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

Yewei Xu, Qin Li

TL;DR

Forward-Euler time discretization for Wasserstein gradient flows can fail even for KL-type energies. The authors construct two counter-examples showing that FE can produce $\rho$ with $\rho \notin D(|\partial F|)$, due to loss of regularity from non-injective pushforwards and derivative consumption. The results highlight stability considerations for time stepping and motivate implicit schemes such as the JKO (minimizing movement) scheme for robustness. The work thus reveals fundamental limitations of explicit discretizations in numerical PDEs on the space of probability measures.

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.

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

TL;DR

Forward-Euler time discretization for Wasserstein gradient flows can fail even for KL-type energies. The authors construct two counter-examples showing that FE can produce with , due to loss of regularity from non-injective pushforwards and derivative consumption. The results highlight stability considerations for time stepping and motivate implicit schemes such as the JKO (minimizing movement) scheme for robustness. The work thus reveals fundamental limitations of explicit discretizations in numerical PDEs on the space of probability measures.

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.
Paper Structure (7 sections, 4 theorems, 43 equations, 2 figures, 1 table)

This paper contains 7 sections, 4 theorems, 43 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basics and notations for gradient flow
  3. Basic Notations
  4. KL Divergence as the energy functional
  5. Counter-examples
  6. Examples
  7. Loss of regularity through \ref{['eqn:forwardEuler_rho']} is generic

Key Result

Proposition 1

Assume the initial distribution $\rho_0(x)$ has the density of then for any $h>0$, the one-time-step solution $\rho_1$ also has a density $p_1$, with $p_1$ being discontinuous, and thus $\rho_1 \not \in D(\vert \partial F \vert)$, stopping the application of eqn:forwardEuler_rho to produce $\rho_2$.

Figures (2)

  • Figure 1: The plot on the left shows the pushforward function $T$. Note that every $y \in (0,\frac{2}{3}\sqrt{\frac{1}{3h}})$ has three pre-image points of $x$. For example, for $y\in[0,\frac{2}{3}\sqrt{\frac{1}{3h}}]$, the different parts of pre-image are bold-lined: $(-\infty,-\sqrt{\frac{1}{3h}}]$, $[-\sqrt{\frac{1}{3h}},\sqrt{\frac{1}{3h}}]$ and $[\sqrt{\frac{1}{3h}},\infty)$ respectively. The plot on the right shows the mapping of $T$. When $T$ is confined in three parts of the region, denoted as $T_1, T_2, T_3$, injectivity is resumed.
  • Figure 2: Plot of pushforward functions $T$ for different step sizes $h$. In the $h\to0$ limit, $T(x)=x$.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 1
  • proof
  • proof : Proof for Proposition \ref{['prop:ex2_property']}
  • Proposition 3
  • proof