Unconditional convergence for discretizations of dynamical optimal transport

TL;DR

This paper establishes a convergence framework for discretized dynamical optimal transport problems, ensuring solutions approach the continuous case as mesh refinement occurs, applicable to various geometries and discretizations.

Contribution

It provides the first unconditional convergence guarantee for discretizations of dynamical optimal transport, independent of mesh ratio and applicable to general measures and manifolds.

Findings

Convergence holds without ratio condition between spatial and temporal steps.

Framework applies to flat spaces and Riemannian manifolds.

Includes discretizations by Gladbach, Kopfer, Maas and others.

Abstract

The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamics formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle \emph{a priori} a vast class of cost functions and geometries. Several discretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space. In this article, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional one for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien and Solomon fit in this framework.