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

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

2008-10-22

arxiv.org/abs/0810.4153v1

Mathematics

Optimization and Control

Scientific paper

A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{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.

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