Computation of Optimal Transport with Finite Volumes

TL;DR

This paper develops finite volume schemes for the dynamic quadratic optimal transport problem, addressing numerical instabilities with nested meshes, and provides convergence analysis and a barrier method for solving the discretized problem.

Contribution

It introduces a novel finite volume discretization for optimal transport, analyzes its convergence, and proposes a practical solution strategy overcoming numerical instabilities.

Findings

Finite volume schemes can be unstable without modifications.

Nested mesh strategies improve numerical stability.

Quantitative convergence estimates are established.

Abstract

We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.