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, funded by EYAWKAJKOS
    Noemi David, post-doc, funded by Labex MILyon
    Anastasiia Hraivoronska, post-doc, funded by EYAWKAJKOS
    Anatole Gallouet, post-doc, funded by EYAWKAJKOS and by PEPR EDP-IA
    Kexin Lin, master intern, soon starting a PhD funded by EYAWKAJKOS
    Sofiane Cherf, master intern, soon starting a PhD funded by ENS Lyon
  • 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.
    A fourth edition will take place in Raitenhaslach in Sept 2024.

    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). See the webpage of the event for more details.