Browsing by Subject "math.AP"
Now showing items 1-20 of 28
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A convergent method for linear half-space kinetic equations
(2017-04-23)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 ... -
A Mean Field Limit for the Vlasov–Poisson System
(Archive for Rational Mechanics and Analysis, 2017-09)© 2017, Springer-Verlag Berlin Heidelberg. We present a probabilistic proof of the mean field limit and propagation of chaos N-particle systems in three dimensions with positive (Coulomb) or negative (Newton) 1/r potentials ... -
A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations
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 ... -
A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue
(2017-04-23)The purpose of this short note 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 $\Omega$ by the largest mean first ... -
An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus
(2017-04-23)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 ... -
Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model
(2017-04-23)In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface ... -
Asymptotic behavior of branching diffusion processes in periodic media
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, ... -
Bloch dynamics with second order Berry phase correction
(2017-04-23)We derive the semiclassical Bloch dynamics with the second order Berry phase correction, based on a two-scale WKB asymptotic analysis. For uniform external electric field, the bi-characteristics system after a positional ... -
Convergence of Phase-Field Free Energy and Boundary Force for Molecular Solvation
(2017-04-23)We study a phase-field variational model for the solvaiton 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 ... -
Defect resonances of truncated crystal structures
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 ... -
Derivation of the Bogoliubov Time Evolution for Gases with Finite Speed of Sound
The derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, ... -
Exterior Differential Systems and Euler-Lagrange Partial Differential Equations
We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first ... -
Global regularity for 1D Eulerian dynamics with singular interaction forces
(2017-12-18)The Euler-Poisson-Alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/repulsion as well as alignment forces. We consider ... -
Irreducible Ginzburg-Landau fields in dimension 2
(2018-01-18)Ginzburg--Landau fields are the solutions of the Ginzburg--Landau equations which depend on two positive parameters, $\alpha$ and $\beta$. We give conditions on $\alpha$ and $\beta$ for the existence of irreducible solutions ... -
Learning interacting particle systems: diffusion parameter estimation for aggregation equations
(2018-02-14)In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation. Specifically, we construct an estimator $\widehat{\nu}$ with partial observed data to approximate the ... -
Microscopic derivation of the Keller-Segel equation in the sub-critical regime
We derive the two-dimensional Keller-Segel equation from a stochastic system of $N$ interacting particles in the case of sub-critical chemosensitivity $\chi < 8 \pi$. The Coulomb interaction force is regularised with a cutoff ... -
Non-Convex Planar Harmonic Maps
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 ... -
On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes
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. ... -
On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics
We provide a new explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance. To achieve this, we first prove a Poincar\'{e}-type inequality with Gibbs measure in space and Gaussian measure in ... -
On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system
We devise and study a random particle blob method for approximating the Vlasov-Poisson-Fokkker-Planck (VPFP) equations by a $N$-particle system subject to the Brownian motion in $\mathbb{R}^3$ space. More precisely, we show ...
