A new flow dynamic approach for Wasserstein gradient flows

TL;DR

This paper introduces a novel regularized flow dynamic method for efficiently computing Wasserstein gradient flows in Lagrangian coordinates, improving accuracy and stability over traditional Eulerian schemes.

Contribution

It proposes a new approach reformulating Wasserstein gradient flows via Benamou-Brenier's flow dynamic, enabling unconstrained minimization and better capturing sharp interfaces.

Findings

Schemes preserve positivity, mass, and energy dissipation.

Numerical simulations validate accuracy and stability.

Method captures sharp interfaces with fewer unknowns.

Abstract

We develop in this paper a new regularized flow dynamic approach to construct efficient numerical schemes for Wasserstein gradient flows in Lagrangian coordinates. Instead of approximating the Wasserstein distance which needs to solve constrained minimization problems, we reformulate the problem using the Benamou-Brenier's flow dynamic approach, leading to algorithms which only need to solve unconstrained minimization problem in $L^2$ distance. Our schemes automatically inherit some essential properties of Wasserstein gradient systems such as positivity-preserving, mass conservative and energy dissipation. We present ample numerical simulations of Porous-Medium equations, Keller-Segel equations and Aggregation equations to validate the accuracy and stability of the proposed schemes. Compared to numerical schemes in Eulerian coordinates, our new schemes can capture sharp interfaces for various Wasserstein gradient flows using relatively smaller number of unknowns.