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

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".