Journée transport optimal, équation de Monge-Ampère et applications

Variational and numerical resolution of det(D²u) = f(u)

by Prof. Filippo SANTAMBROGIO (Université Paris-Sud)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

For a given measure \mu on the Euclidean space, we can look for a convex function u such that the image of the density f(u) through Du is \mu (here f is a given function from R to R_+, typically decreasing enough). In the case where f(t)=e^{-t} we have the moment measures problem, but for negative powers we have interesting problems, linked to other questions in convex and affine geometry. In the case where \mu is uniform on a convex set K in the plane and f(t)=t^{-4}, the convex function u can be used to construct an affine hemi-sphere based on the polar convex set K*, for instance. Also, in the case where \mu is uniform on a set, the problem is equivalent to finding a suitable solution of Det(D^2u)=f(u). In the talk, I will briefly explain how to cast these problems as JKO-like optimization problems involving optimal transport, and explain a first idea of how to use the semidiscrete numerical methods which have been used for steps of the JKO scheme to get an approximation of the solutions of these problems. Then, I will explain how to improve this approach in a way which better fits the problem, thus obtaining a true discretization of this moment measure problem, where the measure \mu has simply been replaced by a finitely supported approximation of it, which is no more specifically linked to optimal transport. For this discretization, I will present numerical results and proofs of convergence, coming from an ongoing work in collaboration with B. Klartag and Q. Mérigot