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

Guillaume Carlier (1) , Alfred Galichon (2) , Filippo Santambrogio (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 X-DEP-ECO - Département d'Économie de l'École Polytechnique

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.

Domaines: Optimisation et contrôle [math.OC]

Guillaume Carlier, Alfred Galichon, Filippo Santambrogio. From Knothe's transport to Brenier's map and a continuation method for optimal transport. SIAM Journal on Mathematical Analysis, 2010, 41 (6), pp.2554-2576. ⟨hal-00360945⟩