EYAWKAJKOS – Everything You Always Wanted to Know About the JKO Scheme

Le projet EYAWKAJKOS porte sur le schéma dit de Jordan-Kinderlehrer-Otto (JKO), une procédure de discrétisation temporelle consistant en une séquence de problèmes d'optimisation itérés impliquant la distance de Wasserstein W_2 entre des mesures de probabilité.

Filippo Santambrogio étudiera l'existence, la régularité et le comportement asymptotique des équations différentielles partielles (EDP) correspondantes. Le projet inclut aussi des estimations uniformes ou des résultats de convergence sur les solutions du schéma JKO, des approximations numériques exploitant le schéma JKO pour des équations bien connues (Fokker-Planck, milieu poreux...) et d'autres moins classiques (de la dynamique des populations, de l'analyse des données...).

Key information / Informations

•    Funding programme / Programme de financement: Horizon Europe - ERC Advanced Grant - 2021
•   Coordinator / Coordinateur: Université Claude Bernard Lyon 1 (UCBL), Prof. Filippo Santambrogio (ICJ)
•   Partners’ list / Liste des partenaires :

o Centre National de la Recherche Scientifique (CNRS)

•   Budget : 2 182 250 €
•   Grant / Subvention : 2 182 250 €
•   Début – Fin : 2023-2028

Context / Contexte

The project EYAWKAJKOS 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 W_2 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 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 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...).

Objectives / Objectifs

The project EYAWKAJKOS will i) systematically consider estimates and properties of the most classical equations which are known for solutions of the continuous-in-time PDEs and try to prove sharp and equivalent analogues in the discrete setting and ii) provide techniques to be applied to the other equations, allowing to prove existence of solutions and to study their qualitative properties.

Expected impact and results / Impact et résultats attendus

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 EYAWKAJKOS, 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 modelling of other phenomena which could take advantage of this techniques.

LIP’s contribution / Rôle de LIP

LIP a accompagné le porteur de projet dans toute la phase de montage, coordination du montage administratif et financier du projet, conseil à la rédaction. LIP accompagne l’Université Lyon 1 pour le suivi administratif et financier du projet.