Séminaire de mathématiques appliquées

During an epidemic outbreak, decision makers crucially need accurate and robust tools to monitor the pathogen propagation. The effective reproduction number, defined as the expected number of secondary infections stemming from one contaminated individual, is a state of the art indicator quantifying the epidemic intensity. Numerous estimators have been developed to precisely track the reproduction number temporal evolution. Yet, COVID 19 pandemic surveillance raised unprecedented challenges due to the poor quality of worldwide reported infection counts. When monitoring the epidemic in different territories simultaneously, leveraging the spatial structure of data significantly enhances both the accuracy and robustness of reproduction number estimates. However, this requires a good estimate of the spatial structure.

Dans cette présentation, nous discutons de résultats asymptotiques, en particulier de variation régulière et de déviations larges, pour certaines classes de processus en clusters de Poisson et pour quelques fonctionnelles associées. L’accent est mis sur le cas du processus de Hawkes (marqué), qui sert de fil conducteur et d’exemple central. Nous illustrons ensuite la pertinence des hypothèses du cadre théorique par une application à des données sismologiques suisses. Il s’agit d’une travail conjoint avec Valérie Chavez-Demoulin et Olivier Wintenberger.

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.

State-Space Models (SSMs) are deterministic or stochastic dynamical systems defined by two processes. The state process, which is not observed directly, models the transformation of the states over time. On another hand, the observation process produces the observables on which model fitting and prediction are based. Ecology frequently uses stochastic SSMs to represent the imperfectly observed dynamics of population sizes or animal movement. However, several simulation-based evaluations of model performance suggest broad identifiability issues in ecological SSMs. Formal SSM identifiability is typically investigated using exhaustive summaries, which are simplified representations of the model. The theory on exhaustive summaries is largely based on continuous-time deterministic modelling and those for ecological SSMs have developed by analogy. While the discreteness of time does not constitute a challenge, finding a good exhaustive summary for a stochastic SSM is more difficult. The strategy adopted so far has been to create exhaustive summaries based on a transfer function of the expectations of the stochastic process. However, this evaluation of identifiability does not allow to take into account the possible dependency between the variance parameters and the process parameters. We show that the output spectral density plays a key role in stochastic SSM identifiability assessment. This allows us to define a new suitable exhaustive summary. Using several ecological examples, we show that usual ecological models are often theoretically identifiable, suggesting that most SSM estimation problems are due to practical rather than theoretical identifiability issues.

Dans cet exposé, on s’intéresse à une stratégie numérique pour reproduire le comportement d’un globule rouge se mouvant dans un milieu fluide et soumis à un stimulus électrique. Pour cela, on s’appuie sur un modèle mécanique dans lequel la cellule est décrite comme une fine membrane élastique séparant le contenu liquide de la cellule de la matrice extérieure. Le défi numérique de ce système est surtout représenté par des conditions de transmission imposées à travers une interface évolutive dans le temps, ainsi que par la manipulation de champs avec des discontinuités à l’interface. La stratégie qu’on propose est basée sur la génération de maillages qui suivent le profil de l’interface, réalisés en découpant un maillage en arrière-plan. L’adoption de maillages adaptés permet de traiter simplement les conditions d’interface, mais l’algorithme de découpage engendre des éléments finis qui ne sont pas des simplexes. Pour faire face à cela, on propose des schémas supportant des maillages génériques.

En particulier, on adopte une méthode de type Hybrid High Order pour résoudre un problème de Stokes avec des conditions de transmission à l’interface, et on introduit un nouveau schéma de type Discrete De Rham pour résoudre le problème elliptique avec interface associé à la variable électrique. Pour ce dernier, on présente un résultat de stabilité ainsi qu’une estimation d’erreur a priori.

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.

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.

TBA

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.