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Splitting Methods in Communication, Imaging, Science, and Engineering pp 577–601Cite as

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

Part of the Scientific Computation book series (SCIENTCOMP)

Abstract

In this chapter, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This problem 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.

Keywords

  • Density Functional Theory
  • Lithium Atom
  • Optimal Transport
  • Transport Plan
  • Alternate Projection

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Author information

Authors and Affiliations

  1. INRIA, MOKAPLAN, Domaine de Voluceau Le Chesnay, France

    Jean-David Benamou, Guillaume Carlier & Luca Nenna

  2. CEREMADE, Université Paris Dauphine, Paris, France

    Jean-David Benamou, Guillaume Carlier & Luca Nenna

Authors
  1. Jean-David Benamou
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  2. Guillaume Carlier
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  3. Luca Nenna
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Corresponding author

Correspondence to Jean-David Benamou .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, California, USA

    Stanley J. Osher

  3. Department of Mathematics, UCLA, Los Angeles, California, USA

    Wotao Yin

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Benamou, JD., Carlier, G., Nenna, L. (2016). A Numerical Method to Solve Multi-Marginal Optimal Transport Problems with Coulomb Cost. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_17

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