# 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 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\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.*

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