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

Rong Lei

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.

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

Abstract

We study the particle method to approximate the gradient flow on the -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 -Wasserstein space over , specifically to the doubly nonlinear diffusion equation in one dimension.
Paper Structure (8 sections, 13 theorems, 74 equations)

This paper contains 8 sections, 13 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Continuum gradient flow
  4. Discrete gradient flow
  5. main results
  6. Tightness and condition on the metric derivatives
  7. Condition on the energy
  8. $\Gamma$-convergence of the discrete energy.

Key Result

proposition 1

There exists a solution to the discrete gradient flow inclusion dis GF. Furthermore, any solution $\boldsymbol{x_N}$ satisfies $j_q \boldsymbol{x}^{\prime}(t)=\partial_w^0 \widetilde{E}_N(\boldsymbol{x}(t))$ for almost every $t\in[0,T]$.

Theorems & Definitions (34)

  • definition 1: Absolute continuity
  • definition 2: Metric derivative
  • definition 3: Strong upper gradient
  • definition 4: Local slope
  • definition 5: Curve of maximal slope
  • remark 1
  • definition 6: Continuum gradient flow
  • definition 7
  • definition 8: Discrete energy
  • definition 9: Discrete gradient flow
  • ...and 24 more