%% Papers Published
@article{fds374631,
Author = {Iyer, G and Lu, E and Nolen, J},
Title = {USING BERNOULLI MAPS TO ACCELERATE MIXING OF A RANDOM WALK
ON THE TORUS},
Journal = {Quarterly of Applied Mathematics},
Volume = {82},
Number = {2},
Pages = {359-390},
Publisher = {American Mathematical Society (AMS)},
Year = {2024},
Month = {January},
url = {http://dx.doi.org/10.1090/qam/1668},
Abstract = {We study the mixing time of a random walk on the torus,
alternated with a Lebesgue measure preserving Bernoulli map.
Without the Bernoulli map, the mixing time of the random
walk alone is O(1/ε2), where ε is the step size. Our main
results show that for a class of Bernoulli maps, when the
random walk is alternated with the Bernoulli map ϕ the
mixing time becomes O(|ln ε|). We also study the
dissipation time of this process, and obtain O(|ln ε|)
upper and lower bounds with explicit constants.},
Doi = {10.1090/qam/1668},
Key = {fds374631}
}
@article{fds365009,
Author = {Tough, O and Nolen, J},
Title = {The Fleming-Viot Process with McKean-Vlasov
Dynamics},
Journal = {Electronic Journal of Probability},
Volume = {27},
Pages = {1-72},
Publisher = {Institute of Mathematical Statistics},
Year = {2022},
Month = {August},
url = {http://dx.doi.org/10.1214/22-EJP820},
Abstract = {The Fleming-Viot particle system consists of N identical
particles diffusing in a domain U⊂Rd. Whenever a particle
hits the boundary ∂U, that particle jumps onto another
particle in the interior. It is known that this system
provides a particle representation for both the
Quasi-Stationary Distribution (QSD) and the distribution
conditioned on survival for a given diffusion killed at the
boundary of its domain. We extend these results to the case
of McKean-Vlasov dynamics. We prove that the law conditioned
on survival of a given McKean-Vlasov process killed on the
boundary of its domain may be obtained from the hydrodynamic
limit of the corresponding Fleming-Viot particle system. We
then show that if the target killed McKean-Vlasov process
converges to a QSD as t→∞, such a QSD may be obtained
from the stationary distributions of the corresponding
N-particle Fleming-Viot system as N→∞.},
Doi = {10.1214/22-EJP820},
Key = {fds365009}
}
@article{fds353872,
Author = {Berestycki, J and Brunet, E and Nolen, J and Penington,
S},
Title = {Brownian bees in the infinite swarm limit},
Journal = {Annals of Probability},
Volume = {50},
Number = {6},
Pages = {2133-2177},
Publisher = {Institute of Mathematical Statistics},
Year = {2022},
url = {http://dx.doi.org/10.1214/22-AOP1578},
Doi = {10.1214/22-AOP1578},
Key = {fds353872}
}
@article{fds353873,
Author = {Berestycki, J and Brunet, É and Nolen, J and Penington,
S},
Title = {A free boundary problem arising from branching Brownian
motion with selection},
Journal = {Transactions of the American Mathematical
Society},
Volume = {374},
Number = {9},
Pages = {6269-6329},
Publisher = {American Mathematical Society (AMS)},
Year = {2021},
Month = {May},
url = {http://dx.doi.org/10.1090/tran/8370},
Abstract = {<p>We study a free boundary problem for a parabolic partial
differential equation in which the solution is coupled to
the moving boundary through an integral constraint. The
problem arises as the hydrodynamic limit of an interacting
particle system involving branching Brownian motion with
selection, the so-called <italic>Brownian bees</italic>
model which is studied in the companion paper (see Julien
Berestycki, Éric Brunet, James Nolen, and Sarah Penington
[<italic>Brownian bees in the infinite swarm limit</italic>,
2020]). In this paper we prove existence and uniqueness of
the solution to the free boundary problem, and we
characterise the behaviour of the solution in the large time
limit.</p>},
Doi = {10.1090/tran/8370},
Key = {fds353873}
}
@article{fds349744,
Author = {Lim, TS and Lu, Y and Nolen, JH},
Title = {Quantitative propagation of chaos in a bimolecular chemical
reaction-diffusion model},
Journal = {SIAM Journal on Mathematical Analysis},
Volume = {52},
Number = {2},
Pages = {2098-2133},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1137/19M1287687},
Abstract = {We study a stochastic system of N interacting particles
which models bimolecular chemical reaction-diffusion. In
this model, each particle i carries two attributes: the
spatial location Xit ∈ Td, and the type [I]it ∈ { 1, . .
., n} . While Xit is a standard (independent) diffusion
process, the evolution of the type [I]it is described by
pairwise interactions between different particles under a
series of chemical reactions described by a chemical
reaction network. We prove that, as N → ∞, the
stochastic system has a mean field limit which is described
by a nonlocal reaction-diffusion partial differential
equation. In particular, we obtain a quantitative
propagation of chaos result for the interacting particle
system. Our proof is based on the relative entropy method
used recently by Jabin and Wang [Invent. Math., 214 (2018),
pp. 523-591]. The key ingredient of the relative entropy
method is a large deviation estimate for a special partition
function, which was proved previously by combinatorial
estimates. We give a simple probabilistic proof based on a
novel martingale argument.},
Doi = {10.1137/19M1287687},
Key = {fds349744}
}
@article{fds353255,
Author = {Hebbar, P and Koralov, L and Nolen, J},
Title = {Asymptotic behavior of branching diffusion processes in
periodic media},
Journal = {Electronic Journal of Probability},
Volume = {25},
Pages = {1-40},
Year = {2020},
Month = {January},
url = {http://dx.doi.org/10.1214/20-EJP527},
Abstract = {We study the asymptotic behavior of branching diffusion
processes in periodic media. For a super-critical branching
process, we distinguish two types of behavior for the
normalized number of particles in a bounded domain,
depending on the distance of the domain from the region
where the bulk of the particles is located. At distances
that grow linearly in time, we observe intermittency (i.e.,
the k-th moment dominates the k-th power of the first moment
for some k), while, at distances that grow sub-linearly in
time, we show that all the moments converge. A key
ingredient in our analysis is a sharp estimate of the
transition kernel for the branching process, valid up to
linear in time distances from the location of the initial
particle.},
Doi = {10.1214/20-EJP527},
Key = {fds353255}
}
@article{fds353871,
Author = {Nolen, JH and Cohn, S and Iyer, G and Pego, R},
Title = {Anomalous diffusion in comb-shaped domains and
graphs},
Journal = {Communications in Mathematical Sciences},
Volume = {18},
Number = {7},
Pages = {1815-1862},
Publisher = {International Press},
Year = {2020},
url = {http://dx.doi.org/10.4310/CMS.2020.v18.n7.a2},
Abstract = {In this paper we study the asymptotic behavior of Brownian
motion in both comb-shaped planar domains, and comb-shaped
graphs. We show convergence to a limiting process when both
the spacing between the teeth and the width of the teeth
vanish at the same rate. The limiting process exhibits an
anomalous diffusive behavior and can be described as a
Brownian motion time-changed by the local time of an
independent sticky Brownian motion. In the two dimensional
setting the main technical step is an oscillation estimate
for a Neumann problem, which we prove here using a
probabilistic argument. In the one dimensional setting we
provide both a direct SDE proof, and a proof using the
trapped Brownian motion framework in Ben Arous et al. (Ann.
Probab. ’15).},
Doi = {10.4310/CMS.2020.v18.n7.a2},
Key = {fds353871}
}
@article{fds318326,
Author = {Nolen, J and Roquejoffre, J-M and Ryzhik, L},
Title = {Refined long-time asymptotics for Fisher–KPP
fronts},
Journal = {Communications in Contemporary Mathematics},
Volume = {21},
Number = {07},
Pages = {1850072-1850072},
Publisher = {World Scientific Pub Co Pte Lt},
Year = {2019},
Month = {November},
url = {http://dx.doi.org/10.1142/s0219199718500724},
Abstract = {<jats:p> We study the one-dimensional Fisher–KPP equation,
with an initial condition [Formula: see text] that coincides
with the step function except on a compact set. A well-known
result of Bramson in [Maximal displacement of branching
Brownian motion, Comm. Pure Appl. Math. 31 (1978)
531–581; Convergence of Solutions of the Kolmogorov
Equation to Travelling Waves (American Mathematical Society,
Providence, RI, 1983)] states that, as [Formula: see text],
the solution converges to a traveling wave located at the
position [Formula: see text], with the shift [Formula: see
text] that depends on [Formula: see text]. Ebert and Van
Saarloos have formally derived in [Front propagation into
unstable states: Universal algebraic convergence towards
uniformly translating pulled fronts, Phys. D 146 (2000)
1–99; Front propagation into unstable states, Phys.
Rep. 386 (2003) 29–222] a correction to the Bramson
shift, arguing that [Formula: see text]. Here, we prove
that this result does hold, with an error term of the size
[Formula: see text], for any [Formula: see text]. The
interesting aspect of this asymptotics is that the
coefficient in front of the [Formula: see text]-term does
not depend on [Formula: see text]. </jats:p>},
Doi = {10.1142/s0219199718500724},
Key = {fds318326}
}
@article{fds346862,
Author = {Henderson, NT and Pablo, M and Ghose, D and Clark-Cotton, MR and Zyla,
TR and Nolen, J and Elston, TC and Lew, DJ},
Title = {Ratiometric GPCR signaling enables directional sensing in
yeast.},
Journal = {PLoS Biol},
Volume = {17},
Number = {10},
Pages = {e3000484},
Year = {2019},
Month = {October},
url = {http://dx.doi.org/10.1371/journal.pbio.3000484},
Abstract = {Accurate detection of extracellular chemical gradients is
essential for many cellular behaviors. Gradient sensing is
challenging for small cells, which can experience little
difference in ligand concentrations on the up-gradient and
down-gradient sides of the cell. Nevertheless, the tiny
cells of the yeast Saccharomyces cerevisiae reliably decode
gradients of extracellular pheromones to find their mates.
By imaging the behavior of polarity factors and pheromone
receptors, we quantified the accuracy of initial
polarization during mating encounters. We found that cells
bias the orientation of initial polarity up-gradient, even
though they have unevenly distributed receptors. Uneven
receptor density means that the gradient of ligand-bound
receptors does not accurately reflect the external pheromone
gradient. Nevertheless, yeast cells appear to avoid being
misled by responding to the fraction of occupied receptors
rather than simply the concentration of ligand-bound
receptors. Such ratiometric sensing also serves to amplify
the gradient of active G protein. However, this process is
quite error-prone, and initial errors are corrected during a
subsequent indecisive phase in which polarity clusters
exhibit erratic mobile behavior.},
Doi = {10.1371/journal.pbio.3000484},
Key = {fds346862}
}
@article{fds343338,
Author = {Lu, J and Lu, Y and Nolen, J},
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},
Month = {January},
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 = {fds343338}
}
@article{fds365308,
Author = {Lu, Y and Lu, J and Nolen, J},
Title = {Accelerating Langevin Sampling with Birth-death},
Year = {2019},
Key = {fds365308}
}
@article{fds339330,
Author = {Nolen, JH and Cristali, I and Ranjan, V and Steinberg, J and Beckman, E and Durrett, R and Junge, M},
Title = {Block size in Geometric(p)-biased permutations},
Journal = {Electronic Communications in Probability},
Volume = {23},
Pages = {1-10},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
url = {http://dx.doi.org/10.1214/18-ECP182},
Abstract = {Fix a probability distribution p = (p1, p2, …) on the
positive integers. The first block in a p-biased permutation
can be visualized in terms of raindrops that land at each
positive integer j with probability pj. It is the first
point K so that all sites in [1, K] are wet and all sites in
(K, ∞) are dry. For the geometric distribution pj = p(1
− p)j−1 we show that p log K converges in probability to
an explicit constant as p tends to 0. Additionally, we prove
that if p has a stretch exponential distribution, then K is
infinite with positive probability.},
Doi = {10.1214/18-ECP182},
Key = {fds339330}
}
@article{fds316609,
Author = {Nolen, J and Mourrat, J-C},
Title = {Scaling limit of the corrector in stochastic
homogenization},
Journal = {Annals of Applied Probability},
Volume = {27},
Number = {2},
Pages = {944-959},
Publisher = {Institute of Mathematical Statistics (IMS)},
Year = {2017},
ISSN = {1050-5164},
url = {http://arxiv.org/abs/1502.07440},
Abstract = {In the homogenization of divergence-form equations with
random coefficients, a central role is played by the
corrector.We focus on a discrete space setting and on
dimension 3 and more. Under a minor smoothness assumption on
the law of the random coefficients, we identify the scaling
limit of the corrector, which is akin to a Gaussian free
field. This completes the argument started in [Ann. Probab.
44 (2016) 3207-3233].},
Doi = {10.1214/16-AAP1221},
Key = {fds316609}
}
@article{fds316662,
Author = {Nolen, J and Roquejoffre, J-M and Ryzhik, L},
Title = {Convergence to a single wave in the Fisher-KPP
equation},
Journal = {Chinese Annals of Mathematics, Series B},
Volume = {38},
Number = {2},
Pages = {629-646},
Publisher = {Springer Nature},
Year = {2017},
url = {http://arxiv.org/abs/1604.02994},
Abstract = {The authors study the large time asymptotics of a solution
of the Fisher-KPP reaction-diffusion equation, with an
initial condition that is a compact perturbation of a step
function. A well-known result of Bramson states that, in the
reference frame moving as 2t−(3/2)log t+x∞, the solution
of the equation converges as t → +∞ to a translate of
the traveling wave corresponding to the minimal speed c* =
2. The constant x∞ depends on the initial condition u(0,
x). The proof is elaborate, and based on probabilistic
arguments. The purpose of this paper is to provide a simple
proof based on PDE arguments.},
Doi = {10.1007/s11401-017-1087-4},
Key = {fds316662}
}
@article{fds320462,
Author = {Hamel, F and Nolen, J and Roquejoffre, JM and Ryzhik,
L},
Title = {The logarithmic delay of KPP fronts in a periodic
medium},
Journal = {Journal of the European Mathematical Society},
Volume = {18},
Number = {3},
Pages = {465-505},
Publisher = {European Mathematical Publishing House},
Year = {2016},
Month = {January},
url = {http://arxiv.org/abs/1211.6173},
Abstract = {We extend, to parabolic equations of the KPP type in
periodic media, a result of Bramson which asserts that, in
the case of a spatially homogeneous reaction rate, the time
lag between the position of an initially compactly supported
solution and that of a traveling wave grows logarithmically
in time.},
Doi = {10.4171/JEMS/595},
Key = {fds320462}
}
@article{fds290937,
Author = {Bhamidi, S and Hannig, J and Lee, CY and Nolen, J},
Title = {The importance sampling technique for understanding rare
events in Erdős-Rényi random graphs},
Journal = {Electronic Journal of Probability},
Volume = {20},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
Month = {October},
url = {http://dx.doi.org/10.1214/EJP.v20-2696},
Abstract = {In dense Erdős-Rényi random graphs, we are interested in
the events where large numbers of a given subgraph occur.
The mean behavior of subgraph counts is known, and only
recently were the related large deviations results
discovered. Consequently, it is natural to ask, can one
develop efficient numerical schemes to estimate the
probability of an Erdős-Rényi graph containing an
excessively large number of a fixed given subgraph? Using
the large deviation principle we study an importance
sampling scheme as a method to numerically compute the small
probabilities of large triangle counts occurring within
Erdős-Rényi graphs. We show that the exponential tilt
suggested directly by the large deviation principle does not
always yield an optimal scheme. The exponential tilt used in
the importance sampling scheme comes from a generalized
class of exponential random graphs. Asymptotic optimality, a
measure of the efficiency of the importance sampling scheme,
is achieved by a special choice of the parameters in the
exponential random graph that makes it indistinguishable
from an Erdős-Rényi graph conditioned to have many
triangles in the large network limit. We show how this
choice can be made for the conditioned Erdős-Rényi graphs
both in the replica symmetric phase as well as in parts of
the replica breaking phase to yield asymptotically optimal
numerical schemes to estimate this rare event
probability.},
Doi = {10.1214/EJP.v20-2696},
Key = {fds290937}
}
@article{fds227095,
Author = {S. Bhamidi and J. Hannig and C. Lee and J. Nolen},
Title = {The importance sampling technique for understanding rare
events in Erdős-Rényi random graphs},
Journal = {Electronic Journal of Probability},
Year = {2015},
Month = {October},
url = {http://ejp.ejpecp.org/article/view/2696},
Doi = {10.1214/EJP.v20-2696},
Key = {fds227095}
}
@article{fds287342,
Author = {Nolen, J and Roquejoffre, JM and Ryzhik, L},
Title = {Power-Like Delay in Time Inhomogeneous Fisher-KPP
Equations},
Journal = {Communications in Partial Differential Equations},
Volume = {40},
Number = {3},
Pages = {475-505},
Publisher = {Informa UK Limited},
Year = {2015},
Month = {March},
ISSN = {0360-5302},
url = {http://math.duke.edu/~nolen/preprints/bigdelay-draft.pdf},
Abstract = {We consider solutions of the KPP equation with a
time-dependent diffusivity of the form σ(t/T). For an
initial condition that has sufficiently fast decay in x, we
show that when σ(s) is increasing in time the front
position at time T is (Formula presented.). That is, X(T)
lags behind the linear front by an amount that is algebraic
in T, not by the Bramson correction (3/2)log T as in the
uniform medium. This refines a result by Fang and
Zeitouni.},
Doi = {10.1080/03605302.2014.972744},
Key = {fds287342}
}
@article{fds287345,
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 = {fds287345}
}
@article{fds287343,
Author = {Huckemann, S and Mattingly, JC and Miller, E and Nolen,
J},
Title = {Sticky central limit theorems at isolated hyperbolic planar
singularities},
Journal = {Electronic Journal of Probability},
Volume = {20},
Pages = {1-34},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
url = {http://hdl.handle.net/10161/9516 Duke open
access},
Abstract = {We derive the limiting distribution of the barycenter bn of
an i.i.d. sample of n random points on a planar cone with
angular spread larger than 2π. There are three mutually
exclusive possibilities: (i) (fully sticky case) after a
finite random time the barycenter is almost surely at the
origin; (ii) (partly sticky case) the limiting distribution
of √nb<inf>n</inf> comprises a point mass at the origin,
an open sector of a Gaussian, and the projection of a
Gaussian to the sector’s bounding rays; or (iii)
(nonsticky case) the barycenter stays away from the origin
and the renormalized fluctuations have a fully supported
limit distribution—usually Gaussian but not always. We
conclude with an alternative, topological definition of
stickiness that generalizes readily to measures on general
metric spaces.},
Doi = {10.1214/EJP.v20-3887},
Key = {fds287343}
}
@article{fds316661,
Author = {Nolen, J},
Title = {Normal approximation for the net flux through a random
conductor},
Journal = {Stochastic Partial Differential Equations: Analysis and
Computations},
Volume = {4},
Number = {3},
Pages = {439-476},
Publisher = {Springer Nature},
Year = {2015},
ISSN = {2194-0401},
url = {http://arxiv.org/abs/1406.2186},
Abstract = {We consider solutions of an elliptic partial differential
equation in Rd with a stationary, random conductivity
coefficient. The boundary condition on a square domain of
width L is chosen so that the solution has a macroscopic
unit gradient. We then consider the average flux through the
domain. It is known that in the limit L →∞, this
quantity converges to a deterministic constant, almost
surely. Our main result is about normal approximation for
this flux when L is large: we give an estimate of the
Kantorovich–Wasserstein distance between the law of this
random variable and that of a normal random variable. This
extends a previous result of the author (Probab Theory Relat
Fields, 2013. doi:10.1007/s00440-013-0517-9) to a much
larger class of random conductivity coefficients. The
analysis relies on elliptic regularity, on bounds for the
Green’s function, and on a normal approximation method
developed by Chatterjee (Ann Probab 36:1584–1610, 2008)
based on Stein’s method.},
Doi = {10.1007/s40072-015-0068-4},
Key = {fds316661}
}
@article{fds316608,
Author = {Gloria, A and Nolen, J},
Title = {A Quantitative Central Limit Theorem for the Effective
Conductance on the Discrete Torus},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {69},
Number = {12},
Pages = {2304-2348},
Publisher = {WILEY},
Year = {2015},
ISSN = {0010-3640},
url = {http://dx.doi.org/10.1002/cpa.21614},
Abstract = {We study a random conductance problem on a d-dimensional
discrete torus of size L > 0. The conductances are
independent, identically distributed random variables
uniformly bounded from above and below by positive
constants. The effective conductance AL of the network is a
random variable, depending on L, that converges almost
surely to the homogenized conductance Ahom. Our main result
is a quantitative central limit theorem for this quantity as
L → ∞. In particular, we prove there exists some σ > 0
such that dK (Ld/2A – Ahom/ σ, g) ≲ L–d/2 logd
L,where dK is the Kolmogorov distance and gis a standard
normal variable. The main achievement of this contribution
is the precise asymptotic description of the variance of
AL.© 2015 Wiley Periodicals, Inc.},
Doi = {10.1002/cpa.21614},
Key = {fds316608}
}
@article{fds223681,
Author = {J. Lu and J. Nolen},
Title = {Reactive trajectories and the transition path
process.},
Journal = {Probability Theory and Related Fields},
Year = {2014},
Month = {January},
url = {http://arxiv.org/abs/1303.1744},
Doi = {10.1007/s00440-014-0547-y},
Key = {fds223681}
}
@article{fds287346,
Author = {Nolen, J},
Title = {Normal approximation for a random elliptic
equation},
Journal = {Probability Theory and Related Fields},
Volume = {159},
Number = {3-4},
Pages = {1-40},
Publisher = {Springer Nature},
Year = {2013},
ISSN = {0178-8051},
url = {http://math.duke.edu/~nolen/preprints/ellipfluctper_rev.pdf},
Abstract = {We consider solutions of an elliptic partial differential
equation in {Mathematical expression} with a stationary,
random conductivity coefficient that is also periodic with
period {Mathematical expression}. Boundary conditions on a
square domain of width {Mathematical expression} are
arranged so that the solution has a macroscopic unit
gradient. We then consider the average flux that results
from this imposed boundary condition. It is known that in
the limit {Mathematical expression}, this quantity converges
to a deterministic constant, almost surely. Our main result
is that the law of this random variable is very close to
that of a normal random variable, if the domain size
{Mathematical expression} is large. We quantify this
approximation by an error estimate in total variation. The
error estimate relies on a second order Poincaré inequality
developed recently by Chatterjee. © 2013 Springer-Verlag
Berlin Heidelberg.},
Doi = {10.1007/s00440-013-0517-9},
Key = {fds287346}
}
@article{fds287350,
Author = {Hotz, T and Huckemann, S and Le, H and Marron, JS and Mattingly, JC and Miller, E and Nolen, J and Owen, M and Patrangenaru, V and Skwerer,
S},
Title = {Sticky central limit theorems on open books},
Journal = {The Annals of Applied Probability},
Volume = {23},
Number = {6},
Pages = {2238-2258},
Publisher = {Institute of Mathematical Statistics},
Year = {2013},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/12-AAP899},
Doi = {10.1214/12-AAP899},
Key = {fds287350}
}
@article{fds330358,
Author = {Nolen, J and Pavliotis, GA and Stuart, AM},
Title = {Multiscale modelling and inverse problems},
Journal = {Lecture Notes in Computational Science and
Engineering},
Volume = {83},
Pages = {1-34},
Booktitle = {Numerical Analysis of Multiscale Problems, Lecture Notes in
Computational Science and Engineering},
Publisher = {Springer Berlin Heidelberg},
Editor = {I.G. Graham and T.Y. Hou and O. Lakkis and R. Scheichl},
Year = {2012},
Month = {January},
ISBN = {9783642220609},
url = {http://arxiv.org/abs/1009.2943},
Abstract = {The need to blend observational data and mathematical models
arises in many applications and leads naturally to inverse
problems. Parameters appearing in the model, such as
constitutive tensors, initial conditions, boundary
conditions, and forcing can be estimated on the basis of
observed data. The resulting inverse problems are usually
ill-posed and some form of regularization is required. These
notes discuss parameter estimation in situations where the
unknown parameters vary across multiple scales. We
illustrate the main ideas using a simple model for
groundwater flow. We will highlight various approaches to
regularization for inverse problems, including Tikhonov and
Bayesian methods.We illustrate three ideas that arise when
considering inverse problems in the multiscale context. The
first idea is that the choice of space or set in which to
seek the solution to the inverse problem is intimately
related to whether a homogenized or full multiscale solution
is required. This is a choice of regularization. The second
idea is that, if a homogenized solution to the inverse
problem is what is desired, then this can be recovered from
carefully designed observations of the full multiscale
system. The third idea is that the theory of homogenization
can be used to improve the estimation of homogenized
coefficients from multiscale data.},
Doi = {10.1007/978-3-642-22061-6_1},
Key = {fds330358}
}
@article{fds287349,
Author = {Hamel, F and Nolen, J and Roquejoffre, JM and Ryzhik,
L},
Title = {A short proof of the logarithmic Bramson correction in
Fisher-KPP equations},
Journal = {Networks and Heterogeneous Media},
Volume = {8},
Number = {1},
Pages = {275-289},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2012},
url = {http://math.duke.edu/~nolen/preprints/hnrr1_submit.pdf},
Doi = {10.3934/nhm.2013.8.275},
Key = {fds287349}
}
@article{fds287351,
Author = {Matic, I and Nolen, J},
Title = {A Sublinear Variance Bound for Solutions of a Random
Hamilton-Jacobi Equation},
Journal = {Journal of Statistical Physics},
Volume = {149},
Number = {2},
Pages = {342-361},
Publisher = {Springer Nature},
Year = {2012},
ISSN = {0022-4715},
url = {http://math.duke.edu/~nolen/preprints/sublinHJ_submit_rev.pdf},
Abstract = {We estimate the variance of the value function for a random
optimal control problem. The value function is the solution
w ε of a Hamilton-Jacobi equation with random Hamiltonian
H(p, x, ω)=K(p)-V(x/ε, ω) in dimension d ≥ 2. It is
known that homogenization occurs as ε → 0, but little is
known about the statistical fluctuations of w ε. Our main
result shows that the variance of the solution w ε is
bounded by O(ε/{pipe}log ε {pipe}). The proof relies on a
modified Poincaré inequality of Talagrand. © 2012 Springer
Science+Business Media, LLC.},
Doi = {10.1007/s10955-012-0590-y},
Key = {fds287351}
}
@article{fds287352,
Author = {Nolen, J and Roquejoffre, JM and Ryzhik, L and Zlatoš,
A},
Title = {Existence and Non-Existence of Fisher-KPP Transition
Fronts},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {203},
Number = {1},
Pages = {217-246},
Publisher = {Springer Nature},
Year = {2012},
ISSN = {0003-9527},
url = {http://arxiv.org/abs/1012.2392},
Abstract = {We consider Fisher-KPP-type reaction-diffusion equations
with spatially inhomogeneous reaction rates. We show that a
sufficiently strong localized inhomogeneity may prevent
existence of transition-front-type global-in-time solutions
while creating a global-in-time bump-like solution. This is
the first example of a medium in which no reaction-diffusion
transition front exists. A weaker localized inhomogeneity
leads to the existence of transition fronts, but only in a
finite range of speeds. These results are in contrast with
both Fisher-KPP reactions in homogeneous media as well as
ignition-type reactions in inhomogeneous media. © 2011
Springer-Verlag.},
Doi = {10.1007/s00205-011-0449-4},
Key = {fds287352}
}
@article{fds287355,
Author = {Mellet, A and Nolen, J},
Title = {Capillary drops on a rough surface},
Journal = {Interfaces and Free Boundaries},
Volume = {14},
Number = {2},
Pages = {167-184},
Publisher = {European Mathematical Publishing House},
Year = {2012},
ISSN = {1463-9963},
url = {http://dx.doi.org/10.4171/IFB/278},
Abstract = {We study liquid drops lying on a rough planar surface. The
drops are minimizers of an energy functional that includes a
random adhesion energy. We prove the existence of minimizers
and the regularity of the free boundary. When the length
scale of the randomly varying surface is small, we show that
minimizers are close to spherical caps which are minimizers
of an averaged energy functional. In particular, we give an
error estimate that is algebraic in the scale parameter and
holds with high probability. © European Mathematical
Society 2012.},
Doi = {10.4171/IFB/278},
Key = {fds287355}
}
@article{fds287356,
Author = {Nolen, J and Novikov, A},
Title = {Homogenization of the G-equation with incompressible random
drift in two dimensions},
Journal = {Communications in Mathematical Sciences},
Volume = {9},
Number = {2},
Pages = {561-582},
Publisher = {International Press of Boston},
Year = {2011},
Month = {January},
ISSN = {1539-6746},
url = {http://math.duke.edu/~nolen/preprints/nolen_novikov_cms.pdf},
Abstract = {We study the homogenization limit of solutions to the
G-equation with random drift. This Hamilton-Jacobi equation
is a model for flame propagation in a turbulent fluid in the
regime of thin flames. For a fluid velocity field that is
statistically stationary and ergodic, we prove sufficient
conditions for homogenization to hold with probability one.
These conditions are expressed in terms of travel times for
the associated control problem. When the spatial dimension
is equal to two and the fluid velocity is divergence-free,
we verify that these conditions hold under suitable
assumptions about the growth of the random stream function.
© 2011 International Press.},
Doi = {10.4310/CMS.2011.v9.n2.a11},
Key = {fds287356}
}
@article{fds287353,
Author = {Nolen, J},
Title = {A central limit theorem for pulled fronts in a random
medium},
Journal = {Networks and Heterogeneous Media},
Volume = {6},
Number = {2},
Pages = {167-194},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2011},
ISSN = {1556-1801},
url = {http://math.duke.edu/~nolen/preprints/nolen_kpp_clt.pdf},
Abstract = {We consider solutions to a nonlinear reaction diffusion
equation when the reaction term varies randomly with respect
to the spatial coordinate. The nonlinearity is the KPP type
nonlinearity. For a stationary and ergodic medium, and for
certain initial condition, the solution develops a moving
front that has a deterministic asymptotic speed in the large
time limit. The main result of this article is a central
limit theorem for the position of the front, in the
supercritical regime, if the medium satisfies a mixing
condition. © American Institute of Mathematical
Sciences.},
Doi = {10.3934/nhm.2011.6.167},
Key = {fds287353}
}
@article{fds287354,
Author = {Nolen, J},
Title = {An invariance principle for random traveling waves in one
dimension},
Journal = {SIAM Journal on Mathematical Analysis},
Volume = {43},
Number = {1},
Pages = {153-188},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2011},
ISSN = {0036-1410},
url = {http://math.duke.edu/~nolen/preprints/clt_front.pdf},
Abstract = {We consider solutions to a nonlinear reaction diffusion
equation when the reaction term varies randomly with respect
to the spatial coordinate. The nonlinearity is either the
ignition nonlinearity or the bistable nonlinearity, under
suitable restrictions on the size of the spatial
fluctuations. It is known that the solution develops an
interface which propagates with a well-defined speed in the
large time limit. The main result of this article is a
functional central limit theorem for the random interface
position. Copyright © 2011 by SIAM.},
Doi = {10.1137/090746513},
Key = {fds287354}
}
@article{fds287357,
Author = {Cardaliaguet, P and Nolen, J and Souganidis, PE},
Title = {Homogenization and Enhancement for the G-Equation},
Journal = {Archive for Rational Mechanics and Analysis},
Volume = {199},
Number = {2},
Pages = {527-561},
Publisher = {Springer Nature},
Year = {2011},
ISSN = {0003-9527},
url = {http://arxiv.org/abs/1003.4160},
Abstract = {We consider the so-called G-equation, a level set
Hamilton-Jacobi equation used as a sharp interface model for
flame propagation, perturbed by an oscillatory advection in
a spatio-temporal periodic environment. Assuming that the
advection has suitably small spatial divergence, we prove
that, as the size of the oscillations diminishes, the
solutions homogenize (average out) and converge to the
solution of an effective anisotropic first-order
(spatio-temporal homogeneous) level set equation. Moreover,
we obtain a rate of convergence and show that, under certain
conditions, the averaging enhances the velocity of the
underlying front. We also prove that, at scale one, the
level sets of the solutions of the oscillatory problem
converge, at long times, to the Wulff shape associated with
the effective Hamiltonian. Finally, we also consider
advection depending on position at the integral scale. ©
2010 Springer-Verlag.},
Doi = {10.1007/s00205-010-0332-8},
Key = {fds287357}
}
@article{fds287359,
Author = {Nolen, J and Xin, J and Yu, Y},
Title = {Bounds on front speeds for inviscid and viscous
G-equations},
Journal = {Methods and Applications of Analysis},
Volume = {16},
Number = {4},
Year = {2009},
Month = {December},
url = {http://math.duke.edu/~nolen/preprints/MAA_16_4_06-2.pdf},
Key = {fds287359}
}
@article{fds287358,
Author = {Nolen, J and Papanicolaou, G},
Title = {Fine scale uncertainty in parameter estimation for elliptic
equations},
Journal = {Inverse Problems},
Volume = {25},
Number = {11},
Pages = {115021-115021},
Publisher = {IOP Publishing},
Year = {2009},
Month = {November},
ISSN = {0266-5611},
url = {http://math.duke.edu/~nolen/preprints/uq_elliptic.pdf},
Abstract = {We study the problem of estimating the coefficients in an
elliptic partial differential equation using noisy
measurements of a solution to the equation. Although the
unknown coefficients may vary on many scales, we aim only at
estimating their slowly varying parts, thus reducing the
complexity of the inverse problem. However, ignoring the
fine-scale fluctuations altogether introduces uncertainty in
the estimates, even in the absence of measurement noise. We
propose a strategy for quantifying the uncertainty due to
the fine-scale fluctuations in the coefficients by modeling
their effect on the solution of the forward problem using
the central limit theorem. When this is possible, the
Bayesian estimation of the coefficients reduces to a
weighted least-squares problem with a covariance matrix
whose rank is low regardless of the number of measurements
and does not depend on the details of the coefficient
fluctuations. © 2009 IOP Publishing Ltd.},
Doi = {10.1088/0266-5611/25/11/115021},
Key = {fds287358}
}
@article{fds287348,
Author = {Nolen, J and Xin, J},
Title = {KPP fronts in a one-dimensional random drift},
Journal = {Discrete and Continuous Dynamical Systems - Series
B},
Volume = {11},
Number = {2},
Pages = {421-442},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2009},
ISSN = {1531-3492},
url = {http://dx.doi.org/10.3934/dcdsb.2009.11.421},
Abstract = {We establish the variational principle of
Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in a one
dimensional random drift which is a mean zero stationary
ergodic process with mixing property and local Lipschitz
continuity. To prove the variational principle, we use the
path integral representation of solutions, hitting time and
large deviation estimates of the associated stochastic
flows. The variational principle allows us to derive upper
and lower bounds of the front speeds which decay according
to a power law in the limit of large root mean square
amplitude of the drift. This scaling law is different from
that of the effective diffusion (homogenization)
approximation which is valid for front speeds in
incompressible periodic advection.},
Doi = {10.3934/dcdsb.2009.11.421},
Key = {fds287348}
}
@article{fds287363,
Author = {Nolen, J and Xin, J},
Title = {Asymptotic spreading of KPP reactive fronts in
incompressible space-time random flows},
Journal = {Annales de l'Institut Henri Poincare. Annales: Analyse Non
Lineaire/Nonlinear Analysis},
Volume = {26},
Number = {3},
Pages = {815-839},
Publisher = {Elsevier BV},
Year = {2009},
ISSN = {0294-1449},
url = {http://math.duke.edu/~nolen/preprints/rand_fronts_sub2.pdf},
Abstract = {We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov
(KPP) fronts in space-time random incompressible flows in
dimension d > 1. We prove that if the flow field is
stationary, ergodic, and obeys a suitable moment condition,
the large time front speeds (spreading rates) are
deterministic in all directions for compactly supported
initial data. The flow field can become unbounded at large
times. The front speeds are characterized by the convex rate
function governing large deviations of the associated
diffusion in the random flow. Our proofs are based on the
Harnack inequality, an application of the sub-additive
ergodic theorem, and the construction of comparison
functions. Using the variational principles for the front
speed, we obtain general lower and upper bounds of front
speeds in terms of flow statistics. The bounds show that
front speed enhancement in incompressible flows can grow at
most linearly in the root mean square amplitude of the
flows, and may have much slower growth due to rapid temporal
decorrelation of the flows. © 2008 Elsevier Masson SAS. All
rights reserved.},
Doi = {10.1016/j.anihpc.2008.02.005},
Key = {fds287363}
}
@article{fds287364,
Author = {Nolen, J and Ryzhik, L},
Title = {Traveling waves in a one-dimensional heterogeneous
medium},
Journal = {Annales de l'Institut Henri Poincare. Annales: Analyse Non
Lineaire/Nonlinear Analysis},
Volume = {26},
Number = {3},
Pages = {1021-1047},
Publisher = {Elsevier BV},
Year = {2009},
ISSN = {0294-1449},
url = {http://math.duke.edu/~nolen/preprints/rand-tw-sub2.pdf},
Abstract = {We consider solutions of a scalar reaction-diffusion
equation of the ignition type with a random, stationary and
ergodic reaction rate. We show that solutions of the Cauchy
problem spread with a deterministic rate in the long time
limit. We also establish existence of generalized random
traveling waves and of transition fronts in general
heterogeneous media. © 2009 Elsevier Masson SAS. All rights
reserved.},
Doi = {10.1016/j.anihpc.2009.02.003},
Key = {fds287364}
}
@article{fds287365,
Author = {Mellet, A and Nolen, J and Roquejoffre, JM and Ryzhik,
L},
Title = {Stability of generalized transition fronts},
Journal = {Communications in Partial Differential Equations},
Volume = {34},
Number = {6},
Pages = {521-552},
Publisher = {Informa UK Limited},
Year = {2009},
ISSN = {0360-5302},
url = {http://math.duke.edu/~nolen/preprints/stab-submit.pdf},
Abstract = {We study the qualitative properties of the generalized
transition fronts for the reaction-diffusion equations with
the spatially inhomogeneous nonlinearity of the ignition
type. We show that transition fronts are unique up to
translation in time and are globally exponentially stable
for the solutions of the Cauchy problem. The results hold
for reaction rates that have arbitrary spatial variations
provided that the rate is uniformly positive and bounded
from above. © Taylor & Francis Group,
LLC.},
Doi = {10.1080/03605300902768677},
Key = {fds287365}
}
@article{fds287371,
Author = {Nolen, J and Xin, J},
Title = {KPP Fronts in 1D Random Drift},
Journal = {Discrete and Continuous Dynamical Systems
B},
Volume = {11},
Number = {2},
Year = {2009},
url = {http://math.duke.edu/~nolen/preprints/nx1D_final.pdf},
Key = {fds287371}
}
@article{fds287373,
Author = {Nolen, J and Xin, J},
Title = {Variational principle and reaction-diffusion front speeds in
random flows},
Journal = {ICIAM07-Proceedings},
Pages = {1040701-1040702},
Year = {2008},
Month = {December},
Key = {fds287373}
}
@article{fds287360,
Author = {Nolen, J and Xin, J},
Title = {Computing reactive front speeds in random flows by
variational principle},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {237},
Number = {23},
Pages = {3172-3177},
Publisher = {Elsevier BV},
Year = {2008},
ISSN = {0167-2789},
url = {http://dx.doi.org/10.1016/j.physd.2008.04.024},
Abstract = {We study reactive front speeds in randomly perturbed
cellular flows using a variational representation for the
front speed. We develop this representation into a
computational tool for computing the front speeds without
resorting to closure approximations. We demonstrate that the
front speeds depend on flow statistics and topologies in a
complex and dramatic manner. © 2008 Elsevier B.V. All
rights reserved.},
Doi = {10.1016/j.physd.2008.04.024},
Key = {fds287360}
}
@article{fds287372,
Author = {Nolen, J and Papanicolaou, G and Pironneau, O},
Title = {A framework for adaptive multiscale methods for elliptic
problems},
Journal = {Multiscale Modeling and Simulation},
Volume = {7},
Number = {1},
Pages = {171-196},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2008},
ISSN = {1540-3459},
url = {http://math.duke.edu/~nolen/preprints/multiscale_npp_final.pdf},
Abstract = {We describe a projection framework for developing adaptive
multiscale methods for computing approximate solutions to
elliptic boundary value problems. The framework is
consistent with homogenization when there is scale
separation. We introduce an adaptive form of the finite
element algorithms for solving problems with no clear scale
separation. We present numerical simulations demonstrating
the effectiveness and adaptivity of the multiscale method,
assess its computational complexity, and discuss the
relationship between this framework and other multiscale
methods, such as wavelets, multiscale finite element
methods, and the use of harmonic coordinates. We prove in
detail that the projection-based method captures
homogenization when there is strong scale separation. ©
2008 Society for Industrial and applied Mathematics.},
Doi = {10.1137/070693230},
Key = {fds287372}
}
@article{fds287366,
Author = {Nolen, J and Xin, J},
Title = {Variational Principle of KPP Front Speeds in Temporally
Random Shear Flows with Applications},
Journal = {Communications in Mathematical Physics},
Volume = {269},
Number = {2},
Pages = {493-532},
Year = {2007},
ISSN = {0010-3616},
url = {http://math.duke.edu/~nolen/preprints/timerand_cmp_current.pdf},
Abstract = {We establish the variational principle of
Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in
temporally random shear flows with sufficiently decaying
correlations. A key quantity in the variational principle is
the almost sure Lyapunov exponent of a heat operator with
random potential. To prove the variational principle, we use
the comparison principle of solutions, the path integral
representation of solutions, and large deviation estimates
of the associated stochastic flows. The variational
principle then allows us to analytically bound the front
speeds. The speed bounds imply the linear growth law in the
regime of large root mean square shear amplitude at any
fixed temporal correlation length, and the sublinear growth
law if the temporal decorrelation is also large enough, the
so-called bending phenomenon. © 2006 Springer-Verlag.},
Doi = {10.1007/s00220-006-0144-8},
Key = {fds287366}
}
@article{fds287369,
Author = {Nolen, J and Xin, J},
Title = {Existence of KPP type fronts in space-time periodic shear
flows and a study of minimal speeds based on variational
principle},
Journal = {Discrete and Continuous Dynamical Systems},
Volume = {13},
Number = {5},
Pages = {1217-1234},
Publisher = {American Institute of Mathematical Sciences
(AIMS)},
Year = {2005},
Month = {January},
url = {http://math.duke.edu/~nolen/preprints/nx2_r6.pdf},
Abstract = {We prove the existence of reaction-diffusion traveling
fronts in mean zero space-time periodic shear flows for
nonnegative reactions including the classical KPP
(Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP
nonlinearity, the minimal front speed is characterized by a
variational principle involving the principal eigenvalue of
a space-time periodic parabolic operator. Analysis of the
variational principle shows that adding a mean-zero space
time periodic shear flow to an existing mean zero
space-periodic shear flow leads to speed enhancement.
Computation of KPP minimal speeds is performed based on the
variational principle and a spectrally accurate
discretization of the principal eigenvalue problem. It shows
that the enhancement is monotone decreasing in temporal
shear frequency, and that the total enhancement from pure
reaction-diffusion obeys quadratic and linear laws at small
and large shear amplitudes.},
Doi = {10.3934/dcds.2005.13.1217},
Key = {fds287369}
}
@article{fds287368,
Author = {Nolen, J and Rudd, M and Xin, J},
Title = {Existence of KPP fronts in spatially-temporally periodic
advection and variational principle for propagation
speeds},
Journal = {Dynamics of PDE},
Volume = {2},
Pages = {1-24},
Year = {2005},
url = {http://math.duke.edu/~nolen/preprints/nruddxin_arxiv0502436.pdf},
Key = {fds287368}
}
@article{fds287370,
Author = {Nolen, J and Xin, J},
Title = {A variational principle based study of KPP minimal front
speeds in random shears},
Journal = {Nonlinearity},
Volume = {18},
Number = {4},
Pages = {1655-1675},
Publisher = {IOP Publishing},
Year = {2005},
url = {http://www.iop.org/EJ/toc/0951-7715/18/4},
Abstract = {The variational principle for Kolmogorov-Petrovsky-Piskunov
(KPP) minimal front speeds provides an efficient tool for
statistical speed analysis, as well as a fast and accurate
method for speed computation. A variational principle based
analysis is carried out on the ensemble of KPP speeds
through spatially stationary random shear flows inside
infinite channel domains. In the regime of small root mean
square (rms) shear amplitude, the enhancement of the
ensemble averaged KPP front speeds is proved to obey the
quadratic law under certain shear moment conditions.
Similarly, in the large rms amplitude regime, the
enhancement follows the linear law. In particular, both laws
hold for the Ornstein-Uhlenbeck (O-U) process in the case of
two-dimensional channels. An asymptotic ensemble averaged
speed formula is derived in the small rms regime and is
explicit in the case of the O-U process of the shear. The
variational principle based computation agrees with these
analytical findings, and allows further study of the speed
enhancement distributions as well as the dependence of the
enhancement on the shear covariance. Direct simulations in
the small rms regime suggest a quadratic speed enhancement
law for non-KPP nonlinearities. © 2005 IOP Publishing Ltd
and London Mathematical Society.},
Doi = {10.1088/0951-7715/18/4/013},
Key = {fds287370}
}
@article{fds287362,
Author = {Boye, DM and Valdes, TS and Nolen, JH and Silversmith, AJ and Brewer,
KS and Anderman, RE and Meltzer, RS},
Title = {Transient and persistent spectral hole burning in
Eu3+-doped sol-gel produced SiO2
glass},
Journal = {Journal of Luminescence},
Volume = {108},
Number = {1-4},
Pages = {43-47},
Publisher = {Elsevier BV},
Year = {2004},
Month = {June},
url = {http://dx.doi.org/10.1016/j.jlumin.2004.01.008},
Abstract = {Transient and persistent spectral hole burning (TSHB and
PSHB) experiments were performed on Eu3+ ions in sol-gel
SiO2 glasses with aluminum co-doping. Differences in the
hole burning behavior were observed among samples made from
two organosilicate precursors that were annealed to a series
of final temperatures. All glasses exhibited persistent
spectral holes when annealed to 800°C but, as the annealing
temperature was raised to 1000°C, an increasing number of
Eu3+ ions exhibited TSHB with a corresponding decrease in
the number of ions showing PSHB behavior. The transient hole
burning behavior is similar in nature to that observed for
Eu3+-doped silicate melt glass. © 2004 Elsevier B.V. All
rights reserved.},
Doi = {10.1016/j.jlumin.2004.01.008},
Key = {fds287362}
}
@article{fds287367,
Author = {Nolen, J and Xin, J},
Title = {Min-Max Variational Principles and Fronts Speeds in Random
Shear Flows},
Journal = {Methods and Applications of Analysis},
Volume = {11},
Number = {4},
Pages = {635-644},
Year = {2004},
url = {http://math.duke.edu/~nolen/preprints/minmax_0501445.pdf},
Key = {fds287367}
}
@article{fds328338,
Author = {Nolen, J and Xin, J},
Title = {Reaction-diffusion front speeds in spatially-temporally
periodic shear flows},
Journal = {Multiscale Modeling and Simulation},
Volume = {1},
Number = {4},
Pages = {554-570},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2003},
Month = {January},
url = {http://epubs.siam.org/sam-bin/dbq/toc/MMS/1/4},
Abstract = {We study the asymptotics of two space dimensional
reaction-diffusion front speeds through mean zero space-time
periodic shears using both analytical and numerical methods.
The analysis hinges on traveling fronts and their estimates
based on qualitative properties such as monotonicity and a
priori integral inequalities. The computation uses an
explicit second order upwind finite difference method to
provide more quantitative information. At small shear
amplitudes, front speeds are enhanced by an amount
proportional to shear amplitude squared. The proportionality
constant has a closed form expression. It decreases with
increasing shear temporal frequency and is independent of
the form of the known reaction nonlinearities. At large
shear amplitudes and for all reaction nonlinearities, the
enhanced speeds grow proportional to shear amplitude and are
again decreasing with increasing shear temporal frequencies.
The results extend previous ones in the literature on front
speeds through spatially periodic shears and show front
speed slowdown due to shear direction switching in
time.},
Doi = {10.1137/S1540345902420234},
Key = {fds328338}
}
@article{fds287361,
Author = {Boye, DM and Silversmith, AJ and Nolen, J and Rumney, L and Shaye, D and Smith, BC and Brewer, KS},
Title = {Red-to-green up-conversion in Er-doped SiO2 and
SiO2-TiO2 sol-gel silicate
glasses},
Journal = {Journal of Luminescence},
Volume = {94-95},
Pages = {279-282},
Publisher = {Elsevier BV},
Year = {2001},
Month = {December},
ISSN = {0022-2313},
url = {http://dx.doi.org/10.1016/S0022-2313(01)00301-5},
Abstract = {Monolithic Er-doped SiO2-TiO2 binary glasses of high optical
quality were used in an investigation of the effects of
different annealing conditions and titanium content on
fluorescence yields and decay times of the 4S3/2 level of
Er3+. In addition, the characteristics of green upconverted
fluorescence from the 4S3/2 level via excitation with red
laser light (655 nm) were studied. Energy transfer
up-conversion involving two ions in the 4I11/2 state was
determined to be the dominant up-conversion mechanism. The
glasses with TiO2 showed enhanced up-conversion and required
lower annealing temperatures for the up-conversion to be
observed when compared to SiO2 glasses doped with Al. ©
2001 Elsevier Science B.V. All rights reserved.},
Doi = {10.1016/S0022-2313(01)00301-5},
Key = {fds287361}
}
%% Papers Submitted
@article{fds299972,
Author = {J. Nolen and J.-M. Roquejoffre and L. Ryzhik},
Title = {Refined long time asymptotics for the Fisher-KPP
equation},
Year = {2015},
url = {http://math.duke.edu/~nolen/preprints/kpp-brezis-v5.pdf},
Key = {fds299972}
}
|
| 
Mathematics Department