EYAWKAJKOS

Everything You Always Wanted to Know About the JKO Scheme

EYAWKAJKOS is a research project which has been awarded an ERC Advanced Grant. Funded from the ERC AdG 2021 call, lead by Université Claude Bernard Lyon 1 in collaboration with CNRS, it started on September 2023 for a duration of 5 years.

Main research objectives

From the proposal summary
The project deals with the so-called Jordan-Kinderlehrer-Otto scheme, a time-discretization procedure consisting in a sequence of iterated optimization problems involving the Wasserstein distance W₂ between probability measures. This scheme allows to approximate the solutions of a wide class of PDEs (including many diffusion equations with possible aggregation effects) which have a variational structure w.r.t. the distance W₂ 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 naturally provides a time-discretization and, if coupled with efficient computational techniques for optimal transport problems, can be used for numerics. This project will cover both equations which are well-studied (Fokker-Planck, for instance) and less classical ones (higher-order equations, crowd motion, cross-diffusion, sliced Wasserstein flow\dots). For the most classical ones, we will mainly consider estimates and properties which are known for solutions of the continuous-in-time PDEs and try to prove sharp and equivalent analogues in the discrete setting: some of these results ($L^p$, Sobolev, BV\dots) have already been proven in the simplest cases ; the results in the classical case will provide techniques to be applied to the other equations, allowing to prove existence of solutions and to study their qualitative properties. Moreover, some estimates proven on each step of the JKO scheme can provide useful information 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-time PDEs, with the effort to produce efficient numerical strategies, and with the attention to the modeling of other phenomena which could take advantage of these techniques.

See here for the full scientific proposal.

Current members of the project

Principal investigator:
Filippo Santambrogio, professor at ICJ

Permanent participants:
Aymeric Baradat, researcher at ICJ
Nicolas Bonneel, researcher at LIRIS
Ivan Gentil, professor at ICJ

Young researchers:
Thibault Caillet, PhD student, funded by UCBL
Fanch Coudreuse, PhD student, funde by EYAWKAJKOS
Noemi David, post-doc, funded by Labex Milyon
Anastasiia Hraivoronska, post-doc, funded by EYAWKAJKOS

Open positions

A call for a 3-years postdoc position is open till January 28th. See here for details.

Events and seminars

Gradient Flows Face-to-Face 2023, Lyon
EYAWKAJKOS funded and organized the third edition of the workshops Gradient Flows Face-to-Face (first edition: Roma 2021, second edition: L'Aquila 2022).
This small workshop took place in Lyon in September 2023 and was the launching event of the project. More details on the webpage of the workshop.

EYAWKAJKOS working group
We organize a working group on gradient flows and related topics, with both reading seminars (i.e. somebody presents a paper of interest for the group that he/she has previously studied) or research seminars by our guests. Talks are usually on Wednesday on a very irregular basis but approximately twice per month.
The program of this working group is available on its webpage.

Optimal transport and applications 2024, Pisa
EYAWKAJKOS will co-fund and co-organize the next edition of the workshops Optimal transport and applications>/i> which take place in Pisa every two years (last editions: 2018, 2022). This event is currently being prepared by the organizing committee.