Séminaire de mathématiques appliquées (archives)

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 :

Gloaguen, Pierre, Marie-Pierre Etienne, and Sylvain Le Corff. "Stochastic differential equation based on a multimodal potential to model movement data in ecology." Journal of the Royal Statistical Society Series C: Applied Statistics 67.3 (2018): 599-619. Patin, Rémi, et al. "Identifying stationary phases in multivariate time series for highlighting behavioural modes and home range settlements." Journal of Animal Ecology 89.1 (2020): 44-56.

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.

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

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. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel-Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of this approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

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. Numerical experiments are given to show the potential of the method.

Adipose cells or adipocytes are the specialized cells composing the adipose tissue in a variety of species.Their role is the storage of energy in the form of a lipid droplet inside their membrane. Based on the amount of lipid they contain, one can consider the distribution of adipocyte per amount of lipid and observe a peculiar feature : the resulting distribution is bimodal, thus having two local maxima. The aim of this talk is to introduce a model built from the work in Soula & al. (2013) that is able to reproduce this bimodal feature using a Lifshitz-Slyozov model. Additionally we present some result on this model and its relation to the Becker-Döring model. We can show that under some assumptions the later converges to the former and by looking at higher order term we can build an extended diffusive Lifshitz-Slyozov model which better describes the dynamics of adipose cells. I will also present some probabilistic insight into this convergence and some numerical simulations

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.

We study the approximation of a square integrable function from a finite number of its evaluations at well-chosen nodes. The function is assumed to belong to a reproducing kernel Hilbert space (RKHS), the approximation to be one of a few natural finite-dimensional approximations, and the nodes to be drawn from one of two probability distributions. Both distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in L2 norm. We show that determinantal point processes and mixtures thereof can yield fast rates, and that they shed some light on how the rate changes as more smoothness is assumed, a phenomenon known as superconvergence. Besides, determinantal sampling generalizes i.i.d. sampling from the Christoffel function, a standard in the literature. In particular, determinantal sampling guarantees the so-called instance optimality property for a smaller number of function evaluations than i.i.d. sampling.

Référence : https://hal.science/hal-04181079/