Forward-Euler time-discretization for Wasserstein gradient flows can be wrong

TL;DR

This paper demonstrates that the forward-Euler time-discretization method can fail when simulating Wasserstein gradient flows, even in simple cases, highlighting potential pitfalls in numerical approaches.

Contribution

The paper provides counter-examples and explanations showing the limitations of forward-Euler discretization for Wasserstein gradient flows.

Findings

Forward-Euler discretization can fail in simple Wasserstein gradient flow cases.

Counter-examples demonstrate the method's limitations.

Discussion of why the failure occurs.

Abstract

In this note, we examine the forward-Euler discretization for simulating Wasserstein gradient flows. We provide two counter-examples showcasing the failure of this discretization even for a simple case where the energy functional is defined as the KL divergence against some nicely structured probability densities. A simple explanation of this failure is also discussed.