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

2024/07/09 by Axel G. R. Turnquist, Turnquist, Axel G. R.
Mathematics · #35J15 #35J60 #35J96 #35Q60 #35R01 #49Q22 #58J05 #78A05 #Advanced Optimization Algorithms Research #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2407.07256

openalex publication_date 2024/07/09 · openalex created_date 2024/07/13 · openalex updated_date 2026/07/28

Abstract

We focus on Optimal Transport PDE on the unit sphere \mathbbS² with a particular type of cost function c(x,y) = F(x ⋅ y, x ⋅ e, y ⋅ e) which we call cost functions with preferential direction, where e ∈ \mathbbS². 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".

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