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.

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.

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.

Natural gradient descent (NGD) can be seen as a preconditioned update where parameter changes are driven by a functional perspective. In a spirit similar to Newton’s method, the NGD update uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. Although its assemblage and inversion is prohibitively expensive in the context of large machine learning models, it becomes not only feasible but necessary when we look at scientific machine learning problems using models not requiring as many parameters. Sill, both gradient and natural gradient descent will get stuck at any local minima. Furthermore, when the approximation class is a non-linear manifold (i.e. neural networks) or the loss function is other than L² distance (KL-divergence for a classification problem, PDE residual as in physics informed learning) even the natural gradient might yield non-optimal directions at each step. The talk will focus on how we can tackle these situations by introducing a Natural version of classical inertial dynamic methods like Nestorov or Heavy-ball.

The term "slender" refers to structures with a very high ratio between their longitudinal lenght and their transverse dimensions, typically, a cylinder with an height significantly larger than its radius. Because of this particular geometry, many models have been developed to provide a simplified description of the kinematics and dynamics of the structure. A standard approach in this context is to account for the distribution of forces and deformations only along the centerline. Consequently, the velocity fields and equilibrium equations of the structure are described in a one-dimensional (1D) setting. However, when a slender structure is immersed in a three-dimensional (3D) fluid, enforcing kinematic and dynamic coupling conditions on a 1D domain requires the introduction of a double trace operator (codimension 2) which demands regularity for the solution within the fluid domain, a condition which is generally not satisfied a priori. In this talk, I will introduce and analyse a new mathematically sound approach for modelling and solving 3D-1D fluid-structure interaction problems. The main idea is to combine a fictitious domain approach with the projection of the kinematic constraint onto a finite-dimensional space defined along the structure's centerline. The discrete formulation is based on the finite element method and a semi-implicit treatment of the Dirichlet-Neumann coupling conditions, employing a partitioned procedure for the resolution of the fluid-structure interaction problem.

Dans cette présentation, nous nous intéresserons à deux méthodes numériques indépen- dantes permettant d’accélérer et d’améliorer la résolution des EDP.

Récemment, ParaOpt, un algorithme parallèle en temps a été proposé pour résoudre les systèmes découlant de problèmes de contrôle optimal associés à des équations différentielles partielles (EDP). Cet algorithme combine la résolution des problèmes d’évolution forward/backward avec la boucle d’optimisation. Sa convergence et la précision de l’approximation numérique dépendent du schéma de pas de temps utilisé pour discrétiser le problème. Une première analyse de convergence a été présentée dans M.J. Gander, F. Kwok et J. Salomon SISC (2020) dans le cas restreint de la méthode implicite d’Euler pour les problèmes de contrôle optimal linéaire-quadratique impliquant des systèmes dissipatifs. Dans cette présentation, nous généraliserons ce résultat au cas où une méthode de Runge-Kutta est utilisée pour discrétiser le problème de contrôle optimal. Nous expliquons en particulier comment la discrétisation de la fonctionnelle doit être adaptée au schéma considéré et comment des conditions supplémentaires s’ajoutent aux conditions d’ordre habituel des méthodes de Runge-Kutta pour obtenir l’ordre de convergence attendu. Aphynity [Yuan Yin et al., J. Stat. Mech. (2021) 124012] est une approche consistant à ajouter un terme correctif sous la forme d’un réseau neurone à une EDP afin de prévoir avec précision l’évolution d’un système et d’identifier correctement ses paramètres physiques. L’apprentissage est alors effectué dans la boucle externe du solveur considéré, de sorte que l’entraînement se fait indirectement par le biais d’un schéma en temps. Nous décrirons l’influence du schéma choisi sur le réseau résultant et montrerons, dans des cas particuliers, que l’ordre d’approximation des termes corrigés correspond à l’ordre du schéma utilisé. Nous expliquerons également comment optimiser l’apprentissage par une approche de type Richardson.

In this talk, I will present several recent results, in collaboration with Nicolas Burq, on the control theory of Schrödinger propagators on tori. Our goal is to address the following conjecture: on a torus of arbitrary dimension, Schrödinger propagators with bounded potentials are observable, and therefore controllable, from arbitrary space-time domains of positive Lebesgue measure.

Using a scheme that combines (1) approximation of rough functions by continuous functions, (2) the cluster structure of lattice points near paraboloids, and (3) mathematical induction on the dimension, we reduce the conjecture to certain integrability bounds for linear Schrödinger waves. These bounds are weaker than Bourgain’s conjectured periodic Strichartz estimates but remain nontrivial. In particular, our criteria imply the observability conjecture for the one-dimensional torus.

Applications of our results include Cantor--Lebesgue type theorems and uniform nonvanishing estimates for quantum limits.