## Journée transport optimal, équation de Monge-Ampère et applications

This is the first meeting organized by the ANR Project MAGA. The topics covered by this first meeting are quite diverse: application of optimal transport in fluid dynamics and in economy; Monge-Ampère equations and convex geometry; (semi-discrete) numerical methods for Monge-Ampère equations.

**Location:** Institut des Hautes études scientifiques, Bures-sur-Yvette (amphithéâtre Léon Motchane)

**Date:** wednesday december 7, 2016

**Registration is free but mandatory, through this form.**

### Program

*9h20: Welcome*- 9h30:
**François-Xavier Vialard (Univ. Paris-Dauphine),***«From unbalanced optimal transport to the Camassa-Holm equation»* *10h20: Coffee break*- 10h40:
**Daniel Han-Kwan**(École Polytechnique/CNRS),*«Des équations de Vlasov aux équations d’Euler généralisées»* - 11h30:
**Dario Cordero-Erausquin**(Univ. Paris 6),*«Moment measures»* *12h20: Lunch*- 13h50:
**Filippo Santambrogio**(Univ. Paris-Sud),*«Variational and numerical resolution of det(D²u) = f(u)»* - 14h40:
**Roland Hildebrand**(Univ. Grenoble-Alpes/CNRS),*«Canonical barriers on convex cones»* *15h30: Coffee break*- 16h00:
**Guillaume Carlier**(Univ. Paris-Dauphine),*«Matching with optimal transport on one side»*

### Abstracts

*Guillaume Carlier*

**Title:** «Matching with optimal transport on one side»

**Abstract:** We consider a matching problem between a population of consumers and a population of producers, we look for equilibrium prices that is prices for which the distribution of demand and supply coincide. Producers minimize production cost minus price which can be described by means of optimal transport. But on the consumers’ side, the picture is slightly different, indeed a realistic assumption is that consumer maximize their utility under a price constraint. I will prove existence of an equilibrium and, formally, discuss connections with some (nonconvex) optimal transport problems which somehow mix L^1 and L^\infty criteria. This is a joint work in progress with Alfred Galichon and Ivar Ekeland.

*Dario Cordero-Erausquin*

**Title:** «Moment measures»

**Abstract:** To every convex function $\psi$ tending to infinity at infinity we can associate its moment measure, which is the image by the gradient of $\psi$ of the measure with density $e^{-\psi}$. We aim at characterizing all the measures that can be obtained as moment measure of some convex function. This will be done by studying a variational problem that is closely related to the one of optimal transportation theory. This variational problem can be studied using tools from the geometry of log-concave measures.

*Daniel Han-Kwan*

**Title**: Des équations de Vlasov aux équations d’Euler généralisées

**Abstract:** L’exposé portera sur la limite quasineutre pour les équations de Vlasov. Il s’agit d’une limite singulière qui permet de dériver, au moins formellement, les équations d’Euler généralisées à la Brenier. On expliquera les phénomènes d’instabilité qui permettent de comprendre quand la limite formelle est valable ou ne l’est pas.

*Roland Hildebrand*

**Title:** «Canonical barriers on convex cones»

**Abstract:** The Calabi theorem states that for every regular convex cone K in R^n, the Monge-Ampère equation log det F” = 2F/n has a unique convex solution on the interior of K which tends to +infty on the boundary of K. It turns out that this solution is self-concordant and logarithmically homogeneous, and thus is a barrier which can be used for conic optimization. We consider different aspects of this barrier:

- affine spheres as level surfaces
- metrization of the interior of K by the Hessian metric F”
- primal-dual symmetry
- interpretation as a minimal Lagrangian submanifold in a certain para-Kähler space form
- complex-analytic structure on 3-dimensional cones.

*Filippo Santambrogio*

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

**Abstract:** 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

*François-Xavier Vialard*

**Title**: «From unbalanced optimal transport to the Camassa-Holm equation»

**Abstract:** We present an extension of the Wasserstein L2 distance to the space of positive Radon measures as an infimal convolution between the Wasserstein L2 metric and the Fisher-Rao metric. In the work of Brenier, optimal transport has been developed in its study of the incompressible Euler equation. For the Wasserstein-Fisher-Rao metric, the corresponding fluid dynamic equation is known as the Camassa-Holm equation (at least in dimension 1), originally introduced as a geodesic flow on the group of diffeomorphisms. This point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L 2 type of cone metric. As a direct consequence, we describe a new polar factorization on the automorphism group of half-densities which can be seen as a constrained version of Brenier’s theorem. The main application consists in writing the Camassa-Holm equation on S^1 as a particular case of the incompressible Euler equation on a group of homeomorphisms of R^2 that preserve a radial density which has a singularity at 0, the cone point.

### Participants

- Jean-David Benamou, Inria
- Nicolas Brosse, CMAP Ecole Polytechnique
- Guillaume Carlier, Ceremade
- Lénaïc Chizat, DMA-ENS/CEREMADE
- Dario Cordero-Erausquin, IMJ-PRG
- Simone Di Marino, INRIA-Paris Dauphine
- Samer Dweik, ANEDP
- Yanbo Fang, IMJ-PRG
- Thomas Gallouët, Université de Liège
- Saeed Hadikhanloo, Paris Dauphine
- Daniel Han-Kwan, CNRS / École Polytechnique
- Roland Hildebrand, CNRS
- Vasilyev Ioann, Université Paris 7 Diderot
- Gerard Kerkyacharian, LPMA
- Hugo Lavenant, Université Paris-Sud
- Bruno Levy, Inria
- Bertrand Maury, LMO
- Guilherme Mazanti, Université Paris-Sud
- Quentin Mérigot, Université Paris-Sud
- Jocelyn Meyron, GIPSA-lab
- Jean-Marie Mirebeau, CNRS / Université Paris-Sud
- Edouard Oudet, UGA
- Paul Pegon, Université Paris-Sud
- Gabriel Peyré, CNRS et Ecole Normale Supérieure
- Cyril Porée, ANSRR
- Bouslamti Samir, upmc
- Filippo Santambrogio, Université Paris-Sud
- Boris Thibert, Laboratoire Jean Kuntzmann
- François-Xavier, Vialard Université Paris-Dauphine