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Foundations of Computational Mathematics

, Volume 19, Issue 5, pp 1113–1143 | Cite as

Second-Order Models for Optimal Transport and Cubic Splines on the Wasserstein Space

  • Jean-David BenamouEmail author
  • Thomas O. Gallouët
  • François-Xavier Vialard
Article

Abstract

On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose a simpler approach based on the relaxation of the variational problem on the path space. We explore two different numerical approaches, one based on multimarginal optimal transport and entropic regularization and the other based on semi-discrete optimal transport.

Keywords

Multimarginal optimal transportation Splines Wasserstein geodesics 

Mathematics Subject Classification

49M99 65D99 

Notes

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Copyright information

© SFoCM 2019

Authors and Affiliations

  • Jean-David Benamou
    • 1
    • 2
    Email author
  • Thomas O. Gallouët
    • 1
    • 2
  • François-Xavier Vialard
    • 3
  1. 1.Project Team MokaplanINRIAParisFrance
  2. 2.CeremadeUniversité Paris-Dauphine, PSL Research UniversityParisFrance
  3. 3.LIGM, UPEMUniversity Paris-EstMarne-la-ValléeFrance

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