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

G. Carlier; A. Galichon; F. Santambrogio

doi:10.1137/080740647

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 journal › Article › peer-review

Abstract

A simple procedure to map two probability measures in ℝ^d 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 language	English (US)
Pages (from-to)	2554-2576
Number of pages	23
Journal	SIAM Journal on Mathematical Analysis
Volume	41
Issue number	6
DOIs	https://doi.org/10.1137/080740647
State	Published - 2009

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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10.1137/080740647

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AU - Galichon, A.

AU - Santambrogio, F.

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AB - 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.

KW - Continuation methods

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KW - Optimal transport

KW - Rearrangement of vector-valued maps

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

Abstract

Keywords

ASJC Scopus subject areas

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