Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The talk will review recent progress on turbulent transport based on Ito-Stratonovich calculus. The general principle is a new way to reformulate turbulent transport as an additional diffusion, by taking a suitable scaling limit based on stochastic calculus. The idea has been applied with success to several problems, like the transport in passive scalars, transport of passive vector fields (like the magnetic field), the effect of turbulence on coalescence of droplets, the effect of turbulent small scales on the large scales of the fluid itself.

After reviewing the principles of so-called sampling methods for imaging the geometry of an inclusion from measurements of scattered waves, we will highlight the role of some specific frequencies where some transparent incident waves may exist. While these frequencies must be avoided for these imaging algorithms, their link to the material properties can be a key property for looking inside highly cluttered media.

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)).