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

G. Carlier, A. Galichon, F. Santambrogio

    Research output: Contribution to journalArticle

    Abstract

    A simple procedure to map two probability measures in ℝd is the so-called Knothe- Rosenblatt rearrangement, which consists of 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.

    Original languageEnglish (US)
    Pages (from-to)2554-2576
    Number of pages23
    JournalSIAM Journal on Mathematical Analysis
    Volume41
    Issue number6
    DOIs
    StatePublished - Jan 1 2009

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    Keywords

    • Continuation methods
    • Knothe-Rosenblatt transport
    • Optimal transport
    • Rearrangement of vector-valued maps

    ASJC Scopus subject areas

    • Analysis
    • Computational Mathematics
    • Applied Mathematics

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