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

Guillaume Carlier (1) , Alfred Galichon (2) , Filippo Santambrogio (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 ECON - Département d'économie (Sciences Po)
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Domains: Humanities and Social Sciences; Economics and Finance

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, 416, pp.2554-2576. ⟨hal-01023796⟩