Discrete Aleksandrov solutions of the Monge-Ampere equation

TL;DR

This paper proves the convergence of a finite difference scheme to the Aleksandrov solution of the Monge-Ampere equation with Dirac masses, aiding optimal transport computations.

Contribution

It introduces a convergence proof for a wide stencil finite difference scheme for the Monge-Ampere equation with singular measures, and demonstrates its application to optimal transport problems.

Findings

Convergence of the scheme to Aleksandrov solutions is established.

The scheme effectively handles Dirac mass right-hand sides.

Application to initial guesses in geometric optimal transport methods.

Abstract

We prove the convergence of a wide stencil finite difference scheme to the Aleksandrov solution of the elliptic Monge-Ampere equation when the right hand side is a sum of Dirac masses. The discrete scheme we analyze for the Dirichlet problem, when coupled with a discretization of the second boundary condition, can be used to get a good initial guess for geometric methods solving optimal transport between two measures.