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

Guillaume Carlier (carlier@ceremade.dauphine.fr), Alfred Galichon and Filippo Santambrogio
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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
Filippo Santambrogio: LMO - Laboratoire de Mathématiques d'Orsay - UP11 - Université Paris-Sud - Paris 11 - CNRS - Centre National de la Recherche Scientifique

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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.

Date: 2010
Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-01023796v1
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Citations: View citations in EconPapers (2)

Published in SIAM Journal on Mathematical Analysis, 2010, 416, pp.2554-2576

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