Séminaire de mathématiques appliquées (archives)

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.

The objective of topological data analysis is to extract information of topological nature (connected components, holes, voids...) from the data and use them as features to perform various machine-learning tasks. We start by building a nested sequence of simplicial complexes on top of the data and track the evolution of its topology along the sequence. Computing the Euler characteristic of each complex in the sequence yields a descriptor called the Euler characteristic curve that we use as a feature vector. We will demonstrate that this descriptor has a very good performance in terms of accuracy, strong explainability in terms of topology, and stability with respect to the input data while having a drastically reduced computational cost.

Classical inference methods fail when applied to data-driven test hypotheses. Selective inference is particularly relevant post-clustering, typically when testing a difference in the mean between two clusters. Thus, dedicated methodologies are required to obtain statistical quarantees for these selective inference problems. In this work, we address convex clustering with l1 penalization, by leveraging related selective inference tools for regression, based on Gaussian vectors conditioned to polyhedral sets.

On présente deux travaux reliés concernant des modèles cinétiques de type saut au point-milieu. Le premier (avec Pierre Degond, Gaël Raoul) est un modèle d’alignement de particules autopropulsées (la version cinétique du modèle de Bertin-Droz-Grégoire). La version homogène en espace et non-bruitée correspond à un modèle de saut au point-milieu sur la sphère unité. Le deuxième (avec Cécile Taing) est le modèle de Fisher infinitésimal en dynamique des populations sexuées, dans le cas où la variabilité est nulle. Ceci correspond également à un modèle de saut au point-milieu mais dans lequel le taux de mortalité variable. Dans ces deux modèles, une des difficultés principales et l’absence de conservation du centre de masse.

Le développement de système de suivi GPS léger et très autonome permet de suivre les déplacements des individus de manière précise précise et peu invasive. Ces données de trajectoires sont comprises par les écologues comme le reflet des besoins internes de l'animal, envisagé comme un problème de segmentation, et de sa réponse à l'environnement. Dans cet exposé, je présenterai plusieurs développement statistiques récents, reposant sur des approches à variables latentes, qui visent à extraire ce type d'information des données déplacement. Ces méthodes posent des questions intéressantes en terme de modélisation, d'inférence et de choix de modèles qui seront détaillées dans l'exposé.

Reférences :

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.

Présentation d'une version préliminaire de sa soutenance de thèse :

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.

The aim of this work is to propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre's hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge.

The aim of this work is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson partial differential equation (PDE) with Neumann boundary condition. Using Barron’s representation of the solution with a probability measure defined on the set of parameter values, the energy is minimized thanks to a gradient curve dynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. Inspired by a work from Bach and Chizat, we prove that if the gradient curve converges, then the represented function is the solution of the elliptic equation considered. In contrast to previous works, the activation function we use here is not assumed to be homogeneous to obtain global convergence of the flow.