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

Santambrogio, Filippo; Carlier, Guillaume; Galichon, Alfred (2010), From Knothe's transport to Brenier's map and a continuation method for optimal transport, SIAM Journal on Mathematical Analysis, 41, 6, p. 2554-2576. http://dx.doi.org/10.1137/080740647

Type
Article accepté pour publication ou publié
Date
2010
Journal name
SIAM Journal on Mathematical Analysis
Volume
41
Number
6
Publisher
SIAM - Society for Industrial and Applied Mathematics
Pages
2554-2576
Publication identifier
Metadata
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Author(s)
Santambrogio, Filippo
Carlier, Guillaume
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Galichon, Alfred
Abstract (EN)
A simple procedure to map two probability measures in R^d is the so-called Knothe-Rosenblatt rearrangement, which consists in rearranging monotonically the marginal distributions of the first 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.
Subjects / Keywords
continuation methods; optimal transport; rearrangement of vector-valued maps; Knothe-Rosenblatt transport