Pointwise Convergence Analysis for Approximations of Optimal Transport Problems with a Target Measure that Has Unbounded Support

TL;DR

This paper analyzes the convergence of optimal transport maps and potentials when approximating a target measure with unbounded support, providing quantitative rates and justifying numerical solvers.

Contribution

It offers the first pointwise convergence rates for optimal transport approximations with unbounded target measures, including radially symmetric and non-symmetric cases.

Findings

Derived non-asymptotic pointwise convergence rates.

Validated cutoff approximation for unbounded target measures.

Supported the use of numerical Monge-Ampère solvers.

Abstract

We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target measure is approximated, with special attention given to a cutoff approximation in which we parametrize the approximation by a ``cutoff" radius $R$ for the target measure. We study both the convergence of the mapping and potential functions for the forward and inverse problem in many cases such as 1) the radially symmetric case with the cutoff approximation for general cost functions and 2) the non-radially symmetric case with the squared distance cost function. We derive quantitative non-asymptotic pointwise convergence rates in special cases, building on the $L^2$ convergence rates established by Delalande and M\`{e}rigot. These results can be used, for instance, to justify the use of certain types of numerical Monge-Amp\`{e}re equation solvers in computationally solving the problem.