periodic potentials},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {38},
   Number = {4},
   Pages = {A2440-A2463},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.1137/15M1040384},
   Abstract = {We develop a gauge-invariant frozen Gaussian approximation
             (GIFGA) method for the Schrödinger equation (LSE) with
             periodic potentials in the semiclassical regime. The method
             generalizes the Herman-Kluk propagator for LSE to the case
             with periodic media. It provides an efficient computational
             tool based on asymptotic analysis on phase space and Bloch
             waves to capture the high-frequency oscillations of the
             solution. Compared to geometric optics and Gaussian beam
             methods, GIFGA works in both scenarios of caustics and beam
             spreading. Moreover, it is invariant with respect to the
             gauge choice of the Bloch eigenfunctions and thus avoids the
             numerical difficulty of computing gauge-dependent Berry
             phase. We numerically test the method by several
             one-dimensional examples; in particular, the first order
             convergence is validated, which agrees with our companion
             analysis paper [Frozen Gaussian Approximation for High
             Frequency Wave Propagation in Periodic Media,
             arXiv:1504.08051, 2015].},
   Doi = {10.1137/15M1040384},
   Key = {fds320188}
}

@article{fds318297,
   Author = {Chen, J and Lu, J},
   Title = {Analysis 0f the divide-and-conquer method for electronic
             structure calculations},
   Journal = {Mathematics of Computation},
   Volume = {85},
   Number = {302},
   Pages = {2919-2938},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2016},
   Month = {January},
   url = {http://dx.doi.org/10.1090/mcom/3066},
   Abstract = {We study the accuracy of the divide-and-conquer method for
             electronic structure calculations. The analysis is conducted
             for a prototypical subdomain problem in the method. We prove
             that the pointwise difference between electron densities of
             the global system and the subsystem decays exponentially as
             a function of the distance away from the boundary of the
             subsystem, under the gap assumption of both the global
             system and the subsystem. We show that the gap assumption is
             crucial for the accuracy of the divide-and-conquer method by
             numerical examples. In particular, we show examples with the
             loss of accuracy when the gap assumption of the subsystem is
             invalid.},
   Doi = {10.1090/mcom/3066},
   Key = {fds318297}
}

@article{fds361459,
   Author = {Lu, J and Zhang, Z and Zhou, Z},
   Title = {Bloch dynamics with second order Berry phase
             correction},
   Year = {2015},
   Month = {December},
   Abstract = {We derive the semiclassical Bloch dynamics with the
             second-order Berry phase correction in the presence of the
             slow-varying scalar potential as perturbation. Our
             mathematical derivation is based on a two-scale WKB
             asymptotic analysis. For a uniform external electric field,
             the bi-characteristics system after a positional shift
             introduced by Berry connections agrees with the recent
             result in previous works. Moreover, for the case with a
             linear external electric field, we show that the extra terms
             arising in the bi-characteristics system after the
             positional shift are also gauge independent.},
   Key = {fds361459}
}

@article{fds243728,
   Author = {Lu, J and Moroz, V and Muratov, CB},
   Title = {Orbital-Free Density Functional Theory of Out-of-Plane
             Charge Screening in Graphene},
   Journal = {Journal of Nonlinear Science},
   Volume = {25},
   Number = {6},
   Pages = {1391-1430},
   Publisher = {Springer Nature},
   Year = {2015},
   Month = {December},
   ISSN = {0938-8974},
   url = {http://dx.doi.org/10.1007/s00332-015-9259-4},
   Abstract = {We propose a density functional theory of
             Thomas–Fermi–Dirac–von Weizsäcker type to describe
             the response of a single layer of graphene resting on a
             dielectric substrate to a point charge or a collection of
             charges some distance away from the layer. We formulate a
             variational setting in which the proposed energy functional
             admits minimizers, both in the case of free graphene layers
             and under back-gating. We further provide conditions under
             which those minimizers are unique and correspond to
             configurations consisting of inhomogeneous density profiles
             of charge carrier of only one type. The associated
             Euler–Lagrange equation for the charge density is also
             obtained, and uniqueness, regularity and decay of the
             minimizers are proved under general conditions. In addition,
             a bifurcation from zero to nonzero response at a finite
             threshold value of the external charge is
             proved.},
   Doi = {10.1007/s00332-015-9259-4},
   Key = {fds243728}
}

@article{fds305048,
   Author = {Li, C and Lu, J and Yang, W},
   Title = {Gentlest ascent dynamics for calculating first excited state
             and exploring energy landscape of Kohn-Sham density
             functionals.},
   Journal = {The Journal of chemical physics},
   Volume = {143},
   Number = {22},
   Pages = {224110},
   Year = {2015},
   Month = {December},
   ISSN = {0021-9606},
   url = {http://dx.doi.org/10.1063/1.4936411},
   Abstract = {We develop the gentlest ascent dynamics for Kohn-Sham
             density functional theory to search for the index-1 saddle
             points on the energy landscape of the Kohn-Sham density
             functionals. These stationary solutions correspond to
             excited states in the ground state functionals. As shown by
             various examples, the first excited states of many chemical
             systems are given by these index-1 saddle points. Our novel
             approach provides an alternative, more robust way to obtain
             these excited states, compared with the widely used ΔSCF
             approach. The method can be easily generalized to target
             higher index saddle points. Our results also reveal the
             physical interest and relevance of studying the Kohn-Sham
             energy landscape.},
   Doi = {10.1063/1.4936411},
   Key = {fds305048}
}

@article{fds361460,
   Author = {Lu, J and Ying, L},
   Title = {Fast algorithm for periodic density fitting for Bloch
             waves},
   Year = {2015},
   Month = {December},
   Abstract = {We propose an efficient algorithm for density fitting of
             Bloch waves for Hamiltonian operators with periodic
             potential. The algorithm is based on column selection and
             random Fourier projection of the orbital functions. The
             computational cost of the algorithm scales as
             $\mathcal{O}\bigl(N_{\text{grid}} N^2 + N_{\text{grid}} NK
             \log (NK)\bigr)$, where $N_{\text{grid}}$ is number of
             spatial grid points, $K$ is the number of sampling
             $k$-points in first Brillouin zone, and $N$ is the number of
             bands under consideration. We validate the algorithm by
             numerical examples in both two and three
             dimensions.},
   Key = {fds361460}
}

@article{fds361461,
   Author = {Lu, J and Otto, F},
   Title = {An isoperimetric problem with Coulomb repulsion and
             attraction to a background nucleus},
   Year = {2015},
   Month = {August},
   Abstract = {We study an isoperimetric problem the energy of which
             contains the perimeter of a set, Coulomb repulsion of the
             set with itself, and attraction of the set to a background
             nucleus as a point charge with charge $Z$. For the
             variational problem with constrained volume $V$, our main
             result is that the minimizer does not exist if $V - Z$ is
             larger than a constant multiple of $\max(Z^{2/3}, 1)$. The
             main technical ingredients of our proof are a uniform
             density lemma and electrostatic screening
             arguments.},
   Key = {fds361461}
}

@article{fds243731,
   Author = {Yang, H and Lu, J and Brown, WP and Daubechies, I and Ying,
             L},
   Title = {Quantitative Canvas Weave Analysis Using 2-D Synchrosqueezed
             Transforms: Application of time-frequency analysis to art
             investigation},
   Journal = {Signal Processing Magazine, IEEE},
   Volume = {32},
   Number = {4},
   Pages = {55-63},
   Publisher = {Institute of Electrical and Electronics Engineers
             (IEEE)},
   Year = {2015},
   Month = {July},
   ISSN = {1053-5888},
   url = {http://hdl.handle.net/10161/12009 Duke open
             access},
   Abstract = {Quantitative canvas weave analysis has many applications in
             art investigations of paintings, including dating,
             forensics, and canvas rollmate identification.
             Traditionally, canvas analysis is based on X-radiographs.
             Prior to serving as a painting canvas, a piece of fabric is
             coated with a priming agent; smoothing its surface makes
             this layer thicker between and thinner right on top of weave
             threads. These variations affect the X-ray absorption,
             making the weave pattern stand out in X-ray images of the
             finished painting. To characterize this pattern, it is
             customary to visually inspect small areas within the
             X-radiograph and count the number of horizontal and vertical
             weave threads; averages of these then estimate the overall
             canvas weave density. The tedium of this process typically
             limits its practice to just a few sample regions of the
             canvas. In addition, it does not capture more subtle
             information beyond weave density, such as thread angles or
             variations in the weave pattern. Signal processing
             techniques applied to art investigation are now increasingly
             used to develop computer-assisted canvas weave analysis
             tools.},
   Doi = {10.1109/MSP.2015.2406882},
   Key = {fds243731}
}

@article{fds243732,
   Author = {Li, Q and Lu, J and Sun, W},
   Title = {Diffusion approximations and domain decomposition method of
             linear transport equations: Asymptotics and
             numerics},
   Journal = {Journal of Computational Physics},
   Volume = {292},
   Pages = {141-167},
   Publisher = {Elsevier BV},
   Year = {2015},
   Month = {July},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2015.03.014},
   Abstract = {In this paper we construct numerical schemes to approximate
             linear transport equations with slab geometry by diffusion
             equations. We treat both the case of pure diffusive scaling
             and the case where kinetic and diffusive scalings coexist.
             The diffusion equations and their data are derived from
             asymptotic and layer analysis which allows general
             scattering kernels and general data. We apply the half-space
             solver in [20] to resolve the boundary layer equation and
             obtain the boundary data for the diffusion equation. The
             algorithms are validated by numerical experiments and also
             by error analysis for the pure diffusive scaling
             case.},
   Doi = {10.1016/j.jcp.2015.03.014},
   Key = {fds243732}
}

@article{fds243733,
   Author = {Lu, J and Mendl, CB},
   Title = {Numerical scheme for a spatially inhomogeneous matrix-valued
             quantum Boltzmann equation},
   Journal = {Journal of Computational Physics},
   Volume = {291},
   Pages = {303-316},
   Publisher = {Elsevier BV},
   Year = {2015},
   Month = {June},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2015.03.020},
   Abstract = {We develop an efficient algorithm for a spatially
             inhomogeneous matrix-valued quantum Boltzmann equation
             derived from the Hubbard model. The distribution functions
             are 2 × 2 matrix-valued to accommodate the spin degree of
             freedom, and the scalar quantum Boltzmann equation is
             recovered as a special case when all matrices are
             proportional to the identity. We use Fourier discretization
             and fast Fourier transform to efficiently evaluate the
             collision kernel with spectral accuracy, and numerically
             investigate periodic, Dirichlet and Maxwell boundary
             conditions. Model simulations quantify the convergence to
             local and global thermal equilibrium.},
   Doi = {10.1016/j.jcp.2015.03.020},
   Key = {fds243733}
}

@article{fds243734,
   Author = {Lu, J and Liu, JG and Margetis, D},
   Title = {Emergence of step flow from an atomistic scheme of epitaxial
             growth in 1+1 dimensions},
   Journal = {Physical Review E - Statistical, Nonlinear, and Soft Matter
             Physics},
   Volume = {91},
   Number = {3},
   Pages = {032403},
   Year = {2015},
   Month = {March},
   ISSN = {1539-3755},
   url = {http://dx.doi.org/10.1103/PhysRevE.91.032403},
   Abstract = {The Burton-Cabrera-Frank (BCF) model for the flow of line
             defects (steps) on crystal surfaces has offered useful
             insights into nanostructure evolution. This model has rested
             on phenomenological grounds. Our goal is to show via scaling
             arguments the emergence of the BCF theory for noninteracting
             steps from a stochastic atomistic scheme of a kinetic
             restricted solid-on-solid model in one spatial dimension.
             Our main assumptions are: adsorbed atoms (adatoms) form a
             dilute system, and elastic effects of the crystal lattice
             are absent. The step edge is treated as a front that
             propagates via probabilistic rules for atom attachment and
             detachment at the step. We formally derive a quasistatic
             step flow description by averaging out the stochastic scheme
             when terrace diffusion, adatom desorption, and deposition
             from above are present.},
   Doi = {10.1103/PhysRevE.91.032403},
   Key = {fds243734}
}

@article{fds243743,
   Author = {Lu, J and Nolen, J},
   Title = {Reactive trajectories and the transition path
             process},
   Journal = {Probability Theory and Related Fields},
   Volume = {161},
   Number = {1-2},
   Pages = {195-244},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2015},
   Month = {February},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/s00440-014-0547-y},
   Abstract = {We study the trajectories of a solution (formula presented)
             to an Itô stochastic differential equation in (formula
             presented), as the process passes between two disjoint open
             sets, (formula presented) and (formula presented). These
             segments of the trajectory are called transition paths or
             reactive trajectories, and they are of interest in the study
             of chemical reactions and thermally activated processes. In
             that context, the sets (formula presented) and (formula
             presented) represent reactant and product states. Our main
             results describe the probability law of these transition
             paths in terms of a transition path process (formula
             presented), which is a strong solution to an auxiliary SDE
             having a singular drift term. We also show that statistics
             of the transition path process may be recovered by empirical
             sampling of the original process (formula presented). As an
             application of these ideas, we prove various representation
             formulas for statistics of the transition paths. We also
             identify the density and current of transition paths. Our
             results fit into the framework of the transition path theory
             by Weinan and Vanden-Eijnden.},
   Doi = {10.1007/s00440-014-0547-y},
   Key = {fds243743}
}

@article{fds243730,
   Author = {Lai, R and Lu, J and Osher, S},
   Title = {Density matrix minimization with ℓ1 regularization},
   Journal = {Communications in Mathematical Sciences},
   Volume = {13},
   Number = {8},
   Pages = {2097-2117},
   Publisher = {International Press of Boston},
   Year = {2015},
   Month = {January},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/CMS.2015.v13.n8.a6},
   Abstract = {We propose a convex variational principle to find sparse
             representation of low-lying eigenspace of symmetric
             matrices. In the context of electronic structure
             calculation, this corresponds to a sparse density matrix
             minimization algorithm with ℓ1 regularization. The
             minimization problem can be efficiently solved by a split
             Bregman iteration type algorithm. We further prove that from
             any initial condition, the algorithm converges to a
             minimizer of the variational principle.},
   Doi = {10.4310/CMS.2015.v13.n8.a6},
   Key = {fds243730}
}

@article{fds243736,
   Author = {Liu, J and Lu, J and Zhou, X},
   Title = {Efficient rare event simulation for failure problems in
             random media},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {37},
   Number = {2},
   Pages = {A609-A624},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2015},
   Month = {January},
   ISSN = {1064-8275},
   url = {http://dx.doi.org/10.1137/140965569},
   Abstract = {In this paper we study rare events associated to the
             solutions of an elliptic partial differential equation with
             a spatially varying random coefficient. The random
             coefficient follows the lognormal distribution, which is
             determined by a Gaussian process. This model is employed to
             study the failure problem of elastic materials in random
             media in which the failure is characterized by the criterion
             that the strain field exceeds a high threshold. We propose
             an efficient importance sampling scheme to compute the small
             failure probability in the high threshold limit. The change
             of measure in our scheme is parametrized by two density
             functions. The efficiency of the importance sampling scheme
             is validated by numerical examples.},
   Doi = {10.1137/140965569},
   Key = {fds243736}
}

@article{fds243737,
   Author = {Lu, J and Marzuola, JL},
   Title = {Strang splitting methods for a quasilinear Schrödinger
             equation: Convergence, instability, and dynamics},
   Journal = {Communications in Mathematical Sciences},
   Volume = {13},
   Number = {5},
   Pages = {1051-1074},
   Publisher = {International Press of Boston},
   Year = {2015},
   Month = {January},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/CMS.2015.v13.n5.a1},
   Abstract = {We study the Strang splitting scheme for quasilinear
             Schrödinger equations. We establish the convergence of the
             scheme for solutions with small initial data. We analyze the
             linear instability of the numerical scheme, which explains
             the numerical blow-up of large data solutions and connects
             to the analytical breakdown of regularity of solutions to
             quasilinear Schrödinger equations. Numerical tests are
             performed for a modified version of the superfluid thin film
             equation.},
   Doi = {10.4310/CMS.2015.v13.n5.a1},
   Key = {fds243737}
}

@article{fds243778,
   Author = {Xian, Y and Thompson, A and Qiu, Q and Nolte, L and Nowacek, D and Lu, J and Calderbank, R},
   Title = {Classification of whale vocalizations using the Weyl
             transform},
   Volume = {2015-August},
   Pages = {773-777},
   Booktitle = {2015 IEEE International Conference on Acoustics, Speech and
             Signal Processing (ICASSP)},
   Year = {2015},
   Month = {January},
   ISBN = {9781467369978},
   ISSN = {1520-6149},
   url = {http://dx.doi.org/10.1109/ICASSP.2015.7178074},
   Abstract = {In this paper, we apply the Weyl transform to represent the
             vocalization of marine mammals. In contrast to other popular
             representation methods, such as the MFCC and the Chirplet
             transform, the Weyl transform captures the global
             information of signals. This is especially useful when the
             signal has low order polynomial phase. We can reconstruct
             the signal from the coefficients obtained from the Weyl
             transform, and perform classification based on these
             coefficients. Experimental results show that classification
             using features extracted from the Weyl transform outperforms
             the MFCC and the Chirplet transform on our collected whales
             data.},
   Doi = {10.1109/ICASSP.2015.7178074},
   Key = {fds243778}
}

@article{fds305050,
   Author = {Yang, H and Lu, J and Ying, L},
   Title = {Crystal image analysis using 2D synchrosqueezed
             transforms},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {13},
   Number = {4},
   Pages = {1542-1572},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2015},
   Month = {January},
   ISSN = {1540-3459},
   url = {http://hdl.handle.net/10161/11648 Duke open
             access},
   Abstract = {We propose efficient algorithms based on a band-limited
             version of 2D synchrosqueezed transforms to extract
             mesoscopic and microscopic information from atomic crystal
             images. The methods analyze atomic crystal images as an
             assemblage of nonoverlapping segments of 2D general
             intrinsic mode type functions, which are superpositions of
             nonlinear wave-like components. In particular, crystal
             defects are interpreted as the irregularity of local energy;
             crystal rotations are described as the angle deviation of
             local wave vectors from their references; the gradient of a
             crystal elastic deformation can be obtained by a linear
             system generated by local wave vectors. Several numerical
             examples of synthetic and real crystal images are provided
             to illustrate the efficiency, robustness, and reliability of
             our methods.},
   Doi = {10.1137/140955872},
   Key = {fds305050}
}

@article{fds243729,
   Author = {Lu, J and Ying, L},
   Title = {Compression of the electron repulsion integral tensor in
             tensor hypercontraction format with cubic scaling
             cost},
   Journal = {Journal of Computational Physics},
   Volume = {302},
   Pages = {329-335},
   Publisher = {Elsevier BV},
   Year = {2015},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2015.09.014},
   Abstract = {© 2015 Elsevier Inc.Electron repulsion integral tensor has
             ubiquitous applications in electronic structure
             computations. In this work, we propose an algorithm which
             compresses the electron repulsion tensor into the tensor
             hypercontraction format with O(nN2logN) computational cost,
             where N is the number of orbital functions and n is the
             number of spatial grid points that the discretization of
             each orbital function has. The algorithm is based on a novel
             strategy of density fitting using a selection of a subset of
             spatial grid points to approximate the pair products of
             orbital functions on the whole domain.},
   Doi = {10.1016/j.jcp.2015.09.014},
   Key = {fds243729}
}

@article{fds351554,
   Author = {Xian, Y and Thompson, A and Qiu, Q and Nolte, L and Nowacek, D and Lu, J and Calderbank, R},
   Title = {CLASSIFICATION OF WHALE VOCALIZATIONS USING THE WEYL
             TRANSFORM},
   Journal = {2015 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND
             SIGNAL PROCESSING (ICASSP)},
   Pages = {773-777},
   Year = {2015},
   Key = {fds351554}
}

@article{fds243754,
   Author = {Lu, J and Otto, F},
   Title = {Nonexistence of a Minimizer for Thomas–Fermi–Dirac–von
             Weizsäcker Model},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {67},
   Number = {10},
   Pages = {1605-1617},
   Publisher = {Wiley},
   Year = {2014},
   Month = {October},
   ISSN = {0010-3640},
   url = {http://dx.doi.org/10.1002/cpa.21477},
   Doi = {10.1002/cpa.21477},
   Key = {fds243754}
}

@article{fds243738,
   Author = {Yang, Y and Peng, D and Lu, J and Yang, W},
   Title = {Excitation energies from particle-particle random phase
             approximation: Davidson algorithm and benchmark
             studies},
   Journal = {The Journal of Chemical Physics},
   Volume = {141},
   Number = {12},
   Pages = {124104-124104},
   Publisher = {AIP Publishing},
   Year = {2014},
   Month = {September},
   ISSN = {0021-9606},
   url = {http://dx.doi.org/10.1063/1.4895792},
   Abstract = {<jats:p>The particle-particle random phase approximation
             (pp-RPA) has been used to investigate excitation problems in
             our recent paper [Y. Yang, H. van Aggelen, and W. Yang, J.
             Chem. Phys. 139, 224105 (2013)]. It has been shown to be
             capable of describing double, Rydberg, and charge transfer
             excitations, which are challenging for conventional
             time-dependent density functional theory (TDDFT). However,
             its performance on larger molecules is unknown as a result
             of its expensive O(N6) scaling. In this article, we derive
             and implement a Davidson iterative algorithm for the pp-RPA
             to calculate the lowest few excitations for large systems.
             The formal scaling is reduced to O(N4), which is comparable
             with the commonly used configuration interaction singles
             (CIS) and TDDFT methods. With this iterative algorithm, we
             carried out benchmark tests on molecules that are
             significantly larger than the molecules in our previous
             paper with a reasonably large basis set. Despite some
             self-consistent field convergence problems with ground state
             calculations of (N − 2)-electron systems, we are able to
             accurately capture lowest few excitations for systems with
             converged calculations. Compared to CIS and TDDFT, there is
             no systematic bias for the pp-RPA with the mean signed error
             close to zero. The mean absolute error of pp-RPA with B3LYP
             or PBE references is similar to that of TDDFT, which
             suggests that the pp-RPA is a comparable method to TDDFT for
             large molecules. Moreover, excitations with relatively large
             non-HOMO excitation contributions are also well described in
             terms of excitation energies, as long as there is also a
             relatively large HOMO excitation contribution. These
             findings, in conjunction with the capability of pp-RPA for
             describing challenging excitations shown earlier, further
             demonstrate the potential of pp-RPA as a reliable and
             general method to describe excitations, and to be a good
             alternative to TDDFT methods.</jats:p>},
   Doi = {10.1063/1.4895792},
   Key = {fds243738}
}

@article{fds243739,
   Author = {Lu, J and Vanden-Eijnden, E},
   Title = {Exact dynamical coarse-graining without time-scale
             separation},
   Journal = {The Journal of Chemical Physics},
   Volume = {141},
   Number = {4},
   Pages = {044109-044109},
   Publisher = {AIP Publishing},
   Year = {2014},
   Month = {July},
   ISSN = {0021-9606},
   url = {http://dx.doi.org/10.1063/1.4890367},
   Abstract = {<jats:p>A family of collective variables is proposed to
             perform exact dynamical coarse-graining even in systems
             without time scale separation. More precisely, it is shown
             that these variables are not slow in general, yet satisfy an
             overdamped Langevin equation that statistically preserves
             the sequence in which any regions in collective variable
             space are visited and permits to calculate exactly the mean
             first passage times from any such region to another. The
             role of the free energy and diffusion coefficient in this
             overdamped Langevin equation is discussed, along with the
             way they transform under any change of variable in
             collective variable space. These results apply both to
             systems with and without inertia, and they can be
             generalized to using several collective variables
             simultaneously. The view they offer on what makes collective
             variables and reaction coordinates optimal breaks from the
             standard notion that good collective variable must be slow
             variable, and it suggests new ways to interpret data from
             molecular dynamics simulations and experiments.</jats:p>},
   Doi = {10.1063/1.4890367},
   Key = {fds243739}
}

@article{fds243740,
   Author = {E, W and Lu, J},
   Title = {Mathematical theory of solids: From quantum mechanics to
             continuum models},
   Journal = {Discrete and Continuous Dynamical Systems},
   Volume = {34},
   Number = {12},
   Pages = {5085-5097},
   Publisher = {American Institute of Mathematical Sciences
             (AIMS)},
   Year = {2014},
   Month = {June},
   ISSN = {1078-0947},
   url = {http://dx.doi.org/10.3934/dcds.2014.34.5085},
   Doi = {10.3934/dcds.2014.34.5085},
   Key = {fds243740}
}

@article{fds243741,
   Author = {Kohn, RV and Lu, J and Schweizer, B and Weinstein,
             MI},
   Title = {A variational perspective on cloaking by anomalous localized
             resonance},
   Journal = {Communications in Mathematical Physics},
   Volume = {328},
   Number = {1},
   Pages = {1-27},
   Publisher = {Springer Nature},
   Year = {2014},
   Month = {March},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/s00220-014-1943-y},
   Abstract = {A body of literature has developed concerning “cloaking by
             anomalous localized resonance.” The mathematical heart of
             the matter involves the behavior of a divergence-form
             elliptic equation in the plane, div (a(x) grad u(x)) = f
             (x). The complex-valued coefficient has a matrix-shell-core
             geometry, with real part equal to 1 in the matrix and the
             core, and −1 in the shell; one is interested in
             understanding the resonant behavior of the solution as the
             imaginary part of a(x) decreases to zero (so that
             ellipticity is lost). Most analytical work in this area has
             relied on separation of variables, and has therefore been
             restricted to radial geometries. We introduce a new approach
             based on a pair of dual variational principles, and apply it
             to some non-radial examples. In our examples, as in the
             radial setting, the spatial location of the source f plays a
             crucial role in determining whether or not resonance
             occurs.},
   Doi = {10.1007/s00220-014-1943-y},
   Key = {fds243741}
}

@article{fds243742,
   Author = {Lin, L and Lu, J and Shao, S},
   Title = {Analysis of time reversible born-oppenheimer molecular
             dynamics},
   Journal = {Entropy},
   Volume = {16},
   Number = {1},
   Pages = {110-137},
   Publisher = {MDPI AG},
   Editor = {G. Ciccotti and M. Ferrario and Ch. Schuette},
   Year = {2014},
   Month = {January},
   url = {http://dx.doi.org/10.3390/e16010110},
   Abstract = {We analyze the time reversible Born-Oppenheimer molecular
             dynamics (TRBOMD) scheme, which preserves the time
             reversibility of the Born-Oppenheimer molecular dynamics
             even with non-convergent self-consistent field iteration. In
             the linear response regime, we derive the stability
             condition, as well as the accuracy of TRBOMD for computing
             physical properties, such as the phonon frequency obtained
             from the molecular dynamics simulation. We connect and
             compare TRBOMD with Car-Parrinello molecular dynamics in
             terms of accuracy and stability. We further discuss the
             accuracy of TRBOMD beyond the linear response regime for
             non-equilibrium dynamics of nuclei. Our results are
             demonstrated through numerical experiments using a
             simplified one-dimensional model for Kohn-Sham density
             functional theory. ©2013 by the author; licensee MDPI,
             Basel, Switzerland.},
   Doi = {10.3390/e16010110},
   Key = {fds243742}
}

@article{fds243744,
   Author = {Lu, J and Ming, P},
   Title = {Stability of a force-based hybrid method with planar sharp
             interface},
   Journal = {SIAM Journal on Numerical Analysis},
   Volume = {52},
   Number = {4},
   Pages = {2005-2026},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2014},
   Month = {January},
   ISSN = {0036-1429},
   url = {http://dx.doi.org/10.1137/130904843},
   Abstract = {We study a force-based hybrid method that couples an
             atomistic model with a Cauchy-Born elasticity model with
             sharp transition interface. We identify stability conditions
             that guarantee the convergence of the hybrid scheme to the
             solution of the atomistic model with second order accuracy,
             as the ratio between lattice parameter and the
             characteristic length scale of the deformation tends to
             zero. Convergence is established for hybrid schemes with
             planar sharp interface for systems without defects, with
             general finite range atomistic potential and simple lattice
             structure. The key ingredients of the proof are regularity
             and stability analysis of elliptic systems of difference
             equations. We apply the results to atomistic-to-continuum
             scheme for a two-dimensional triangular lattice with planar
             interface.},
   Doi = {10.1137/130904843},
   Key = {fds243744}
}

@article{fds243746,
   Author = {Lu, J-F and Yang, X},
   Title = {Asymptotic analysis of quantum dynamics in crystals: the
             Bloch-Wigner transform, Bloch dynamics and Berry
             phase},
   Journal = {Acta Mathematicae Applicatae Sinica, English
             Series},
   Volume = {29},
   Number = {3},
   Pages = {465-476},
   Year = {2013},
   Month = {July},
   ISSN = {0168-9673},
   url = {http://dx.doi.org/10.1007/s10255-011-0095-5},
   Abstract = {We study the semi-classical limit of the Schrödinger
             equation in a crystal in the presence of an external
             potential and magnetic field. We first introduce the
             Bloch-Wigner transform and derive the asymptotic equations
             governing this transform in the semi-classical setting. For
             the second part, we focus on the appearance of the Berry
             curvature terms in the asymptotic equations. These terms
             play a crucial role in many important physical phenomena
             such as the quantum Hall effect. We give a simple derivation
             of these terms in different settings using asymptotic
             analysis. © 2013 Institute of Applied Mathematics, Academy
             of Mathematics and System Sciences, Chinese Academy of
             Sciences and Springer-Verlag Berlin Heidelberg.},
   Doi = {10.1007/s10255-011-0095-5},
   Key = {fds243746}
}

@article{fds243745,
   Author = {Lu, J and Vanden-Eijnden, E},
   Title = {Infinite swapping replica exchange molecular dynamics leads
             to a simple simulation patch using mixture
             potentials.},
   Journal = {The Journal of chemical physics},
   Volume = {138},
   Number = {8},
   Pages = {084105},
   Year = {2013},
   Month = {February},
   url = {http://www.ncbi.nlm.nih.gov/pubmed/23464138},
   Abstract = {Replica exchange molecular dynamics (REMD) becomes more
             efficient as the frequency of swap between the temperatures
             is increased. Recently Plattner et al. [J. Chem. Phys. 135,
             134111 (2011)] proposed a method to implement infinite
             swapping REMD in practice. Here we introduce a natural
             modification of this method that involves molecular dynamics
             simulations over a mixture potential. This modification is
             both simple to implement in practice and provides a better,
             energy based understanding of how to choose the temperatures
             in REMD to optimize efficiency. It also has implications for
             generalizations of REMD in which the swaps involve other
             parameters than the temperature.},
   Doi = {10.1063/1.4790706},
   Key = {fds243745}
}

@article{fds243747,
   Author = {Lu, J and Ming, P},
   Title = {Convergence of a Force‐Based Hybrid Method in Three
             Dimensions},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {66},
   Number = {1},
   Pages = {83-108},
   Publisher = {Wiley},
   Year = {2013},
   Month = {January},
   ISSN = {0010-3640},
   url = {http://dx.doi.org/10.1002/cpa.21429},
   Abstract = {<jats:title>Abstract</jats:title><jats:p>We study a
             force‐based hybrid method that couples an atomistic model
             with the Cauchy‐Born elasticity model. We show that the
             proposed scheme converges to the solution of the atomistic
             model with second‐order accuracy, since the ratio between
             lattice parameter and the characteristic length scale of the
             deformation tends to 0. Convergence is established for the
             three‐dimensional system without defects, with general
             finite‐range atomistic potential and simple lattice
             structure. The proof is based on consistency and stability
             analysis. General tools for stability analysis are developed
             in the framework opseudodifference operators in arbitrary
             dimensions. © 2012 Wiley Periodicals, Inc.</jats:p>},
   Doi = {10.1002/cpa.21429},
   Key = {fds243747}
}

@article{fds243748,
   Author = {Yang, X and Lu, J and Fomel, S},
   Title = {Seismic modeling using the frozen Gaussian
             approximation},
   Journal = {SEG Technical Program Expanded Abstracts
             2013},
   Pages = {4677-4682},
   Booktitle = {SEG Technical Program Expanded Abstracts
             2013},
   Publisher = {Society of Exploration Geophysicists},
   Year = {2013},
   ISBN = {9781629931883},
   url = {http://library.seg.org/doi/abs/10.1190/segam2013-1225.1},
   Abstract = {We adopt the frozen Gaussian approximation (FGA) for
             modeling seismic waves. The FGA method belongs to the
             category of ray-based beam methods. It decomposes the
             seismic wavefield into a set of Gaussian functions and
             propagates these functions along appropriate ray paths. As
             opposed to the classic Gaussian-beam method, FGA keeps the
             Gaussians frozen (at a fixed width) during the propagation
             process and adjusts their amplitudes in order to produce an
             accurate approximation after summation. We perform the
             initial decomposition of seismic data using a fast version
             of the FBI (Fourier- Bros-Iagolnitzer) transform and
             propagate the frozen Gaussian beams numerically using ray
             tracing. A test using a smoothed Marmousi model confirms the
             validity of FGA for accurate modeling of seismic
             wavefields.},
   Doi = {10.1190/segam2013-1225.1},
   Key = {fds243748}
}

@article{fds243750,
   Author = {E, W and Lu, J and Yao, Y},
   Title = {The landscape of complex networks: Critical nodes and a
             hierarchical decomposition},
   Journal = {Methods and Applications of Analysis},
   Volume = {20},
   Number = {4},
   Pages = {383-404},
   Publisher = {International Press of Boston},
   Year = {2013},
   ISSN = {1073-2772},
   url = {http://dx.doi.org/10.4310/MAA.2013.v20.n4.a5},
   Doi = {10.4310/MAA.2013.v20.n4.a5},
   Key = {fds243750}
}

@article{fds243756,
   Author = {E, W and Lu, J},
   Title = {Stability and the continuum limit of the spin-polarized
             Thomas-Fermi-Dirac-von Weizsäcker model},
   Journal = {Journal of Mathematical Physics},
   Volume = {53},
   Number = {11},
   Pages = {115615-115615},
   Publisher = {AIP Publishing},
   Year = {2012},
   Month = {November},
   ISSN = {0022-2488},
   url = {http://dx.doi.org/10.1063/1.4755952},
   Abstract = {<jats:p>The continuum limit of the spin-polarized
             Thomas-Fermi-Dirac-von Weizsäcker model in an external
             magnetic field is studied. An extension of the classical
             Cauchy-Born rule for crystal lattices is established for the
             electronic structure under sharp stability conditions on
             charge density and spin density waves. A Landau-Lifshitz
             type of micromagnetic energy functional is
             derived.</jats:p>},
   Doi = {10.1063/1.4755952},
   Key = {fds243756}
}

@article{fds243753,
   Author = {Lu, J and Yang, X},
   Title = {Frozen gaussian approximation for general linear strictly
             hyperbolic systems: Formulation and eulerian
             methods},
   Journal = {Multiscale Modeling and Simulation},
   Volume = {10},
   Number = {2},
   Pages = {451-472},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2012},
   Month = {September},
   ISSN = {1540-3459},
   url = {http://dx.doi.org/10.1137/10081068X},
   Abstract = {The frozen Gaussian approximation, proposed in [J. Lu and X.
             Yang, Commun. Math. Sci., 9 (2011), pp. 663-683], is an
             efficient computational tool for high frequency wave
             propagation. We continue in this paper the development of
             frozen Gaussian approximation. The frozen Gaussian
             approximation is extended to general linear strictly
             hyperbolic systems. Eulerian methods based on frozen
             Gaussian approximation are developed to overcome the
             divergence problem of Lagrangian methods. The proposed
             Eulerian methods can also be used for the Herman-Kluk
             propagator in quantum mechanics. Numerical examples verify
             the performance of the proposed methods. © 2012 Society for
             Industrial and Applied Mathematics.},
   Doi = {10.1137/10081068X},
   Key = {fds243753}
}

@article{fds243752,
   Author = {Lu, J and Yang, X},
   Title = {Convergence of frozen Gaussian approximation for
             high-frequency wave propagation},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {65},
   Number = {6},
   Pages = {759-789},
   Publisher = {WILEY},
   Year = {2012},
   Month = {June},
   ISSN = {0010-3640},
   url = {http://dx.doi.org/10.1002/cpa.21384},
   Abstract = {The frozen Gaussian approximation provides a highly
             efficient computational method for high-frequency wave
             propagation. The derivation of the method is based on
             asymptotic analysis. In this paper, for general linear
             strictly hyperbolic systems, we establish the rigorous
             convergence result for frozen Gaussian approximation. As a
             byproduct, higher-order frozen Gaussian approximation is
             developed. © 2011 Wiley Periodicals, Inc.},
   Doi = {10.1002/cpa.21384},
   Key = {fds243752}
}

@article{fds243755,
   Author = {Lin, L and Lu, J and Ying, L and E, W},
   Title = {Optimized local basis set for Kohn–Sham density functional
             theory},
   Journal = {Journal of Computational Physics},
   Volume = {231},
   Number = {13},
   Pages = {4515-4529},
   Publisher = {Elsevier BV},
   Year = {2012},
   Month = {May},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2012.03.009},
   Abstract = {We develop a technique for generating a set of optimized
             local basis functions to solve models in the Kohn-Sham
             density functional theory for both insulating and metallic
             systems. The optimized local basis functions are obtained by
             solving a minimization problem in an admissible set
             determined by a large number of primitive basis functions.
             Using the optimized local basis set, the electron energy and
             the atomic force can be calculated accurately with a small
             number of basis functions. The Pulay force is systematically
             controlled and is not required to be calculated, which makes
             the optimized local basis set an ideal tool for ab initio
             molecular dynamics and structure optimization. We also
             propose a preconditioned Newton-GMRES method to obtain the
             optimized local basis functions in practice. The optimized
             local basis set is able to achieve high accuracy with a
             small number of basis functions per atom when applied to a
             one dimensional model problem. © 2012 Elsevier
             Inc.},
   Doi = {10.1016/j.jcp.2012.03.009},
   Key = {fds243755}
}

@article{fds243751,
   Author = {Lin, L and Lu, J and Ying, L and E, W},
   Title = {Adaptive local basis set for Kohn–Sham density functional
             theory in a discontinuous Galerkin framework I: Total energy
             calculation},
   Journal = {Journal of Computational Physics},
   Volume = {231},
   Number = {4},
   Pages = {2140-2154},
   Publisher = {Elsevier BV},
   Year = {2012},
   Month = {February},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2011.11.032},
   Abstract = {Kohn-Sham density functional theory is one of the most
             widely used electronic structure theories. In the
             pseudopotential framework, uniform discretization of the
             Kohn-Sham Hamiltonian generally results in a large number of
             basis functions per atom in order to resolve the rapid
             oscillations of the Kohn-Sham orbitals around the nuclei.
             Previous attempts to reduce the number of basis functions
             per atom include the usage of atomic orbitals and similar
             objects, but the atomic orbitals generally require fine
             tuning in order to reach high accuracy. We present a novel
             discretization scheme that adaptively and systematically
             builds the rapid oscillations of the Kohn-Sham orbitals
             around the nuclei as well as environmental effects into the
             basis functions. The resulting basis functions are localized
             in the real space, and are discontinuous in the global
             domain. The continuous Kohn-Sham orbitals and the electron
             density are evaluated from the discontinuous basis functions
             using the discontinuous Galerkin (DG) framework. Our method
             is implemented in parallel and the current implementation is
             able to handle systems with at least thousands of atoms.
             Numerical examples indicate that our method can reach very
             high accuracy (less than 1. meV) with a very small number
             (4-40) of basis functions per atom. © 2011 Elsevier
             Inc.},
   Doi = {10.1016/j.jcp.2011.11.032},
   Key = {fds243751}
}

@article{fds243749,
   Author = {E, W and Lu, J},
   Title = {The Kohn-Sham equation for deformed crystals},
   Journal = {Memoirs of the American Mathematical Society},
   Volume = {221},
   Number = {1040},
   Pages = {1-1},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2012},
   ISSN = {0065-9266},
   url = {http://dx.doi.org/10.1090/s0065-9266-2012-00659-9},
   Abstract = {The solution to the Kohn-Sham equation in the density
             functional theory of the quantum many-body problem is
             studied in the context of the electronic structure of
             smoothly deformed macroscopic crystals. An analog of the
             classical Cauchy-Born rule for crystal lattices is
             established for the electronic structure of the deformed
             crystal under the following physical conditions: (1) the
             band structure of the undeformed crystal has a gap, i.e. the
             crystal is an insulator, (2) the charge density waves are
             stable, and (3) the macroscopic dielectric tensor is
             positive definite. The effective equation governing the
             piezoelectric effect of a material is rigorously derived.
             Along the way, we also establish a number of fundamental
             properties of the Kohn-Sham map. © 2012 by the American
             Mathematical Society. All rights reserved.},
   Doi = {10.1090/s0065-9266-2012-00659-9},
   Key = {fds243749}
}

@article{fds243764,
   Author = {Daubechies, I and Lu, J and Wu, H-T},
   Title = {Synchrosqueezed wavelet transforms: An empirical mode
             decomposition-like tool},
   Journal = {Applied and Computational Harmonic Analysis},
   Volume = {30},
   Number = {2},
   Pages = {243-261},
   Publisher = {Elsevier BV},
   Year = {2011},
   Month = {March},
   ISSN = {1063-5203},
   url = {http://dx.doi.org/10.1016/j.acha.2010.08.002},
   Abstract = {The EMD algorithm is a technique that aims to decompose into
             their building blocks functions that are the superposition
             of a (reasonably) small number of components, well separated
             in the time-frequency plane, each of which can be viewed as
             approximately harmonic locally, with slowly varying
             amplitudes and frequencies. The EMD has already shown its
             usefulness in a wide range of applications including
             meteorology, structural stability analysis, medical studies.
             On the other hand, the EMD algorithm contains heuristic and
             ad hoc elements that make it hard to analyze mathematically.
             In this paper we describe a method that captures the flavor
             and philosophy of the EMD approach, albeit using a different
             approach in constructing the components. The proposed method
             is a combination of wavelet analysis and reallocation
             method. We introduce a precise mathematical definition for a
             class of functions that can be viewed as a superposition of
             a reasonably small number of approximately harmonic
             components, and we prove that our method does indeed succeed
             in decomposing arbitrary functions in this class. We provide
             several examples, for simulated as well as real data. ©
             2010 Elsevier Inc. All rights reserved.},
   Doi = {10.1016/j.acha.2010.08.002},
   Key = {fds243764}
}

@article{fds243763,
   Author = {Lin, L and Yang, C and Meza, JC and Lu, J and Ying, L and E,
             W},
   Title = {SelInv---An Algorithm for Selected Inversion of a Sparse
             Symmetric Matrix},
   Journal = {ACM Transactions on Mathematical Software},
   Volume = {37},
   Number = {4},
   Pages = {1-19},
   Publisher = {Association for Computing Machinery (ACM)},
   Year = {2011},
   Month = {February},
   ISSN = {0098-3500},
   url = {http://dx.doi.org/10.1145/1916461.1916464},
   Abstract = {<jats:p> We describe an efficient implementation of an
             algorithm for computing selected elements of a general
             sparse symmetric matrix <jats:italic>A</jats:italic> that
             can be decomposed as <jats:italic>A</jats:italic> =
             <jats:italic>LDLT</jats:italic> , where <jats:italic>L</jats:italic>
             is lower triangular and <jats:italic>D</jats:italic> is
             diagonal. Our implementation, which is called
             <jats:italic>SelInv</jats:italic> , is built on top of an
             efficient supernodal left-looking <jats:italic>LDLT</jats:italic>
             factorization of <jats:italic>A</jats:italic> . We discuss
             how computational efficiency can be gained by making use of
             a relative index array to handle indirect addressing. We
             report the performance of SelInv on a collection of sparse
             matrices of various sizes and nonzero structures. We also
             demonstrate how SelInv can be used in electronic structure
             calculations. </jats:p>},
   Doi = {10.1145/1916461.1916464},
   Key = {fds243763}
}

@article{fds243765,
   Author = {E, W and Lu, J},
   Title = {The Electronic Structure of Smoothly Deformed Crystals:
             Wannier Functions and the Cauchy–Born Rule},
   Journal = {Archive for Rational Mechanics and Analysis},
   Volume = {199},
   Number = {2},
   Pages = {407-433},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2011},
   Month = {February},
   ISSN = {0003-9527},
   url = {http://dx.doi.org/10.1007/s00205-010-0339-1},
   Abstract = {The electronic structure of a smoothly deformed crystal is
             analyzed for the case when the effective Hamiltonian is a
             given function of the nuclei by considering the regime when
             the scale of the deformation is much larger than the lattice
             parameter. Wannier functions are defined by projecting the
             Wannier functions for the undeformed crystal to the space
             spanned by the wave functions of the deformed crystal. The
             exponential decay of such Wannier functions is proved for
             the case when the undeformed crystal is an insulator. The
             celebrated Cauchy-Born rule for crystal lattices is extended
             to the present situation for electronic structure analysis.
             © 2010 Springer-Verlag.},
   Doi = {10.1007/s00205-010-0339-1},
   Key = {fds243765}
}

@article{fds243757,
   Author = {Lin, L and Yang, C and Lu, J and Ying, L and E, W},
   Title = {A Fast Parallel Algorithm for Selected Inversion of
             Structured Sparse Matrices with Application to 2D Electronic
             Structure Calculations},
   Journal = {SIAM Journal on Scientific Computing},
   Volume = {33},
   Number = {3},
   Pages = {1329-1351},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {2011},
   Month = {January},
   ISSN = {1064-8275},
   url = {http://dx.doi.org/10.1137/09077432x},
   Abstract = {An efficient parallel algorithm is presented for computing
             selected components of A-1 where A is a structured symmetric
             sparse matrix. Calculations of this type are useful for
             several applications, including electronic structure
             analysis of materials in which the diagonal elements of the
             Green's functions are needed. The algorithm proposed here is
             a direct method based on a block LDLT factorization. The
             selected elements of A -1 we compute lie in the nonzero
             positions of L+LT . We use the elimination tree associated
             with the block LDLT factorization to organize the parallel
             algorithm, and reduce the synchronization overhead by
             passing the data level by level along this tree using the
             technique of local buffers and relative indices. We
             demonstrate the efficiency of our parallel implementation by
             applying it to a discretized two dimensional Hamiltonian
             matrix. We analyze the performance of the parallel algorithm
             by examining its load balance and communication overhead,
             and show that our parallel implementation exhibits an
             excellent weak scaling on a large-scale high performance
             distributed-memory parallel machine. © 2011 Society for
             Industrial and Applied Mathematics.},
   Doi = {10.1137/09077432x},
   Key = {fds243757}
}

@article{fds243759,
   Author = {Lin, L and Lu, J and Ying, L},
   Title = {Fast construction of hierarchical matrix representation from
             matrix-vector multiplication},
   Journal = {Journal of Computational Physics},
   Volume = {230},
   Number = {10},
   Pages = {4071-4087},
   Publisher = {Elsevier BV},
   Year = {2011},
   Month = {January},
   ISSN = {0021-9991},
   url = {http://dx.doi.org/10.1016/j.jcp.2011.02.033},
   Abstract = {We develop a hierarchical matrix construction algorithm
             using matrix-vector multiplications, based on the randomized
             singular value decomposition of low-rank matrices. The
             algorithm uses O(logn) applications of the matrix on
             structured random test vectors and O(nlogn) extra
             computational cost, where n is the dimension of the unknown
             matrix. Numerical examples on constructing Green's functions
             for elliptic operators in two dimensions show efficiency and
             accuracy of the proposed algorithm. © 2011 Elsevier
             Inc.},
   Doi = {10.1016/j.jcp.2011.02.033},
   Key = {fds243759}
}

@article{fds243758,
   Author = {E, W and Lu, J and Yang, X},
   Title = {Effective Maxwell equations from time-dependent density
             functional theory},
   Journal = {Acta Math. Sin.},
   Volume = {32},
   Number = {2},
   Pages = {339-339},
   Publisher = {Springer Nature},
   Year = {2011},
   url = {http://dx.doi.org/10.1007/s10114-011-0555-0},
   Doi = {10.1007/s10114-011-0555-0},
   Key = {fds243758}
}

@article{fds243760,
   Author = {Lu, J and Yang, X},
   Title = {Frozen Gaussian approximation for high frequency wave
             propagation},
   Journal = {Communications in Mathematical Sciences},
   Volume = {9},
   Number = {3},
   Pages = {663-683},
   Publisher = {International Press of Boston},
   Year = {2011},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/CMS.2011.v9.n3.a2},
   Abstract = {We propose the frozen Gaussian approximation for computation
             of high frequency wave propagation. This method approximates
             the solution to the wave equation by an integral
             representation. It provides a highly efficient computational
             tool based on the asymptotic analysis on the phase plane.
             Compared to geometric optics, it provides a valid solution
             around caustics. Compared to the Gaussian beam method, it
             overcomes the drawback of beam spreading. We give several
             numerical examples to verify that the frozen Gaussian
             approximation performs well in the presence of caustics and
             when the Gaussian beam spreads. Moreover, it is observed
             numerically that the frozen Gaussian approximation exhibits
             better accuracy than the Gaussian beam method. © 2011
             International Press.},
   Doi = {10.4310/CMS.2011.v9.n3.a2},
   Key = {fds243760}
}

@article{fds243762,
   Author = {E, W and Lu, J},
   Title = {Multiscale modeling},
   Journal = {Scholarpedia},
   Volume = {6},
   Number = {10},
   Pages = {11527},
   Publisher = {Scholarpedia},
   Year = {2011},
   url = {http://www.scholarpedia.org/article/Multiscale_modeling},
   Doi = {10.4249/scholarpedia.11527},
   Key = {fds243762}
}

@article{fds344665,
   Author = {Noe, F and Sarich, M and Schutte, C and Vanden-Eijnden,
             E},
   Title = {Markov state models based on milestoning},
   Journal = {J. Chem. Phys.},
   Volume = {134},
   Pages = {204105},
   Year = {2011},
   Key = {fds344665}
}

@article{fds243766,
   Author = {E, W and Lu, J},
   Title = {Electronic structure of smoothly deformed crystals:
             Cauchy‐born rule for the nonlinear tight‐binding
             model},
   Journal = {Communications on Pure and Applied Mathematics},
   Volume = {63},
   Number = {11},
   Pages = {1432-1468},
   Publisher = {Wiley},
   Year = {2010},
   Month = {November},
   ISSN = {0010-3640},
   url = {http://dx.doi.org/10.1002/cpa.20330},
   Abstract = {<jats:title>Abstract</jats:title><jats:p>The electronic
             structure of a smoothly deformed crystal is analyzed using a
             minimalist model in quantum many‐body theory, the
             nonlinear tight‐binding model. An extension of the
             classical Cauchy‐Born rule for crystal lattices is
             established for the electronic structure under sharp
             stability conditions. A nonlinear elasticity model is
             rigorously derived. The onset of instability is briefly
             examined. © 2010 Wiley Periodicals, Inc.</jats:p>},
   Doi = {10.1002/cpa.20330},
   Key = {fds243766}
}

@article{fds243767,
   Author = {E, W and Li, T and Lu, J},
   Title = {Localized bases of eigensubspaces and operator
             compression},
   Journal = {Proceedings of the National Academy of Sciences},
   Volume = {107},
   Number = {4},
   Pages = {1273-1278},
   Publisher = {Proceedings of the National Academy of Sciences},
   Year = {2010},
   Month = {January},
   ISSN = {0027-8424},
   url = {http://dx.doi.org/10.1073/pnas.0913345107},
   Abstract = {<jats:p>Given a complex local operator, such as the
             generator of a Markov chain on a large network, a
             differential operator, or a large sparse matrix that comes
             from the discretization of a differential operator, we would
             like to find its best finite dimensional approximation with
             a given dimension. The answer to this question is often
             given simply by the projection of the original operator to
             its eigensubspace of the given dimension that corresponds to
             the smallest or largest eigenvalues, depending on the
             setting. The representation of such subspaces, however, is
             far from being unique and our interest is to find the most
             localized bases for these subspaces. The reduced operator
             using these bases would have sparsity features similar to
             that of the original operator. We will discuss different
             ways of obtaining localized bases, and we will give an
             explicit characterization of the decay rate of these basis
             functions. We will also discuss efficient numerical
             algorithms for finding such basis functions and the reduced
             (or compressed) operator.</jats:p>},
   Doi = {10.1073/pnas.0913345107},
   Key = {fds243767}
}

@article{fds243770,
   Author = {Lin, L and Lu, J and Car, R and E, W},
   Title = {Multipole representation of the Fermi operator with
             application to the electronic structure analysis of metallic
             systems},
   Journal = {Physical Review B},
   Volume = {79},
   Number = {11},
   Pages = {115133},
   Publisher = {American Physical Society (APS)},
   Year = {2009},
   Month = {March},
   ISSN = {1098-0121},
   url = {http://dx.doi.org/10.1103/physrevb.79.115133},
   Abstract = {We propose a multipole representation of the Fermi-Dirac
             function and the Fermi operator and use this representation
             to develop algorithms for electronic structure analysis of
             metallic systems. The algorithm is quite simple and
             efficient. Its computational cost scales logarithmically
             with βΔ where β is the inverse temperature and Δ is the
             width of the spectrum of the discretized Hamiltonian matrix.
             © 2009 The American Physical Society.},
   Doi = {10.1103/physrevb.79.115133},
   Key = {fds243770}
}

@article{fds243769,
   Author = {García-Cervera, CJ and Lu, J and Xuan, Y and E, W},
   Title = {Linear-scaling subspace-iteration algorithm with optimally
             localized nonorthogonal wave functions for Kohn-Sham density
             functional theory},
   Journal = {Physical Review B},
   Volume = {79},
   Number = {11},
   Pages = {115110},
   Publisher = {American Physical Society (APS)},
   Year = {2009},
   Month = {March},
   ISSN = {1098-0121},
   url = {http://dx.doi.org/10.1103/physrevb.79.115110},
   Abstract = {We present a linear-scaling method for electronic structure
             computations in the context of Kohn-Sham density functional
             theory (DFT). The method is based on a subspace iteration,
             and takes advantage of the nonorthogonal formulation of the
             Kohn-Sham functional, and the improved localization
             properties of nonorthogonal wave functions. A
             one-dimensional linear problem is presented as a benchmark
             for the analysis of linear-scaling algorithms for Kohn-Sham
             DFT. Using this one-dimensional model, we study the
             convergence properties of the localized subspace-iteration
             algorithm presented. We demonstrate the efficiency of the
             algorithm for practical applications by performing fully
             three-dimensional computations of the electronic density of
             alkane chains. © 2009 The American Physical
             Society.},
   Doi = {10.1103/physrevb.79.115110},
   Key = {fds243769}
}

@article{fds243768,
   Author = {Lin, L and Lu, J and Ying, L and Car, R and E, W},
   Title = {Fast algorithm for extracting the diagonal of the inverse
             matrix with application to the electronic structure analysis
             of metallic systems},
   Journal = {Commun. Math. Sci.},
   Volume = {7},
   Number = {3},
   Pages = {755-777},
   Publisher = {International Press of Boston},
   Year = {2009},
   url = {http://dx.doi.org/10.4310/cms.2009.v7.n3.a12},
   Doi = {10.4310/cms.2009.v7.n3.a12},
   Key = {fds243768}
}

@article{fds243771,
   Author = {Lin, L and Lu, J and Ying, L and E, W},
   Title = {Pole-based approximation of the Fermi-Dirac
             function},
   Journal = {Chin. Ann. Math. Ser. B},
   Volume = {30},
   Number = {6},
   Pages = {729-742},
   Publisher = {Springer Nature},
   Year = {2009},
   url = {http://dx.doi.org/10.1007/s11401-009-0201-7},
   Doi = {10.1007/s11401-009-0201-7},
   Key = {fds243771}
}

@article{fds243772,
   Author = {Garcia-Cervera, CJ and Ren, W and Lu, J and E, W},
   Title = {Sequential multiscale modelling using sparse
             representation},
   Journal = {Commun. Comput. Phys.},
   Volume = {4},
   Number = {5},
   Pages = {1025-1033},
   Year = {2008},
   Abstract = {The main obstacle in sequential multiscale modeling is the
             pre-computation of the constitutive relation which often
             involves many independent variables. The constitutive
             relation of a polymeric fluid is a function of six
             variables, even after making the simplifying assumption that
             stress depends only on the rate of strain. Precomputing such
             a function is usually considered too expensive. Consequently
             the value of sequential multiscale modeling is often limited
             to "parameter passing". Here we demonstrate that sparse
             representations can be used to drastically reduce the
             computational cost for precomputing functions of many
             variables. This strategy dramatically increases the
             efficiency of sequential multiscale modeling, making it very
             competitive in many situations. © 2008 Global-Science
             Press.},
   Key = {fds243772}
}

@article{fds243776,
   Author = {E, W and Lu, J},
   Title = {The Elastic Continuum Limit of the Tight Binding
             Model*},
   Journal = {Chinese Annals of Mathematics, Series B},
   Volume = {28},
   Number = {6},
   Pages = {665-676},
   Publisher = {Springer Science and Business Media LLC},
   Year = {2007},
   Month = {December},
   ISSN = {0252-9599},
   url = {http://dx.doi.org/10.1007/s11401-006-0447-2},
   Abstract = {The authors consider the simplest quantum mechanics model of
             solids, the tight binding model, and prove that in the
             continuum limit, the energy of tight binding model converges
             to that of the continuum elasticity model obtained using
             Cauchy-Born rule. The technique in this paper is based
             mainly on spectral perturbation theory for large matrices.
             © 2007 The Editorial Office of CAM and Springer-Verlag
             Berlin Heidelberg.},
   Doi = {10.1007/s11401-006-0447-2},
   Key = {fds243776}
}

@article{fds243773,
   Author = {E, W and García-Cervera, CJ and Lu, J},
   Title = {A sub-linear scaling algorithm for computing the electronic
             structure of materials},
   Journal = {Communications in Mathematical Sciences},
   Volume = {5},
   Number = {4},
   Pages = {999-1026},
   Publisher = {International Press of Boston},
   Year = {2007},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/cms.2007.v5.n4.a14},
   Abstract = {We introduce a class of sub-linear scaling algorithms for
             analyzing the electronic structure of crystalline solids
             with isolated defects. We divide the localized orbitals of
             the electrons into two sets: one set associated with the
             atoms in the region where the deformation of the material is
             smooth (smooth region), and the other set associated with
             the atoms around the defects (non-smooth region). The
             orbitals associated with atoms in the smooth region can be
             approximated accurately using asymptotic analysis. The
             results can then be used in the original formulation to find
             the orbitals in the non-smooth region. For orbital-free
             density functional theory, one can simply partition the
             electron density into a sum of the density in the smooth
             region and a density in the non-smooth region. This kind of
             partition is not used for Kohn-Sham density functional
             theory and one uses instead the partition of the set of
             orbitals. As a byproduct, we develop the necessary real
             space formulations and we present a formulation of the
             electronic structure problem for a subsystem, when the
             electronic structure for the remaining part is known. ©
             2007 International Press.},
   Doi = {10.4310/cms.2007.v5.n4.a14},
   Key = {fds243773}
}

@article{fds243774,
   Author = {E, W and Lu, J},
   Title = {Seamless multiscale modeling via dynamics on fiber
             bundles},
   Journal = {Communications in Mathematical Sciences},
   Volume = {5},
   Number = {3},
   Pages = {649-663},
   Publisher = {International Press of Boston},
   Year = {2007},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/cms.2007.v5.n3.a7},
   Abstract = {We present a general mathematical framework for modeling the
             macroscale behavior of a multiscale system using only
             microscale models, by formulating the effective macroscale
             models as dynamic models on the underlying fiber bundles.
             This framework allows us to carry out seamless multiscale
             modeling using traditional numerical techniques. At the same
             time, they give rise to an interesting mathematical
             structure and new interesting mathematical problems. We
             discuss several examples from homogenization problems,
             continuum modeling of solids based on atomistic or
             electronic structure models, macroscale behavior of
             interacting diffusion, and continuum modeling of complex
             fluids based on kinetic and Brownian dynamics models. ©
             2007 International Press.},
   Doi = {10.4310/cms.2007.v5.n3.a7},
   Key = {fds243774}
}

@article{fds243775,
   Author = {E, W and Lu, J},
   Title = {The continuum limit and QM-continuum approximation of
             quantum mechanical models of solids},
   Journal = {Communications in Mathematical Sciences},
   Volume = {5},
   Number = {3},
   Pages = {679-696},
   Publisher = {International Press of Boston},
   Year = {2007},
   ISSN = {1539-6746},
   url = {http://dx.doi.org/10.4310/cms.2007.v5.n3.a9},
   Abstract = {We consider the continuum limit for models of solids that
             arise in density functional theory and the QM-continuum
             approximation of such models. Two different versions of
             QM-continuum approximation are proposed, depending on the
             level at which the Cauchy-Born rule is used, one at the
             level of electron density and one at the level of energy.
             Consistency at the interface between the smooth and the
             non-smooth regions is analyzed. We show that if the
             Cauchy-Born rule is used at the level of electron density,
             then the resulting QM-continuum model is free of the
             so-called "ghost force" at the interface. We also present
             dynamic models that bridge naturally the Car-Parrinello
             method and the QM-continuum approximation. © 2007
             International Press.},
   Doi = {10.4310/cms.2007.v5.n3.a9},
   Key = {fds243775}
}

@article{fds243777,
   Author = {E, W and Lu, J and Yang, JZ},
   Title = {Uniform accuracy of the quasicontinuum method},
   Journal = {Physical Review B},
   Volume = {74},
   Number = {21},
   Pages = {214115},
   Publisher = {American Physical Society (APS)},
   Year = {2006},
   Month = {December},
   ISSN = {1098-0121},
   url = {http://dx.doi.org/10.1103/physrevb.74.214115},
   Abstract = {The accuracy of the quasicontinuum method is studied by
             reformulating the summation rules in terms of reconstruction
             schemes for the local atomic environment of the
             representative atoms. The necessary and sufficient condition
             for uniform first-order accuracy and, consequently, the
             elimination of the "ghost force" is formulated in terms of
             the reconstruction schemes. The quasi-nonlocal approach is
             discussed as a special case of this condition. Examples of
             reconstruction schemes that satisfy this condition are
             presented. Transition between atom-based and element-based
             summation rules are studied. © 2006 The American Physical
             Society.},
   Doi = {10.1103/physrevb.74.214115},
   Key = {fds243777}
}