We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a
limit from a class of nonlocal partial differential equations. The nonlocal equations are …

This paper studies the derivation of the quadratic porous medium equation and a class of
cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a …

Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we
study a continuum birth–death dynamics. We improve results in previous works (Liu et al …

Motivated by recent work on approximation of diffusion equations by deterministic interacting
particle systems, we develop a nonlocal approximation for a range of linear and nonlinear …

In recent years, there has been a spike in interest in multiphase tissue growth models.
Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law …

The purpose of this paper is to answer a few open questions in the interface of kernel
methods and PDE gradient flows. Motivated by recent advances in machine learning …

We construct deterministic particle solutions for linear and fast diffusion equations using a
nonlocal approximation. We exploit the $2 $-Wasserstein gradient flow structure of the …

This chapter reviews (and expands) some recent results on the modeling of aggregation-
diffusion phenomena at various scales, focusing on the emergence of collective dynamics …

The primary contribution of this thesis is to advance the theory of complexity for sampling
from a continuous probability density over R^ d. Some highlights include: a new analysis of …

We consider stochastic approximations of sampling algorithms, such as Stochastic Gradient
Langevin Dynamics (SGLD) and the Random Batch Method (RBM) for Interacting Particle …