Mathematics, Computation and Geometry of Data

Volume 2 (2022)

Number 2

A geometric variational framework for computing spherical optimal transportation maps II

Pages: 117 – 149

DOI: https://dx.doi.org/10.4310/MCGD.2022.v2.n2.a2

Authors

Zhou Zhao (State University of New York at Stony Brook)

Na Lei (Dalian University of Technology)

Li Cui (Beijing Normal University)

Kehua Su (Wuhan University)

Xiaoyin Xu (Harvard University)

Feng Luo (Rutgers University)

Xianfeng Gu (State University of New York at Stony Brook)

Shing-Tung Yau (Tsinghua University, Harvard University)

Abstract

Optimal transportation maps play fundamental roles in many engineering and medical fields. The computation of optimal transportation maps can be reduced to solve highly non-linear Monge–Ampère equations.

This work summarizes the geometric variational frameworks for spherical optimal transportation maps, which offers solutions to the Minkowski problem in convex differential geometry, reflector design and refractor design problems in optics. The method is rigorous, robust and efficient. The algorithm can directly generalized to higher dimensions.

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Received 5 July 2022

Published 21 March 2023