Nonlocal Approximation of Slow and Fast Diffusion
Abstract
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 diffusion equations and prove convergence of the method in the slow, linear, and fast diffusion regimes. A key ingredient of our approach is a novel technique for using the 2-Wasserstein and dual Sobolev gradient flow structures of the diffusion equations to recover the duality relation characterizing the pressure in the nonlocal-to-local limit. Due to the general class of internal energy densities that our method is able to handle, a byproduct of our result is a novel particle method for sampling a wide range of probability measures, which extends classical approaches based on the Fokker-Planck equation beyond the log-concave setting.
Cite
@article{arxiv.2312.11438,
title = {Nonlocal Approximation of Slow and Fast Diffusion},
author = {Katy Craig and Matt Jacobs and Olga Turanova},
journal= {arXiv preprint arXiv:2312.11438},
year = {2024}
}