Journal of Optimization Theory and Applications

, Volume 111, Issue 2, pp 255–271 | Cite as

Mixed L2-Wasserstein Optimal Mapping Between Prescribed Density Functions

  • J. D. Benamou
  • Y. Brenier


A time-dependent minimization problem for the computation of a mixed L2-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 L2-Wasserstein distance between two densities and the corresponding optimal mapping.

Monge-Kantorovitch mass transfer problem Wasserstein distance least-square distance optimal control conjugate gradient algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benamou, J. D., and Brenier, Y., A Computational Fluid Mechanics Solution to the MongeKantorovich Mass Transfer Problem, Numerische Mathematik, Vol. 84, pp. 375–393, 2000.Google Scholar
  2. 2.
    Douglas, R. J., Decomposition of Weather Forecast Error Using Rearrangements of Functions, Preprint, 1998.Google Scholar
  3. 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. 4.
    Godlewski, E., and Raviart, P. A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer Verlag, New York, NY, 1996.Google Scholar
  5. 5.
    Lions, J. L., Contrôle Optimal de Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, France, 1968.Google Scholar
  6. 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. 7.
    Whitam, G. B., Linear and Nonlinear Waves, John Wiley and Sons, New York, NY, 1999.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • J. D. Benamou
    • 1
  • Y. Brenier
    • 2
  1. 1.Domaine de VoluceauINRIALe ChesnayFrance
  2. 2.Laboratoire d'Analyse NumériqueUniversité Paris 6ParisFrance

Personalised recommendations