Nonlocal Approximation of Slow and Fast Diffusion

Katy Craig; Matt Jacobs; Olga Turanova

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arXiv · 2312.11438

Nonlocal Approximation of Slow and Fast Diffusion

Katy Craig·Matt Jacobs·Olga Turanova

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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.

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BibTeX RIS

Katy Craig, Matt Jacobs, Olga Turanova. 2024-04-04. Nonlocal Approximation of Slow and Fast Diffusion. https://arxiv.org/abs/2312.11438

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