Calculus of Variations and Geometric Measure Theory

G. Carlier - A. Galichon - F. Santambrogio

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

created by santambro on 24 Oct 2008
modified on 04 Nov 2009


Accepted Paper

Inserted: 24 oct 2008
Last Updated: 4 nov 2009

Journal: SIAM J. Math. An.
Year: 2009

attention : this new - accepted - version includes the corrction of an error found out by T. Mikami, don't trust the previous one !


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.

Keywords: Optimal transport, Knothe-Rosenblatt transport, rearrangement of vector-valued maps, continuation methods