Seminars

In this presentation we focus on the numerical simulation of granular materials: collections of rigid, non-Brownian macroscopic grains (sand, grain, sugar, rubble, etc.). The contacts between grains lead to singular interactions for which adapted numerical schemes must be developed. We place ourselves in the framework of "Contact Dynamics" type models developed by J.J. Moreau, based on non-smooth convex analysis. The frictional models between grains lead to complex non-convex optimisation problems. The objective of the presentation is to show how algorithms based, at each instant, on convex optimization problems allow to obtain numerical long time simulations of large numbers of particles. In these schemes, the contact forces are obtained implicitly, as Lagrange multipliers associated with the constraints. The fundamental principle of dynamics is then obtained (in a discretised version) from the Euler (optimality) equations of the minimisation problem. In the case of frictionless grains, we come back to minimisation problems under affine constraints. In order to introduce friction (Coulomb model) between grains, we have to consider minimisation problems under conical constraint, for which the constraint is therefore non-derivable. We will illustrate this work with numerical simulations, showing that the resulting schemes can be used to study the macroscopic behaviour of granular materials. Part of the work was done in collaboration with Sylvain Faure, Philippe Gondret, Yvon Maday, Anne Mangeney, Hugo Martin, Bertrand Maury and Antoine Seguin.

In this talk, we consider the nodal numerical flux extension of classical Eulerian edge numerical flux schemes for linear and nonlinear systems of two dimensional hyperbolic conservation laws. In a first part, we present properties based both on mesh geometry and state interaction implying local conservativity and consistency. Such criteria apply to flux based schemes (VFFC, Roe,..) as well as viscosity based schemes (Rusanov, HLL, VF-Roe,..). This enables us to define composite edges/nodal fluxes in a natural way. In a second part, we deal with the stability (of discrete unknowns) and admissibility (in flux evaluation) of Euler system of gas dynamics. We focus on a nonlinear reconstruction for massic variables and propose a way to circumvent the direct velocity limitation which is not physically relevant. Hence, we show how to deduce from algebraic relation of total energy E = eps + 1/2 |U|^2 a limited velocity reconstruction induced by the direct limitation of internal massic energy eps.

A story about rheology, and the stability of solutions of some system of PDEs. Featuring Fourier, Chapman-Enskog expansions and the Poincaré-Bendixon theorem.

I will present in this talk some variations around a classical model in mathematical epidemiology, the (spatial) SIR model. This model is a rather simple system of parabolic equations. The goal of SIR-like models is to understand how a disease spreads in a population. I will start with a thorough analysis of the standard SIR system (long-time behavior of solutions, convergence, speed of propagation...). Then, we will present other models and the mathematical challenges they bear (models with delay, heterogeneous models.). Finally, I will show some recent results on a more complex model that takes into accounts variant (or strains, or mutations) of a disease.

In this talk, we will present stochastic numerical methods for mean field games and mean field control problems (also called optimal control of McKean-Vlasov dynamics). These problems arise as the limit of Nash equilibria or social optima in games when the number of players grows to infinity. We will mostly focus on methods based on neural networks to compute solutions when the model is fully known, motivated by applications in high dimension or with common noise. If time permits, we will also discuss model-free methods based on reinforcement learning.

The modelisation of diffusion phenomena in solid crystalline alloys is very important in a wide variety of applications, one of which being the modelisation of the fabrication process of thin film solar cells. At the microscopic scale, atoms of various chemical species are located at the sites of the underlying crystalline lattice and may jump to another neighbouring site following a random Markov process, which is usually modeled as a symmetric multi-species exclusion process. In most realistic cases, the number of particles composing the system of interest is much too large for the movement of each atom to be fully simulated, and hydrodynamic limits (limits as the number of particles go to infinity) are thus considered instead. The latter read as cross-diffusion systems, i.e. deterministic systems modeling the evolution of the local concentrations or volumic fractions of each chemical species in the mixture. These systems read as nonlinear degenerate parabolic systems, which make their simulation very intricate. Another difficulty, which is the main focus of this talk, is that they involve the computation of some non-explicit terms, the so-called auto-diffusion matrix of the system, which can be expressed as the solution of a very high-dimensional optimization problem. Standard numerical approaches for the computation of this auto-diffusion matrix suffer from the so-called curse of dimensionality. As an alternative, we consider here a new numerical strategy, based on tensor decompositions, which proves to be highly effective in the present context. The aim of this talk is to present various theoretical and numerical results linked to this new approach.

The minimization of a relative entropy (with respect to the Wiener measure) is a very old problem which dates back to Schrödinger. Strong connections and analogies between this problem and the Monge-Kantorovich problem with quadratic cost (namely the standard Optimal Transport problem) have been recently established. In particular, the entropic interpolation leads to a system of PDEs which presents strong analogies with the Mean Field Game (MFG) system with a quadratic Hamiltonian. In this talk, we will explain how such systems can indeed be obtained by minimization of a relative entropy at the level of measures on paths with an additional term involving the marginal in time. Moreover we will review all the links between Optimal transport and Variational MFG focusing both on theoretical and numerical aspects. If time permits we will also offer a glimpse on the case of multi-population MFG and its connection with some equations arising in quantum mechanics.