Schrödinger Problem and Mean-field PDE Systems: Computational and Theoretical Advances
Problème de Schrödinger et systèmes d’EDP à champ moyen: avancées numériques et théoriques
15 – 19 November, 2021
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Scientific Committee
Comité scientifique Yann Brenier (CNRS, Université Côte d’Azur) |
Organizing Committee
Comité d’organisation Julio Backhoff (University of Twente & University of Wien) |
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Monge’s question of how to optimally move sand-pile stands at the origin of the modern theory of optimal transport. The development of this theory over the last decades led to impressive advances in analysis and geometry, reaching out to applied fields such as economics and machine learning. One and a half century after Monge, Schrödinger asked: “What is the most likely evolution of a cloud of random particles conditionally on the observation of their initial and final configurations? « This question, going under the name of Schödinger Problem, initiated a research line that has growne normously in the last years since it was understood that Schrödinger’s and Monge’s questions are the same when the fluctuations of the random particles are very small. This discovery offered a natural ground for the theories of optimal transport and large deviations to meet and thrive together. At the same time, it also inspired the use of entropic regularization techniques in machine learning and numerics for PDEs, achieving major computational advantages. Asking Schrödinger’s question for strategic particles is the gateway to connect the Schrödinger problem and large deviations with the theory of mean field stochastic control and planning mean field games. This brings to light new mathematical questions. Can entropic regularization speed up the computation of Nash equilibria? Are there new functional inequalities that capture the ergodic behaviour of mean field games? Bringing together researchers from the areas of probability, optimal transport, statistical machine learning and mean field games, this conference aims to strengthen the interplay between the Schrödinger problem and mean field systems, and facilitate the birth of novel methodologies and results.
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La théorie du transport optimal a pour origine une question de Monge concernant le déplacement à moindre coût de monticules de terre. Ces dernières décennies ont connu un développement spectaculaire de cette théorie qui a contribué à des progrès importants en analyse et en géométrie, ainsi que dans des domaines appliqués comme l’ ́economie et l’apprentissage automatique. Un siècle et demi après Monge, Schrödinger posait la question : « Quelle est l’évolution la plus probable d’un nuage de particules aléatoires sachant que ses configurations initiale et finale sont fixées ?” Cette question qui n’est autre que l’énoncé de ce qui est appelé de nos jours le problème de Schrödinger, est à l’origine de nombreux travaux de recherche récents, surtout depuis qu’il a été compris que le problème de Monge s’obtient comme limite du problème de Schrödinger lorsque les fluctuations aléatoires s’atténuent. Cette découverte qui met en lumière un lien naturel entre le transport optimal et les grandes déviations, a donné un cadre analytique aux techniques de régularisation entropique utilisées en apprentissage automatique et en analyse numérique de certaines EDP, améliorant quelques algorithmes de façon spectaculaire. Se poser la question de Schrödinger pour des particules stratégique s‘établit une passerelle entre les grandes déviations associées au problème de Schrödinger et les théories du contrôle stochastique en champ moyen et des jeux à champ moyen sous contraintes marginales. De nouvelles questions surgissent naturellement. La régularisation en tropique accélère-t’elle le calcul numérique des équilibres de Nash ? Existe-t’il de nouvelles inégalités fonctionnelles qui encoderaient le comportement des jeux à champ moyen? En rassemblant des mathématicien.ne.s issu.e.s des probabilités, du transport optimal, de l’apprentissage statistique et des jeux à champ moyen, cette conférence a pour but de consolider notre compréhension des liens existant entre le problème de Schrödinger et les systèmes à champ moyen, d’offrir l’opportunité de nouvelles découvertes et de favoriser l’émergence de nouvelles méthodes.
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Beatrice Acciaio (ETH Zürich) Generative adversarial learning with adapted distances
Aymeric Baradat (CNRS Institut Camille Jordan) From the branching Brownian motion to regularized unbalanced optimal transport
Charles Bertucci (CNRS CMAP Ecole Polytechnique) Master equation for the mean field planning problem in finite state space
Elisa Calzola (Sapienza University of Rome) A second order Lagrange-Galerkin scheme for Fokker-Planck-Kolmogorov equations and applications to MFGs
Fabio Camilli (University of Rome La Sapienza) A Mean Field Games approach to cluster analysis
Elisabetta Carlini (University of Rome La Sapienza) A Semi-Lagrangian Scheme for Hamilton-Jacobi-Bellman Equations with Oblique Boundary Conditions
Alekos Cecchin (CMAP École polytechnique) Finite state N-agent and mean field control problems
Annalisa Cesaroni (University of Padova) Multi-agent optimal control and mean feld limits with density constraints
Gauthier Clerc (Université Lyon 1) An entropic interpolation proof of the Li-Yau inequality
Marco Cuturi (Google Brain and ENSAE) TBA
Valentin De Bortoli (University of Oxford) Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling
Alex Delalande (Université Paris-Saclay) Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport
François Delarue (Université Nice-Sophia Antipolis) Weak solutions to the master equation of a potential mean field game
Goncalo Dos Reis (University of Edinburgh) Simulation of McKean Vlasov SDEs: the super measure case
Max Fathi (Université de Paris) Stability of the spectral gap under a curvature-dimension condition
Augusto Gerolin (VU Amsterdam) A Duality Approach for Entropic-Regularized Optimal Transport and convergence of the Sinkhorn Algorithm
Promit Ghosal (MIT) Geometry and large deviation of entropic optimal transport
Nathael Gozlan (Univeristé Paris-Descartes) Transport-Entropy formulations of Inverse Santaló inequalities
Giacomo Greco (Eindhoven University of Technology) Entropic turnpike estimates for the kinetic Schrödinger problem
Arnaud Guillin (Université Clermont-Auvergne) Uniform in time propagation of chaos in non convex or singular cases
Martin Huesmann (University of Vienna) Bipartite matching, invariance, and regularity of optimal transport
Nicolas Juillet (Université de Haute-Alsace) A martingale exactly fitting an infinite family of given marginals
Young-Heon Kim (University of British Columbia) The Stefan problem and free targets of optimal Brownian martingale transport
Christian Léonard (Université Paris Nanterre) Feynman-Kac formula under a finite entropy condition
Giulia Luise (University College London) Statistical Insights on Implicit Generative Models with Sinkhorn Divergence
Gonzalo Mena (University of Oxford) Some Statistical and Computational Benefits of Entropic Optimal Transport
Leonard Monsaingeon (University of Lisbon) The dynamical Schrödinger problem on metric spaces: Gamma-convergence and convexity
Luca Nenna (Université Paris-Saclay) Grand Canonical optimal Transport
Carlo Orrieri (University of Trento) Lagrangian, Eulerian and Kantorovich formulations of controlled interacting particles
Soumik Pal (University of Washington, Seattle) A permanental representation of discrete Schrödinger bridges
Gabriel Peyré (CNRS, ENS-Paris) Scaling Optimal Transport for High dimensional Learning
Dylan Possamaï (ETH Zürich) Non-asymptotic convergence rates for mean-field games: weak formulation and McKean–Vlasov BSDEs.
Francisco José Silva Alvarez (Université de Limoges) Approximation of determistic mean field games with control-affine dynamics
Luca Tamanini (Université Paris-Dauphine PSL) Classical and mean field Schrödinger problems: Long time behaviour
Oliver Tse (Eindhoven University of Technology) Adjoint-based mean-field optimal control
Axel Turn
quist (New Jersey Institute of Technology) Convergent Numerical Schemes for Optimal Transport with Applications on the Sphere and Beyond
Johannes Wiesel (Columbia University) Entropic optimal transport: convergence of potentials
Jean-Claude Zambrini (University of Lisboa) Solution of Schrödinger’s problem via Madelung transform and separation
of variables in Hamilton-Jacobi-Bellman equation
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 PROJET EFI
Entropy, Flows, Inequalities |
 MOKAPLAN
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