Année 2021-2022

Congestion penalization as an unbalanced transport problem

by Clément Sarrazin

3L15 (Université Paris Saclay)


Université Paris Saclay


I will present a quantifiable way to identify high concentration of mass for a singular measure, for which the more "standard" definitions in term of a penalization on its (non-existing) density w.r.t. the Lebesgue measure cannot be properly applied. This new quantity can be defined as a cost of transport (unbalanced in general) between the singular measure and the better-concentrated Lebesgue measure, with an asymmetrical penalization of the marginals reflecting what type of behavior one wants to avoid.

This approach based on optimal transport is very well adapted, among others, to the Lagrangian discretization of congestion-minimizing motions for probability measures, such as gradient flows or some mean field games, when the minimized energy is ill-defined for discrete measures.