Unidimensional and Evolution Method for Optimal Transportation

In dimension one, optimal transportation is rather straightforward. The easiness with which a solution can be obtained in that setting has recently been used to tackle more general situations, each time thanks to the same method [4, 18, 45]. First, disintegrate your problem to go back to the unidimensional case, and apply the available 1d methods to get a rst result; then, improve it gradually using some evolution process. This dissertation explores that direction more thoroughly. Looking back at two problems only partially solved this way, I show how this viewpoint in fact allows to go even further. The rst of these two problems concerns the computation of Yann Brenier’s optimal map. Guillaume Carlier, Alfred Galichon, and Filippo Santambrogio [18] found a new way to obtain it, thanks to an di erential equation for which an initial condition is given by the Knothe–Rosenblatt rearrangement. (The latter is precisely de ned by a series of unidimensional transformations.) However, they only dealt with discrete target measures; I generalize their approach to a continuous setting [10]. By di erentiation, the Monge–Ampère equation readily gives a pde satis ed by the Kantorovich potential; but to get a proper initial condition, it is necessary to use the Nash– Moser version of the implicit function theorem. The basics of optimal transport are recalled in chapter 1, and the Nash–Moser theory is exposed in chapter 2. My results are presented in chapter 3, and numerical experiments in chapter 4. The last chapter deals with the idt algorithm, devised by François Pitié, Anil C. Kokaram, and Rozenn Dahyot [45]. It builds a transport map that seems close enough to the optimal map for most applications [46]. A complete mathematical understanding of the procedure is, however, still lacking. An interpretation as a gradient ow in the space of probability measures is proposed, with the sliced Wasserstein distance as the functional. I also prove the equivalence between the sliced and usual Wasserstein distances.