Numerical Analysis of the 1-D Parabolic Optimal Transport Problem

TL;DR

This paper investigates a finite difference numerical scheme for solving the 1-D optimal transport problem via a time-dependent approach, demonstrating convergence and providing error analysis and numerical examples.

Contribution

It introduces a finite difference method for 1-D optimal transport, analyzing its convergence and error, and presents numerical experiments to validate the approach.

Findings

The scheme converges exponentially fast to the optimal transport solution.

Error estimates for the finite difference scheme are established.

Numerical examples illustrate the effectiveness of the method.

Abstract

Numerical methods for the optimal transport problem is an active area of research. Recent work of Kitagawa and Abedin shows that the solution of a time-dependent equation converges exponentially fast as time goes to infinity to the solution of the optimal transport problem. This suggests a fast numerical algorithm for computing optimal maps; we investigate such an algorithm here in the 1-dimensional case. Specifically, we use a finite difference scheme to solve the time-dependent optimal transport problem and carry out an error analysis of the scheme. A collection of numerical examples is also presented and discussed.