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From Knothe's transport to Brenier's map and a continuation method for optimal transport

Author

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  • Guillaume Carlier

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

  • Alfred Galichon

    (ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique)

  • Filippo Santambrogio

    (LM-Orsay - Laboratoire de Mathématiques d'Orsay - UP11 - Université Paris-Sud - Paris 11 - CNRS - Centre National de la Recherche Scientifique)

Abstract

A simple procedure to map two probability measures in Rd is the so-called Knothe-Rosenblatt rearrangement, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.

Suggested Citation

  • Guillaume Carlier & Alfred Galichon & Filippo Santambrogio, 2010. "From Knothe's transport to Brenier's map and a continuation method for optimal transport," Post-Print hal-01023796, HAL.
    • Handle: RePEc:hal:journl:hal-01023796
    Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-01023796
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    Cited by:

    1. Dmitry Arkhangelsky, 2019. "Dealing with a Technological Bias: The Difference-in-Difference Approach," Working Papers wp2019_1903, CEMFI.
    2. Alfred Galichon & Arthur Charpentier & Marc Henry, 2012. "Local Utility and Risk Aversion," Post-Print hal-03569250, HAL.

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