Résumés de la conférence

Ci-dessous, vous trouverez les résumés des différentes présentations de jeudi - vendredi :

• Stratified Problems arising from Homogenization of Hamilton-Jacobi equations (Yves Achdou): 

  We describe several cases in which the homogenization of stationary Hamilton-Jacobi equations leads to stratified problems (the class of stratified problems has been introduced by A. Bressan and Y. Hong and later studied by G. Barles and E. Chasseigne). We first recall results obtained with S. Oudet and N. Tchou, in which the limiting problem in |R^d is a stratified problemin which the stratification is of the type (|R^d \ M_{d-1}) \cup M_{d-1}, where M_{d-1} is a d-1 dimensional subspace. Next, we consider a class of Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian (collaboration with C. Le Bris). The limiting problem involves the stratification |(R^d \ {0}) \cup {0}.
  Finally, we study homogenization for a class of bidimensional stationary Hamilton-Jacobi equations where the Hamiltonian is obtained by perturbing near a half-line a Hamiltonian which does not depend on the fast variable, or depends on the fast variable in a periodic manner. We prove that the limiting problem involves a stratification, made of a submanifold of dimension zero, namely the endpoint of the half-line, a submanifold of dimension one, the open half-line, and the complement of the latter two sets which is a submanifold of dimension two. The limiting problem involves effective Hamiltonians which are associated to the abovementioned three submanifolds and keep track of the perturbation.

• Adipose tissue size distribution modeling (Chloé Audebert) : Adipose tissue, made of adipocyte cells (also known as fat cells), is central in many physiological pathways involved in obesity-related diseases. Clarifying the dynamics of adipocytes size and number evolution is crucial to better understand the pathophysiological basis of those diseases. We will study adipocyte size dynamics using mathematical models based on partial differential equations. We aim to mimic characteristic cell size distributions in order to provide information on their connections with health conditions.
  
  A mathematical model describing adipocyte size dynamics, based on exchange of lipid, will be presented. Parameter estimation from adipocyte size distribution will be performed as well as a study of parameter identifiability. Parameter estimation is performed with CMAES and will be presented first on synthetic data, then results on measured adipocyte size distribution of several rats will be discussed.

• Test function approach to fully nonlinear equations in thin domains (Isabeu Birindelli) : In this seminar we will illustrate a work in collaboration with Ariela Briani and Hitoshi Ishii that extents the well known result on thin domains of Hale and Raugel \cite{HR}. The test function approach of C. Evans is very powerful and gives new results even in the case of the Laplacian.

• Un problème de frontière libre en jeux à champ moyen (Pierre Cardaliaguet) : Les jeux à champ moyen modélisent des systèmes avec un grand nombre de contrôleurs en interaction. Dans de nombreux cas, le problème se réduit à un système (le système MFG) couplant une équation de Hamilton-Jacobi à une équation de continuité. Dans une séries de travaux avec Sebastian Munoz (Chicago) et Alessio Porretta (Rome Tor Vergata), nous étudions le comportement des solutions du système MFG lorsque la densité initiale est une fonction supportée de manière compacte sur la droite réelle. Nos résultats montrent que la solution est lisse dans les régions où la densité est strictement positive, et que la densité elle-même est globalement continue. De plus, pour un couplage de type puissance, nous établissons une vitesse de propagation finie, conduisant à la formation d’une frontière libre. Cette frontière libre est strictement convexe et jouit d'une régularité $C^{1,1}$. Nos méthodes sont basées sur l'analyse d'une nouvelle équation elliptique satisfaite par le flux de trajectoires optimales. Les résultats sont également s'appliquent aux problèmes de planification de champ moyen, caractérisant la structure des minimiseurs d'une classe de problèmes de transport optimaux avec congestion.

• Modeling and numerical simulations of irreversible electroporation for cardiac ablation (Annabelle Collin) : In arrhythmia, the electrical waves of the heart become chaotic and disrupt its function. In the most important treatment, catheter ablation, small heart tissue is destroyed to isolate the irregular beats. This is usually done with thermal radiofrequency (RFA), but this work focuses on a non-thermal method: pulsed electric field ablation (PFA). PFA uses intense electrical pulses to induce cell death through irreversible electroporation. Although PFA has been used in oncology for over a decade, it is still in its infancy in cardiology due to its technical complexity.

  In collaboration with IHU Liryc, we are investigating how mathematical models and numerical strategies could support the development of this therapeutic strategy. This leads to several mathematical challenges that I will present in this talk. I will present a bidomain model that includes a nonlinear transport term derived thanks to a two-scale approach that can account for the different time and length scales between cardiac electrophysiology and electroporation. Numerical simulations with industrial catheter geometries will be presented, showing first interesting results for the determination of the electroporated area. I will then present an electrophysiological model of a cardiac area containing a region ablated by PFA. By considering the modeling assumptions and performing a rigorous asymptotic analysis, the transmission conditions at the interface between the two regions were determined. Numerical simulations performed thanks to well-designed Schwarz algorithms allow a numerical explanation for the higher rate of fibrillation recurrence after RFA compared to PFA.

• Joint inviscid-incompressible limit for tissue growth models: from Brinkman's law to the Hele-Shaw problem (Noemi David) : Cancer growth modelling has seen an increasing application of fluid-dynamics concepts to describe the mechanical properties of living tissue. The biomechanical pressure plays a central role, both as the driving force of cell movement and as an inhibitor of cell proliferation. Singular limits that can build a bridge between models with different pressure-velocity or pressure-density relations have attracted great interest in recent years. In particular, the theory on the inviscid limit from a visco-elastic model to porous-medium-like equations and the incompressible limit that links the latter to a Hele-Shaw problem with density constraint is nowadays quite well understood. In this talk, I will address the question of passing to the joint limit from a Brinkman compressible model to the Hele-Shaw free boundary problem. To this end, we exploit the gradient flow structure of the limit problem and a family of energy inequalities satisfied by the Brinkman model in order to prove the strong compactness of the velocity field. This is a work in progress in collaboration with Matt Jacobs and Inwon Kim.

• Adaptative observers for wave equations and associated discretization: formulations and analyses (Tiphaine Delaunay) : The context of this talk is the study of inverse problems for wave propagation phenomena from a control theory perspective, more specifically using observation theory. Our goal is to formalize, analyze, and discretize strategies called sequential in data assimilation, where measurements are taken into account as they become available. The resulting system, called an observer (or a sequential estimator), stabilizes on the observed trajectory hence reconstructs the state and possibly some unknown parameters of the system. Here we focus in particular on source reconstruction on the right-hand side of our wave equation, an estimation problem intermediate in complexity between state estimation (or initial condition estimation) and the more general problem of parameter identification. In this context, we propose to define in a deterministic framework in infinite dimension a so-called Kalman estimator that sequentially estimates the source term to be identified. Using dynamic programming tools, we show that this sequential estimator is equivalent to the minimization of a functional. This equivalence allows us to propose convergence analysis under observability conditions. We then demonstrate observability inequalities for different source types by combining functional analysis, multiplier methods, and Carleman estimates. In particular, these inequalities inform us about the ill-poseness nature of the inverse reconstruction problems we study and allow us to quantify the degree of ill-posedness hence to propose adapted regularizations. Concerning discretization issues and their analysis, we defend the idea of redefining these observers associated with the minimization of the functional once the direct model has been discretized. This discretize-then-optimize approach is advantageous for the analysis compared to optimize-then-discretize approach. Nevertheless, the observability inequalities need to be extended to discretized systems. In particular, in this context, we extend uniform exponential stabilization results to the discretization for high-order finite element discretizations of the wave equation.

• What is new in Domain Decomposition? (Martin J. Gander) : 

  Domain decomposition research intensified in the early nineties, and there is still substantial research activity in this field. There has been however an important shift in domain decomposition, and I will explain three new interesting research directions that are pursued very actively at the moment, and give newest results:

  - Iterative solvers for time harmonic wave propagation: time harmonic wave propagation problems are very hard to solve by iterative methods. All classical iterative methods, like Krylov methods, multigrid, and also domain decomposition methods, fail for the key model problem, the Helmholtz equation. There are new, highly promising domain decomposition methods for such problems, which I will present, and I will also state precisely under which conditions they can work well, and when they still fail.

  - Coarse space components: domain decomposition analysis has lacked behind multigrid in the precise understanding of the interaction between the domain decomposition smoother and coarse space solver, and all classical domain decomposition solvers need Krylov acceleration to be effective, while multigrid does not. I will present a new spectral analysis of the Schwarz iteration operator, which allows us to achive as an accurate understanding of two level Schwarz methods as the seminal Fourier analysis of multigrid methods.

  - Time parallelization: new computing architectures have too many computing cores to parallelize only in space for evolution problems. I will present time and space-time domain decomposition methods and explain which can be effective for parabolic and hyperbolic problems.

• Some qualitative results on parabolic periodic eigenvalue problem (Idriss Mazari): 
  Parabolic periodic eigenvalue problems are important in the study of reaction-diffusion equations, and so is their optimisation with respect to the potential. The main question under consideration is the following: \emph{how to choose $m$ so as to minimise the eigenvalue $\lambda$?} Naturally we would need to specify the proper constraints, but, at a qualitative level, there are two main questions. The first one is the \emph{symmetry} of optimisers: is it true that it always better to replace $m$ with another potential (that satisfies the same constraints) but that is also symmetric in time and in space? The second one, has to do with the \emph{monotonicity} of the optimisers: provided the answer to the first question is positive, is it true that the optimiser is not only symmetric, but also monotonous? Let us emphasise that these questions are answered positively when considering the symmetry and monotonicity with respect to the space variable only.
   

  In this talk, we will present some recent time symmetrisation results for parabolic operators, which, to the best of our knowledge, are the first of the kind. This is a joint work with G. Nadin and B. Bogosel.

• High order cubature for iterated function system (Zoïs Moitier) : The talk is motivated by application to fractal antenna engineering, where antennas with self-similar shapes operate across multiple frequencies. Recent works by Chandler-Wilde, Gibbs, Hewett, Moiola and their co-workers propose boundary integral formulations for solving Helmholtz scattering problems on fractal screens with Dirichlet boundary conditions. These boundary integral equations utilize the Hausdorff measure on fractals instead of the standard Lebesgue's measure. This makes the evaluation of the boundary integral difficult from a numerical viewpoint. To tackle this point, we propose new interpolatory high-order tensor cubature formula on fractals, based on Chebyshev points on an interval. These formulas allow computing integrals of restrictions of regular functions to fractals with a high accuracy. We will discuss the construction of such cubature (in particular computation of the weights) and their properties (asymptotic behavior of weights).

• Le problème de Riemann sur réseau (Régis Monneau) :  Motivé par des questions de trafic routier sur réseau, on considère des lois de conservations scalaires sur une jonction constituée de n+m branches, avec n entrées et m sorties.En toute généralité, on montre qu'il y a une correspondence univoque entre 3 objets différents: les solvers de Riemann à la jonction, les germes de Riemann (constitués des solutions stationnaires à la jonction), et les flux de Godunov à la jonction. De plus, pour les solvers L^1-contractants (qu'on appelle solvers de Kruzkov), on montre qu'il y a une bonne théorie d'existence et d'unicité des solutions des EDP.

• Front propagation in a transport model with a nonlocal nonlinear condition at the boundary (Jean-Michel Roquejoffre) : The model under study is a linear transport equation in the upper half plane, together with a nonlinear and nonlocal Dirichlet condition that couples the values of the unknown function at the boundary to those inside. Its primary motivation is the study of the nonlocal Kermack-McKendrick model for the spread of epidemics, and it has received a  great deal of attention in the 1980's. It can be reduced to a nonlinear integral equation, from which one can infer the development of nasion fronts, whose asymptotic propagation speed can be computed.
  
  The goal of the talk is to present a fresh look at this model and to understand its sharp asymptotics, something that had not previously been done. While the relevance of such an undertaking may be questionned from the epidemiological point of view, its structure presents specificities that make it worth studying. In particular, it is reminiscent of that of the "Road-field model" introduced by Berestycki, Rossi and the author, an analogy that will be discussed.
  
  Joint work with G. Faye and M. Zhang.

• Logarithmic lag for solutions with unbounded initial support (Luca Rossi) : We discuss the question of determining the location of the level sets of solutions of reaction-diffusion equations of the Fisher-KPP type. The answer is known in the literature in two cases: compactly supported and almost planar initial data. Compared to the position of the level sets of planar travelling fronts, there is a lag  which is logarithmic in time, with a multiplicative factor which is different in the two cases. We will consider intermediate situations between compactly supported and almost planar initial data and search for the corresponding lag. As a by-product, we will show the convergence to a planar front for solutions whose initial support is a strictly convex set. This is a joint work with François Hamel.

• Laplacian estimates for the parabolic-elliptic Keller-Segel model (Filippo Santambrogio) : This talk, stemming from an ongoing collaboration with Charles Elbar (Lyon) and Alejandro Jimenez-Fernandez (Oxford), lies at the intersection of two well-known phenomena in parabolic equations. The first concerns nonlinear estimates on the second derivatives of solutions to diffusion equations: by looking at the PDE satisfied by the Laplacian or by the Hessian of the logarithm of the solution of the heat equation (or of suitable powers of the solution for non-linear diffusion such as porous media equations) one can obtain lower bounds in the form of \Delta(\log\rho)\geq -1/t. The second, instead, concerns the critical mass in the parabolic-elliptic Keller-Segel chemotactic system where linear diffusion is coupled with advection by a potential generated by the convolution of the solution with the Poisson Kernel: it is well-known for this nonlinear equation that explosion in finite time or global existence depends on the mass (which is preserved in time) and the best estimates are obtained for small mass. In the talk I will show how, under small assumptions, it is also possible to obtain estimates, with instantaneous regularization, on the Laplacian of the log of the solution. The results that I will show will not be the sharpest one in terms of smallness of the mass, but we are currently working to improve them.