Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprises, among the others, equations involving both standard and Caputo time-derivative, and diffusion operators as the classic and fractional Laplacian, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.

Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems, whose basic concept is to embed physical laws to constrain/inform neural networks, with the need of less data for training a reliable model. In this presentation, we introduce physics-informed neural networks (PINNs) and we explore the resolution of the Stefan problem using this method.

Game Theory is a discipline that we hear more and more about in different areas. This survey aims to study two concepts in this discipline, generalized Nash equilibrium and unawareness in static games. Generalized Nash equilibrium problems model situations in which each player's strategy space depends on the other agents' choices. While unawareness refers to models that represent the situation in which players don't have full knowledge of the game, leading to the emergence of subjective games.

Une épidémie est définie par la propagation rapide d'une maladie contagieuse aux effets significatifs, touchant simultanément un grand nombre de personnes. Deux cas sont possibles : une augmentation d'une maladie endémique (c'est à dire où la présence de la maladie est permanente mais contenue à un taux constant dans une région ou une population particulière), ou l'apparition d'un grand nombre de malades là où il n'y avait rien avant. Le terme de pandémie n'a malheureusement aujourd'hui plus de secret pour personne.

This work addresses second-order stochastic optimization for estimating the minimizer of a convex function written as an expectation. A direct recursive estimation technique for the inverse Hessian matrix using a Robbins-Monro procedure is introduced. This approach enables to drastically reduces computational complexity. Above all, it allows to develop universal stochastic Newton methods and investigate the asymptotic efficiency of the proposed approach.

The $(\max ,+)$ semialgebra relies on a particular choice of operations that belongs to the realm of idempotent analysis. In this short talk, we introduce these operations and the dialog between them and classical algebra. As an application, we give the $(\max ,+)$ parallel to the heat equation.

Obtaining macroscopic laws (e.g., Euler equation/ Heat equation) from interacting particle systems governed by microscopic laws (e.g., Newton/Schrödinger eq.) in proper scaling limits is a very challenging task and has received much attention in recent years. In this talk, I will present the first example (to the best of my knowledge), where one can do this task rigorously for a simple interacting quantum system without any apriori assumptions (CMP, 390, 349–423 (2022)).

The mathematical modelling of cancer has been increasingly applying fluid-dynamics concepts to describe the mechanical properties of tissue growth. The biomechanical pressure plays a central role in these models, both as the driving force of cell movement and as an inhibitor of cell proliferation. In this talk, I will present how it is possible to build a bridge between models that have different pressure-velocity or pressure-density relations.

Dans cet exposé, je présenterai la notion de problème inverse ainsi que différentes applications de ces problèmes. Je commencerai par définir cette notion, puis je donnerai quelques exemples en mentionnant les applications de ces résultats. Enfin, je vous présenterai plus en détails un de mes travaux, en collaboration avec Gunther Uhlmann, lié à une application des problèmes inverses à la tomographie photoacoustique qui est une modalité de l'imagerie biomédicale.