Convergence framework for the second boundary value problem for the Monge-Ampère equation

Document Type

Article

Publication Date

1-1-2019

Abstract

It is well known that the quadratic-cost optimal transportation problem is formally equivalent to the second boundary value problem for the Monge-Ampère equation. Viscosity solutions are a powerful tool for analyzing and approximating fully nonlinear elliptic equations. However, we demonstrate that this nonlinear elliptic equation does not satisfy a comparison principle and thus existing convergence frameworks for viscosity solutions are not valid. We introduce an alternative PDE that couples the usual Monge-Ampère equation to a Hamilton-Jacobi equation that restricts the transportation of mass. We propose a new interpretation of the optimal transport problem in terms of viscosity subsolutions of this PDE. Using this reformulation, we develop a framework for proving convergence of a large class of approximation schemes for the optimal transport problem. Examples of existing schemes that fit within this framework are discussed.

Identifier

85065499753 (Scopus)

Publication Title

SIAM Journal on Numerical Analysis

External Full Text Location

https://doi.org/10.1137/18M1201913

ISSN

00361429

First Page

945

Last Page

971

Issue

2

Volume

57

Grant

1751996

Fund Ref

National Science Foundation

This document is currently not available here.

Share

COinS