Séminaire de mathématiques appliquées (archives)

Les systèmes hyperboliques avec relaxation apparaissent dans de nombreux modèles issus de la mécanique des fluides, lorsque l’évolution hyperbolique est couplée à un mécanisme de relaxation qui ramène le système vers un état d’équilibre. Dans cet exposé, je m’intéresserai à ce phénomène à travers le modèle de relaxation linéaire de Jin-Xin, considéré comme un modèle prototype. L’objectif est d’utiliser l’entropie relative pour étudier comment la solution du système de relaxation se rapproche du modèle limite à l’équilibre, aussi bien au niveau continu qu’au niveau discret. Je présenterai en particulier un schéma asymptotic-preserving de type staggered, ainsi que les différents régimes observés numériquement lorsque le paramètre de relaxation varie.

We study the optimal power flow (OPF) problem in electrical distribution networks. The objective is to minimize total line losses subject to the physical constraints governing the network. This optimization problem is inherently non-convex, making it challenging to solve in practice. In particular, classical solution methods may fail to converge to a global minimum.

A common strategy to address this difficulty is to consider a convex relaxation, where the original non-convex feasible set is replaced by a larger convex set. In this work, we focus on a semidefinite programming (SDP) relaxation obtained by dropping a rank constraint on certain matrix variables.

The main goal of this talk is to show that this relaxation is in fact exact, meaning that it yields the same solution as the original OPF problem. To establish this result, we rely on two key ingredients: the characterization of the Pareto front of the feasible sets and the exploitation of the tree structure of the network.

Spatial Lambda-Fleming-Viot processes, or SLFVs, are a family of stochastic processes that have been developed to overcome some issues arising when defining stochastic population dynamics models in a spatial continuum of arbitrary dimension. Their main characteristic is their ''reproduction event-driven'' dynamics that controls local reproduction rates and ensures the existence of a dual process, which is a key tool to the analysis of SLFV processes. In this talk, after giving an overview of this modelling framework, I will present a recent extension to the modelling of epidemics, giving rise to a stochastic SIS-type model in continuous space. I will show to what extent the central concept of basic reproduction number can be extended to this model, using the duality relation along with the characterization of the process as the unique solution to a martingale problem. Based on a joint work with Bastian Wiederhold (LMU).

Understanding how cells migrate through confined environments is crucial for elucidating fundamental biological processes, including cancer invasion, immune surveillance, and tissue morphogenesis. The nucleus, as the largest and stiffest cellular organelle, often limits cellular deformability, making it a key factor in navigating narrow pores or highly constrained spaces. In this talk, I will present a novel geometric surface partial differential equation (GS-PDE) framework in which the cell plasma membrane and the nuclear envelope are modeled as evolving energetic closed surfaces governed by force-balance equations. To validate the model, we replicate a biophysical experiment using a microfluidic device that imposes compressive stresses on cells driven through narrow microchannels under a controlled pressure gradient. I will discuss the results of our parametric sensitivity analysis, which highlights the dominant influence of specific parameters, such as surface tension and confinement geometry, as key determinants of translocation efficiency. Finally, I will show how this framework, while tailored to a specific experimental setup for validation, provides a robust, flexible, and generalizable tool for investigating the broader interplay between cell mechanics and confinement, laying the groundwork for integrating more complex biochemical processes like active migration.

The problem of testing linear hypotheses for the means of random functions is considered. This includes checking if the mean is zero, checking if two sample means are the same, and checking if the two means have a constant difference or ratio. The random function is defined on a multidimensional compact domain and several independent realizations are observed at random design points, possibly with heteroscedastic error. The number of design points of each realization of the random function can be bounded or arbitrarily large. For two-sample tests, the samples are allowed to be unbalanced and dependent. The testing approach is based on a non-asymptotic Gaussian approximation bound for the estimated Fourier coefficients. A pivotal chi-square type statistics is proposed.

Dans ce travail, nous étudions un modèle de régression visant à décrire le comportement des valeurs extrêmes d’une variable Y à partir de covariables X. Nous proposons une méthode de réduction de dimension spécialement conçue pour les queues de distribution, permettant de surmonter le fléau de la grande dimension et d’améliorer l’estimation de l’indice des valeurs extrêmes conditionnel.

L'hémodialyse est un procédé médical consistant à rééquilibrer certaines espèces chimiques dans le sang d'un patient via l'usage d'une machine appelée dialyseur. Cette dernière est un groupement de fibres très fines dans lesquelles passe le sang du patient ainsi qu'un fluide (le dialysat) permettant des échanges d'espèces chimiques au travers d'une membrane poreuse. Du point de vue mathématique, cet échange est modélisé par un système de type convection-réaction-diffusion avec des conditions frontières mixtes

Certaines approches médicales modernes ont pour objectif d'apporter des soins davantage personnalisés pour chaque patient ce qui, dans le cadre de l'hémodialyse, nécessite une connaissance approfondie du fonctionnement du dialyseur et en particulier de la manière dont sont échangées les espèces chimiques lors de la dialyse.

Dans cet exposé, nous nous intéresserons dans un premier temps à la modélisation des écoulements fluides dans le dialyseur, puis détaillerons les méthodes numériques pour la résolution de problèmes inverses permettant d'identifier des coefficients de diffusion d'espèces chimiques clés au travers de la membrane poreuse, ce qui offre une connaissance plus détaillée du mécanisme du dialyseur. Nous conclurons l'exposé par plusieurs perspectives, notamment l’usage de réseaux de neurones informés par la physique (PINNs) pour réduire le coût computationnel, ainsi que la formulation d’un problème de contrôle optimal en vue d’une adaptation personnalisée des protocoles de dialyse.

Many computational methods across Bayesian statistics aim at constructing parametric probability distributions with properties of interest. One can think of variational inference or adaptive importance sampling for instance. The construction of such distributions can often be formulated as the minimization of a statistical divergence, such as the Kullback-Leibler divergence, over a family of approximating distributions. Depending on the choice of statistical divergence and approximating distributions, the resulting divergence-minimization problem may promote solutions with different features and may be more or less hard to solve. For instance, some problems are convex, promote mass-covering approximations, but may induce hard-to-approximate gradients, while other are non-convex, promote mode-seeking approximating distributions, and have easy-to-approximate gradients.

In this talk, I will discuss some considerations related to the choice of statistical divergence and approximating distributions and present recent results and algorithms to solve some particular classes of divergence-minimization problems. These results leverage tools from stochastic optimization and convex analysis that allow to capture the geometric features of divergence-minimization problems.

Spline interpolation is a widely used class of methods for solving interpolation problems by constructing smooth interpolants that minimize a regularized energy functional involving the Laplacian operator. While many existing approaches focus on Euclidean domains or the sphere, relying on the spectral properties of the Laplacian, this work introduces a method for spline interpolation on general manifolds by exploiting its equivalence with kriging. Specifically, the proposed approach uses finite element approximations of random fields defined over the manifold, based on Gaussian Markov Random Fields and a discretization of the Laplace-Beltrami operator on a triangulated mesh. This framework enables the modeling of spatial fields with local anisotropies through domain deformation. The method is first validated on the sphere using both analytical test cases and a pollution-related study, and is compared to the classical spherical harmonics-based method. Additional experiments on the surface of a cylinder further illustrate the generality of the approach.