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

G. Carlier, Alfred 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 language English (US) 2554-2576 23 SIAM Journal on Mathematical Analysis 41 6 https://doi.org/10.1137/080740647 Published - 2009

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Mass transportation
Optimal Transport
Continuation Method
Transportation Problem
Marginal Distribution
Conditional Distribution
Rearrangement
Probability Measure
Costs

Keywords

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

ASJC Scopus subject areas

• Analysis
• Applied Mathematics
• Computational Mathematics

Cite this

From Knothe's transport to Brenier's map and a continuation method for optimal transport. / Carlier, G.; Galichon, Alfred; Santambrogio, F.

In: SIAM Journal on Mathematical Analysis, Vol. 41, No. 6, 2009, p. 2554-2576.

Research output: Contribution to journalArticle

Carlier, G. ; Galichon, Alfred ; Santambrogio, F. / From Knothe's transport to Brenier's map and a continuation method for optimal transport. In: SIAM Journal on Mathematical Analysis. 2009 ; Vol. 41, No. 6. pp. 2554-2576.
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