Mathematical analysis and convergence of an algorithm for optimal transport problem

Keywords :

• convergence et consistance d'un algorithme
• problèmes multiphysiques

In the beginning of the 2000 years, J. D. Benamou and Y. Brenier have proposed a dynamical formulation of the optimal transport problem, corresponding to the  time-space search of a density and a momentum minimizing a transport energy between two densities. They proposed, in order to solve this problem in practice, to deal with it by looking for a saddle point of some Lagrangian by an augmented Lagrangian algorithm. Using  the theory of non-expansive operators, we will study the convergence of this algorithm to a saddle point of the Lagrangian introduced, in the most general feasible conditions, particularly in cases where initial and final densities are canceling on some areas of the transportation domain. The principal difficulty of our study will consist of the proof, in these conditions, of the existence of a saddle point, and especially in the uniqueness of the density-momentum component. Indeed, these conditions imply to have to deal with non-regular optimal transportation maps: that is why an important part of our works will have for object a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space. To finish, we will explore different approaches for introducing physical priors in the dynamical formulation of optimal transport, based on penalization of the transportation domain or of the velocity field.

Directors:

• Mr Nicolas Papadakis (Chargé de recherche - CNRS Université de Bordeaux )
• Mr Emmanuel Maitre (Professeur - Grenoble INP )

Raporteurs:

• Mr Filipo Santambrogio (Professeur - Université Paris-Sud )
• Mr Stéphane Descombes

Examinators:

• Mr Didier Bresch (Directeur de Recherche - CNRS, Université de Savoie-Mont-Blanc )
• Mr Vincent Duval (Chargé de recherche - INRIA, Université Paris-Dauphine )