Everything You Always Wanted to Know About the JKO Scheme

The project deals with the so-called Jordan-Kinderlehrer-Otto scheme, a time-discretization procedure consisting in a sequence ofiterated optimization problems involving the Wasserstein distance W_2 between probability measures. This scheme allows toapproximate the solutions of a wide class of PDEs (including many diffusion equations with possible aggregation effects) which havea variational structure w.r.t. the distance W_2 but not w.r.t. Hilbertian distances. It has been used both for theoretical purposes(proving existence of solutions for new equations and studying their properties) and for numerical applications. Indeed, it naturallyprovides a time-discretization and, if coupled with efficient computational techniques for optimal transport problems, can be used fornumerics.This project will cover both equations which are well-studied (Fokker-Planck, for instance) and less classical ones (higher-orderequations, crowd motion, cross-diffusion, sliced Wasserstein flow...). For the most classical ones, we will systematically considerestimates and properties which are known for solutions of the continuous-in-time PDEs and try to prove sharp and equivalentanalogues in the discrete setting: some of these results (L^p, Sobolev, BV...) have already been proven in the simplest cases ; theresults in the classical case will provide techniques to be applied to the other equations, allowing to prove existence of solutions andto study their qualitative properties. Moreover, some estimates proven on each step of the JKO scheme can provide usefulinformation for the numerical schemes, reducing the computational complexity or improving the quality of the convergence.During the project, the study of the JKO scheme will be of course coupled with a deep study of the corresponding continuous-in-timePDEs, with the effort to produce efficient numerical strategies, and with the attention to the modeling of other phenomena whichcould take advantage of this techniques.