A Numerical Method to Solve Multi-Marginal Optimal Transport Problems with Coulomb Cost

Benamou, Jean-David; Carlier, Guillaume; Nenna, Luca

Benamou, Jean-David; Carlier, Guillaume; Nenna, Luca (2017), A Numerical Method to Solve Multi-Marginal Optimal Transport Problems with Coulomb Cost, in Glowinski R., Osher S., Yin W., Splitting Methods in Communication, Imaging, Science, and Engineering, Springer : Berlin Heidelberg, p. 577-601. https://doi.org/10.1007/978-3-319-41589-5_17

Type

Chapitre d'ouvrage

External document link

https://hal.inria.fr/hal-01148954

Date

2017

Book title

Splitting Methods in Communication, Imaging, Science, and Engineering

Book author

Glowinski R., Osher S., Yin W.

Publisher

Springer

Published in

Berlin Heidelberg

ISBN

978-3-319-41587-1

Pages

577-601

Abstract (EN)

In this chapter, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This is related to the strong interaction limit of Density Functional Theory. The first idea is to introduce an entropic regularization of the Kantorovich formulation of the Optimal Transport problem. The regularized problem then corresponds to the projection of a vector on the intersection of the constraints with respect to the Kullback-Leibler distance. Iterative Bregman projections on each marginal constraint are explicit which enables us to approximate the optimal transport plan. We validate the numerical method against analytical test cases.

Subjects / Keywords

coulomb cost