Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem

TL;DR

This paper introduces a fast adaptive numerical method for solving the quadratic Monge-Kantorovich optimal transport problem using minimal convex extensions and finite difference discretization, with proven convergence and ability to capture subtle properties.

Contribution

It adapts the MA-LBR scheme to the Monge-Ampère equation with boundary conditions, providing a novel, convergent numerical approach for optimal transport with convex support.

Findings

The method converges as grid stepsize approaches zero.

It accurately reproduces subtle properties of the optimal transport problem.

The approach effectively captures a specific minimal Brenier solution.

Abstract

We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the Optimal Transport problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid stepsize vanishes and we show with numerical experiments that it is able to reproduce subtle properties of the Optimal Transport problem.