Skip to main content
SpringerLink
Log in

Mixed L 2-Wasserstein Optimal Mapping Between Prescribed Density Functions

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

A time-dependent minimization problem for the computation of a mixed L 2-Wasserstein distance between two prescribed density functions is introduced in the spirit of Ref. 1 for the classical Wasserstein distance. The optimum of the cost function corresponds to an optimal mapping between prescribed initial and final densities. We enforce the final density conditions through a penalization term added to our cost function. A conjugate gradient method is used to solve this relaxed problem. We obtain an algorithm which computes an interpolated L 2-Wasserstein distance between two densities and the corresponding optimal mapping.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benamou, J. D., and Brenier, Y., A Computational Fluid Mechanics Solution to the Monge–Kantorovich Mass Transfer Problem, Numerische Mathematik, Vol. 84, pp. 375–393, 2000.

    Google Scholar 

  2. Douglas, R. J., Decomposition of Weather Forecast Error Using Rearrangements of Functions, Preprint, 1998.

  3. Glowinski, R., Ensuring Well-Posedness by Analogy: Stokes Problem and Boundary Control for the Wave Equation, Journal of Computational Physics, Vol. 103, pp. 189–221, 1992.

    Google Scholar 

  4. Godlewski, E., and Raviart, P. A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer Verlag, New York, NY, 1996.

    Google Scholar 

  5. Lions, J. L., Contrôle Optimal de Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, France, 1968.

    Google Scholar 

  6. Saad, Y., and Schultz, M. H., Conjugate Gradient-Like Algorithms for Solving Nonsymmetric Linear Systems, Mathematics of Computation, Vol. 44, pp. 417–424, 1985.

    Google Scholar 

  7. Whitam, G. B., Linear and Nonlinear Waves, John Wiley and Sons, New York, NY, 1999.

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    J. D. Benamou (Directeur de Recherche)

  2. Laboratoire d'Analyse Numérique, Université Paris 6, Paris, France

    Y. Brenier (Professeur)

Authors
  1. J. D. Benamou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Brenier
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benamou, J.D., Brenier, Y. Mixed L 2-Wasserstein Optimal Mapping Between Prescribed Density Functions. Journal of Optimization Theory and Applications 111, 255–271 (2001). https://doi.org/10.1023/A:1011926116573

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011926116573

Search

Navigation