Convergence Rates for Discretized Monge-Ampere Equations and Quantitative Stability of Optimal Transport
Journal article, 2020

In recent works-both experimental and theoretical-it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge-Ampere equations. This yields H-1-converge rates, in terms of the corresponding spatial resolution h, of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on new quantitative stability results for optimal transport maps, shown using complex geometry.

Monge-Ampere equations

Numerical analysis

Optimal transport

Complex differential geometry

Author

Robert Berman

University of Gothenburg

Chalmers, Mathematical Sciences, Algebra and geometry

Foundations of Computational Mathematics

1615-3375 (ISSN) 1615-3383 (eISSN)

Vol. In Press

Subject Categories

Computational Mathematics

Other Physics Topics

Mathematical Analysis

DOI

10.1007/s10208-020-09480-x

More information

Latest update

1/8/2021 4