%% 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} }