Séminaire de mathématiques appliquées (archives)

Dormancy, whether in ecology, bacteriology or cell dynamics, is a reversible reduction of metabolic activity reducing resource consumption and increasing resistance against exterior pressure. We first consider a simple bistable ODE equation to model how dormancy affects survival of a population after either a catastrophic event or after arrival in a new environment. We observe the classical result that dormancy protects a species against uncertain environments and catastrophic events, but we also notice that dormancy strengthens the Allee effect, effectively reducing invasiveness and increasing the inertia of the environment. We then present a bistable coupled ODE-reaction diffusion model for dormancy and Go-or-Grow, and prove the sharp (or non-sharp for some Go-or Grow dynamics) threshold property. Studying its variation with respect to system parameters, we see how the resilience-invasiveness trade-off inherent to dormancy affects the spatial dynamics and survival of a species. We also investigate properties on the optimal strategy for dormancy or reactivation to minimize the survival threshold.

Les modèles FARIMA constituent un outil classique pour la modélisation de séries temporelles présentant des dépendances à court et à long terme. La sélection des ordres autorégressif et moyenne mobile joue alors un rôle essentiel dans la spécification du modèle. Les résultats théoriques disponibles reposent toutefois généralement sur des hypothèses fortes concernant le processus d’innovation, supposé notamment i.i.d. Dans ce travail, nous considérons un cadre plus général dans lequel les innovations sont simplement supposées décorrélées, mais peuvent présenter des dépendances non linéaires. L’objectif est d’étudier si les critères usuels de sélection de modèles restent valides dans ce contexte, sans introduire de nouvelle pénalisation. Nous analysons ainsi le comportement asymptotique du critère d’information bayésien (BIC) et d’un critère d’information quasi-bayésien (QBIC), fondé sur une estimation par quasi-maximum de vraisemblance. Nous établissons la consistance de ces deux critères pour la sélection des ordres autorégressif et moyenne mobile. Ces résultats étendent les théorèmes de consistance existants à une classe plus large de modèles FARIMA faibles et fournissent ainsi une justification théorique à l’utilisation de critères de type BIC lorsque les innovations sont décorrélées mais temporellement dépendantes.

The isotropic helicoid is a peculiar object imagined by Kelvin in 1871: it has octahedral chiral symmetry, but behaves isotropically when sinking in a viscous fluid. Since then, successive studies of particle dynamics in flow have repeatedly highlighted the same intriguing observation: the symmetry properties of the particle’s hydrodynamic behaviour are distinct from the geometric symmetry of the particle itself. In this work, I propose to characterise precisely the distinction between hydrodynamic and geometric symmetry, using classical tools of representation theory applied to this fluid mechanics problem. Hence, I will consider a smooth bounded particle in a fluid governed by the Stokes equation, with prescribed Dirichlet conditions corresponding to an imposed ambient flow. The resistance problem then consists in determining the hydrodynamic drag force moments exerted on the particle from the flow velocity data. By linearity of the Stokes equations, these quantities are organised into a hierarchy of linear operators (Rk) called resistance tensors, living in finite-dimensional O(3)-representation spaces. Then, one can define invariant subspaces of Rk for each symmetry subgroup of O(3), and in particular distinguish the visible classes at each level and their dimension. This systematic approach shows in particular that polyhedral symmetries are isotropic at low level and become visible in strict order: beyond the first (linear) level k=1, Kelvin’s helicoid is not isotropic anymore. Further, I will show how this framework can be applied to derive normal forms and new equations of motion, with interesting applications to classical problems of Stokes flow dynamics, such as the Jeffery equations and effective viscosity of dilute suspensions.

On cherche à approcher une fonction inconnue f à partir d’évaluations en des points x1,…,xn choisis. Si l’on sait que f est proche d’un espace linéaire de dimension m, on peut construire une approximation dans cet espace par moindres carrés dès que n ≥ m. Cependant, la qualité de cette approximation dépend énormément de la distribution des points, et nous discuterons différents algorithmes d’échantillonnage garantissant une reconstruction stable avec n de l’ordre de m.

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.