Optimal Transport Using Cost Functions with Preferential Direction with Applications to Optics Inverse Problems

TL;DR

This paper develops a regularity theory for a class of optimal transport problems on the sphere with cost functions having a preferential direction, motivated by optics inverse problems, and proves existence and uniqueness results under certain conditions.

Contribution

It introduces a new class of cost functions with preferential direction for optimal transport on the sphere and establishes regularity and existence results relevant to optics applications.

Findings

Cost-sectional curvature condition does not hold for the reflector problem.

Existence of a unique solution is proven when source and target intensities are close.

Regularity theory is constructed under specific hypotheses on the cost functions.

Abstract

We focus on Optimal Transport PDE on the unit sphere $\mathbb{S}^2$ with a particular type of cost function $c(x,y) = F(x \cdot y, x \cdot \hat{e}, y \cdot \hat{e})$ which we call cost functions with preferential direction, where $\hat{e} \in \mathbb{S}^2$. This type of cost function arises in an optics application which we call the point-to-point reflector problem. We define basic hypotheses on the cost functions with preferential direction that will allow for the Ma-Trudinger-Wang (MTW) conditions to hold and construct a regularity theory for such cost functions. For the point-to-point reflector problem, we show that the negative cost-sectional curvature condition does not hold. We will nevertheless prove the existence of a unique solution of the point-to-point reflector problem, up to a constant, provided that the source and target intensity are "close enough".