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

  • Jean-David Benamou
  • Guillaume Carlier
  • Luca Nenna
Part of the Scientific Computation book series (SCIENTCOMP)


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.


Density Functional Theory Lithium Atom Optimal Transport Transport Plan Alternate Projection 


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Jean-David Benamou
    • 1
    • 2
  • Guillaume Carlier
    • 1
    • 2
  • Luca Nenna
    • 1
    • 2
  1. 1.INRIAMOKAPLANDomaine de Voluceau Le ChesnayFrance
  2. 2.CEREMADEUniversité Paris DauphineParisFrance

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