A continuation multiple shooting method for Wasserstein geodesic equation

TL;DR

This paper introduces a numerical multiple shooting method with continuation for solving the Wasserstein geodesic equation, effectively addressing the boundary value problem in optimal transport while preserving Hamiltonian structure.

Contribution

It presents a novel multiple shooting algorithm combined with continuation for Wasserstein geodesic equations, maintaining Hamiltonian structure and improving numerical solution accuracy.

Findings

Successful numerical examples demonstrating method effectiveness

Preservation of Hamiltonian structure in solutions

Improved solution stability and accuracy

Abstract

In this paper, we propose a numerical method to solve the classic $L^2$-optimal transport problem. Our algorithm is based on use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the transport problem. We exploit the viewpoint of Wasserstein Hamiltonian flow with initial and target densities, and our method is designed to retain the underlying Hamiltonian structure. Several numerical examples are presented to illustrate the performance of the method.