I will present a work in collaboration with M. Bessemoulin-Chatard and T. Rey, in which we consider a non-linear kinetic model describing a two-species generation-recombination reaction that can be considered as a simplified version of the models describing the generation and recombination of electron-hole pairs in semiconductors. I will introduce a finite volume discretization of this model for which we can prove an exponential decay towards the steady state using discrete hypocoercivity methods. After presenting the ideas of the proof in the continuous framework, I will highlight the main difficulties induced by the discretization process. The properties of the method will then be illustrated by several numerical examples.

In healthy hearts, the propagation of electrical waves follows a predictable pattern, whereas in people suffering from arrhythmia, the electrical waves can become chaotic and affect the heart's pumping function. The main treatment is catheter ablation, during which small areas of heart tissue are destroyed to isolate the cause of the irregular heartbeats. Most catheter ablations are performed thermally through the application of a radiofrequency electromagnetic field (RFA), but in this work, we focus on the study of a non-thermal ablation technique: pulsed electric field ablation (PFA), which utilizes irreversible electroporation, a complex cell death phenomenon that occurs when biological tissue is exposed to very intense electrical pulses. This technique has been used in oncology for more than a decade, but in cardiology it is still in its infancy due to the technical complexity of this novel approach. Mathematical models and numerical strategies could improve the understanding of PFA on the cardiac signal. In this talk, I will introduce the various mathematical challenges that we would like to address. Second, I will present a bidomain model incorporating 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 also be presented, showing first interesting results for the determination of the electroporated area. Third, I will present a cardiac electrophysiological model of a cardiac domain containing a region ablated by PFA. Considering modeling assumptions and performing a rigorous asymptotic analysis, we determine the transmission conditions at the interface between the two regions. Numerical simulations performed thanks to well-designed Schwarz algorithms will be presented in the context of atrial fibrillation. A numerical comparison between PFA and RFA will allow to propose a numerical explanation for the higher rate of fibrillation recurrence after RFA compared to PFA

In this work, we construct a structure-preserving reduced-order model for the resolution of parametric cross-diffusion systems. Cross-diffusion systems model the evolution of the concentrations or volumic fractions of mixtures composed of different species and often read as nonlinear degenerated parabolic partial differential equations whose numerical resolutions are highly expensive from a computational point of view. We are interested here in cross-diffusion systems which exhibit a so-called entropic structure, in the sense that they can be formally written as gradient flows of a certain entropy functional which is actually a Lyapunov functional of the system. In this work, we propose a new reduced-order modelling method, based on a reduced basis paradigm, in order to accelerate the resolution of parameter-dependent cross-diffusion systems, which preserves, at the level of the reduced-order model, the main mathematical properties of the continuous solution, namely mass conservation, non-negativeness, preservation of the volume-filling property and entropy-entropy dissipation relationship. The theoretical advantages of our approach are confirmed by several numerical experiments.

Dans cet exposé, nous présentons des modèles asymptotiques obtenus à l'aide de développements multi-échelles qui permettent de décrire des phénomènes électromagnétiques ou géophysiques. Dans une première partie, nous présentons des modèles d’impédance pour la résolution de problèmes de couche limite ou de couche mince. La précision et la performance des modèles obtenus sont illustrées par des résultats numériques. Dans une seconde partie, nous présentons une méthode de paramétrisation du potentiel magnétique pour un problème de courant de Foucault dans des matériaux ferromagnétiques. Nous montrons que cette méthode est bien adaptée à la résolution du problème pour une gamme de fréquences assez large sans adaptation de maillage et qu'elle fournit le même ordre d'approximation qu'une méthode d'impédance de surface à moindre coût.

The statistical query (SQ) framework consists in replacing the usual access to samples from a distribution, by the access to adversarially perturbed expected values of functions interactively chosen by the learner. This framework provides a natural estimation complexity measure, enforces robustness through adversariality, and is closely related to differential privacy. In this talk, we study the SQ complexity of estimating d-dimensional submanifolds in R^n. We propose a purely geometric algorithm called Manifold Propagation, that reduces the problem to three local geometric routines: projection, tangent space estimation, and point detection. We then provide constructions of these geometric routines in the SQ framework. Given an adversarial STAT(τ) oracle and a target precision ε=O(τ^{2/(d+1)}) for the Hausdorff distance, the resulting SQ manifold reconstruction algorithm has query complexity O(npolylog(n)τ^{d/2}), which is proved to be nearly optimal. In the process, we will present low-rank matrix completion results for SQ's and lower bounds for (randomized) SQ estimators in general metric spaces.

Travail publié dans : Adversarial Manifold Estimation (https://arxiv.org/pdf/2011.04259v2.pdf) Il combine deux champs assez indépendants : - Pour le volet "Statistical queries" : https://arxiv.org/pdf/2004.00557.pdf - Pour le volet "Manifold estimation" : https://arxiv.org/pdf/1512.02857.pdf ou bien https://arxiv.org/pdf/2108.03135.pdf

Dans cet exposé, nous considérerons le laplacien magnétique en dimension deux. Nous supposerons que le champ magnétique est strictement positif et constitué de deux puits radiaux symétriques (entre lesquels le champ magnétique est constant). Nous nous intéresserons à l'écart entre les deux premières valeurs propres dans la limite semiclassique. Nous établirons en particulier un équivalent de cet écart.

Programme :

14:30-14:45. Introduction (François Plantade)

14:45-15:35. Retour sur les ouvrages de Liouville et Briot & Bouquet et étude de la « Note sur la théorie des fonctions elliptiques » de Hermite, 1862. (André-Jean Glière)

15:35-15:50. Pause elliptique

15:50-16:40. Étude de « Die Lehre von den elliptischen Integralen und der Theta-Functionen » de Karl-Heinrich Schellbach, 1864 et de la deuxième édition de la « Theorie der elliptischen Functionen. Versuch einer elementaren Darstellung » de Heinirich Durège, de 1868. (François Plantade)

16:40-17:10. Comparaison de la première et de la deuxième édition de l'ouvrage de Briot & Bouquet (André-Jean Glière)

17:10-17:30. Discussion et comparaison des différents traités présentés. Questions.