Séminaire de mathématiques appliquées (archives)

In this talk, we will study a complex network composed of fibers having the ability to cross-link or unlink each other and to align with each other at the cross links. This model aims to describe networks of collagen fibers in a fibrous tissue. We will first present a microscopic model which features the following basic rules: We assume the existence of a fiber unit element (or monomer) modeled as a line segment of fixed length. We suppose that two fiber elements that cross each other may form a link, thereby creating a longer fiber. The fibers have the ability to branch off and to achieve complex network topologies. We include fiber resistance to bending by assuming the existence of a torque which, in the absence of any other force, makes the two linked fiber elements align with each other. Fibers are also subject to random positional and orientational noise and to external positional and orientational potential forces. Finally, cross-links may also be removed to model possible fiber breakage or depolymerization. We then formally derive a kinetic model for the fiber and cross-links distribution functions, and consider the fast linking/unlinking regime in which the model can be reduced to the fiber distribution function only. Then, we investigate its diffusion limit. The resulting macroscopic model consists of a system of nonlinear diffusion equations for the fiber density and mean orientation. In the case of a homogeneous fiber density, we show that the model is elliptic. We finally will present simulation results which show the good correspondence between the microscopic and the macroscopic models.

Durant cet exposé, je présenterai d'abord le système de Vlasov-Poisson avec un champ magnétique extérieur fort qui permet de confiner le plasma. Ensuite, je rappellerai différents modèles mathématiques que l'on obtient lorsque l'amplitude de de champ magnétique tend vers l'infini. Je proposerai enfin des méthodes numériques qui permettent de capturer cette asymptotique.

Les séries chronologiques à coefficients aléatoires sont devenues très populaires au fil des années, en raison de leur grande flexibilité. Il est cependant contre-intuitif de supposer l'indépendance des coefficients lorsque le modèle est chronologique, ce qui est généralement fait. Dans cet exposé, on proposera une première approche dans laquelle les coefficients sont corrélés. On montrera en particulier dans le cadre d'un exemple autorégressif simple que l'estimateur usuel, tout en restant asymptotiquement normal, n'est plus consistant, ce qui pose certains problèmes d'interprétation sur les données réelles. On souhaitera également par ce travail motiver l'étude de dépendances plus générales dans les coefficients.

The collapse transition of self-interacting random walks is a challenging issue, arising in the study of an homopolymer dipped in a repulsive solvent. Different mathematical models have been built by physicists to try and improve their understanding of this phenomenon. For such models, the possible spatial configurations of the polymer are provided by self-avoiding walk trajectories. However, self avoiding walks, especially in dimension 2 and 3, are complicated objects.

This is the reason why, in the mathematical literature, collapse transition models were rather built by either relaxing the self-avoiding feature of the walk or by considering partially directed path. In our main result, we prove the existence of a collapse transition for a non-directed model built with prudent paths, i.e., non-directed self-avoiding paths which can not take a step towards a previously visited lattice site. The prudent self-avoiding walk has attracted the attention of the combinatorics community over recent years and an open problem is to find out the exponential growth rate of the set of prudent paths. As a by product of our results we are able to solve this problem.

In the first part of the talk we discuss the main models already studied in the literature, while in the second part of the talk we present our main results.

The presentation will be kept at an elementary level: only a basic probabilistic background is assumed.

(Joint work with Nicolas Pétrélis)

The so-called pair correlation function is a fundamental spatial point process characteristic that, given the intensity function, determines second order moments of the point process. Computation of a non-parametric estimate of the pair correlation function is a typical initial step of a statistical analysis of a spatial point pattern. Kernel estimates are popular non-parametric estimates but especially for clustered point patterns suffer from bias for small spatial lags. We introduce a new orthogonal series estimate which is much less biased for clustered point patterns. We consider consistency and asymptotic normality of the new estimate and also finite sample properties in a simulation study. Estimates are finally compared in an application to a data set of tropical rain forest tree locations.

Les systèmes de réaction-diffusion croisée ont été introduits en dynamique des populations par Shigesada et al. pour tenir compte de l'influence des populations sur les taux de diffusion inter et intra spécifiques (on parle de diffusion croisée et auto-diffusion). Nous nous intéresserons à une gamme de systèmes généralisant ceux de Shigesada et. al. pour lesquels nous établirons un résultat d'existence de solutions faibles globales, en toute dimension d'espace. Ce résultat d'existence repose sur une fonctionnelle d'entropie " cachée " du système et sur un Lemme de dualité de Michel Pierre. On présentera également le schéma d'approximation permettant de conserver la positivité des solutions et les estimations précédentes.

I will present anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. They have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian and 'intermediate' infinitely divisible random fields. I will also give the asymptotic form of the covariance function of the random grain model and present some applications to superposed network traffic. The talk is based on a joint work with Donatas Surgailis (Vilnius University, Lithuania).

The extreme value theory is used by many authors to model exceedances in several fields such as hydrology, insurance, finance and environmental science, see Furlan (2010), Coles and Sparks (2006), Moscadelli (2004). Since Balkema-DeHaan (1974) and Pickands (1975), it is well known that the conditional distribution of any random variable over a high threshold has approximately a generalized Pareto distribution (GPD). However, the theory shows some surprises in practical applications. The goal is to find distributions so close to the Pareto distribution as determined by the data, but with greater flexibility in some sense. A new statistical approach to the issue is provided for non-light tails.

Je présenterai la problématique de l'approximation de la distribution invariante d'Equations aux Dérivés Partielles Stochastiques (EDPS), de type parabolique, semilinéaires, et des techniques d'analyse de l'erreur de discrétisation en temps et en espace. En utilisant des travaux récents concernant le cas des Equations Différentielles Stochastiques (EDS), je présenterai un nouveau schéma pour ces EDPS, qui augmente l'ordre de convergence par rapport à la discrétisation en temps. Enfin, je présenterai les challenges théoriques pour prouver ce résultat en toute généralité. L'exposé est basé sur des travaux en collaboration avec Gilles Vilmart (Genève).