Mathematics, Computation and Geometry of Data

Volume 1 (2021)

Number 2

A geometric variational framework for computing optimal transportation maps, I

Pages: 207 – 253



Dongsheng An (State University of New York, Stony Brook, N.Y., U.S.A.)

Na Lei (Dalian University of Technology, Dalian, Liaoning, China)

Li Cui (Beijing Normal University, Beijing, China)

Kehua Su (Wuhan University, Wuhan, Hubei, China)

Xiaoyin Xu (Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts, U.S.A.)

Feng Luo (Rutgers University, Piscataway, New Jersey, U.S.A.)

Xianfeng Gu (State University of New York, Stony Brook, N.Y., U.S.A.)

Shing-Tung Yau (Tsinghua University, Beijing, China; and Harvard University, Cambridge, Massachusetts, U.S.A.)


Optimal transportation (OT) 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. In this work, we summarize the geometric variational framework to solve optimal transportation maps in Euclidean spaces.We generalize the method to solve worst transportation maps and discuss about the symmetry between the optimal and the worst transportation maps. Many algorithms from computational geometry are incorporated into the method to improve the efficiency, the accuracy and the robustness of computing optimal transportation.

Received 5 March 2021

Published 2 August 2022