Convergence of a particle method for gradient flows on the $L^p$-Wasserstein space

TL;DR

This paper analyzes a particle-based numerical method for approximating gradient flows in the $L^p$-Wasserstein space, demonstrating its convergence to the continuum solution of a nonlinear diffusion equation in one dimension.

Contribution

It introduces a discretization scheme that preserves the gradient flow structure at the particle level and proves its convergence to the continuum gradient flow.

Findings

The particle method converges to the continuum gradient flow.

The method accurately approximates the doubly nonlinear diffusion equation.

Preserves the gradient flow structure at the discrete level.

Abstract

We study the particle method to approximate the gradient flow on the $L^p$-Wasserstein space. This method relies on the discretization of the energy introduced by [3] via nonoverlapping balls centered at the particles and preserves the gradient flow structure at the particle level. We prove the convergence of the discrete gradient flow to the continuum gradient flow on the $L^p$-Wasserstein space over $\mathbb R$, specifically to the doubly nonlinear diffusion equation in one dimension.