Convergence rates of particle approximation of forward-backward splitting algorithm for granular medium equations

TL;DR

This paper analyzes the convergence rates of particle approximation methods for the granular medium equation using forward-backward splitting algorithms, without requiring global Lipschitz conditions on potentials, and provides sharp Wasserstein distance convergence rates.

Contribution

The paper introduces efficient forward-backward splitting algorithms for particle approximation of the granular medium equation with non-Lipschitz potentials and establishes sharp convergence rates.

Findings

Established convergence rates in Wasserstein distance.

Designed algorithms applicable to non-Lipschitz potentials.

Provided theoretical guarantees for particle approximation accuracy.

Abstract

We study the spatially homogeneous granular medium equation \[\partial_t\mu=\rm{div}(\mu\nabla V)+\rm{div}(\mu(\nabla W \ast \mu))+\Delta\mu\,,\] within a large and natural class of the confinement potentials $V$ and interaction potentials $W$. The considered problem do not need to assume that $\nabla V$ or $\nabla W$ are globally Lipschitz. With the aim of providing particle approximation of solutions, we design efficient forward-backward splitting algorithms. Sharp convergence rates in terms of the Wasserstein distance are provided.