Unbalanced optimal transport: models, numerical methods and applications

The optimal transport (OT) problem is often described as that of finding the most efficient way of moving a pile of dirt from one configuration to another. Once stated formally, OT provides extremely useful tools for comparing, interpolating and processing objects such as probability measures, histograms or densities. In many applications, there is a need to relax the equality of mass constraint between the two measures being transported.

In this talk I will present how OT can be extended in a meaningful way beyond the classical "balanced" setting of probability distributions, while preserving its key geometrical and numerical properties. I will first review simple aspects of the classical theory of OT and introduce models of unbalanced OT, including a generalization of the p-Wasserstein distances. I will then present numerical methods to solve unbalanced OT that extend Sinkhorn's algorithm, and finally review some applications to histogram manipulation and the study of reaction-diffusion PDEs.

This talk is based on joint work with G. Peyré, B. Schmitzer and F-X. Vialard.