Titre : Optimal discretization for high contrast multi-layered media: it’s (all) about the boundaries

Résumé : Controlling wave propagation in multilayered media with high contrast (aka where material properties exhibit strong variations) is of particular importance in optical cloaking, where multiple concentric thin material layers are stacked to conceal an object. Such large variations in wavenumbers induce strong reflections and complex field behavior within the medium, making the design of such multi-layered structures an active area of research. In this work, we propose to use boundary integral equations to model the problem. These methods are highly efficient, as they only require the evaluation of the field at the material interfaces. However, in multilayered media with high contrasts, several challenges arise: (i) the methods suffer from the so-called close evaluation problem, and (ii) the need for adaptive methods to optimize computational costs. We introduce a modified boundary integral formulation that addresses the close-evaluation problem, and an adaptive strategy that ensures uniform accuracy while maintaining computational efficiency across such media. Numerical experiments on a variety of multilayer configurations demonstrate the scalability and robustness of the proposed approach. This work is carried out in collaboration with Stéphanie Chaillat (ENSTA Paris), Elsie Cortes, and Chrysoula Tsogka (UC Merced).

Titre : Analysis of the anisotropic elastic tensor recovery problem from internal data

Résumé : Elastography is a medical imaging technique that exploits variations in tissue stiffness to detect early disease. Using an approach called the Reverse Weak Formulation, we investigate the stability of this reconstruction under minimal data assumptions. Our plan is to transform the whole problem into a first-order linear system and obtain a closed range property and Lipschitz stability estimates. We further characterize the operator's null space through the introduction of the notion of conservative tensor fields. Finally, through a suitable finite elements discretization, we demonstrate stable reconstructions for both isotropic and anisotropic media under static and time-harmonic excitations.

Titre : Numerical resolution of the JKO scheme for a crowd motion model

Résumé : The JKO scheme is an implicit Euler scheme for the time discretization of equations that have a gradient flow structure in the Wasserstein space. Recently, Santambrogio and Hraivoronska proved the convergence of the fully discrete JKO scheme (with space discretization on a grid), for specific energies. Among those energies, the crowd motion case is particularly tractable since the upper bound on the density allows to rewrite the JKO step as a linear programming problem on a graph, which can be solved using the network simplex algorithm, as in classical discrete optimal transport problems. The aim of this talk is to present this convergence result for the crowd motion equation along with new numerical simulations exploiting its linear programming structure.

Titre : Modelling relaxation experiments

Résumé : We model here some experiments conducted by our collaborators on relaxation. This is an experiment showing that even if you sort the cells according to their stemness, you will recover the initial repartition after some time. We studied for this coupled transport PDE structured by protein concentration.

Titre : The dissipation rate of internal functionals along the JKO scheme

Résumé : The Jordan–Kinderlehrer–Otto (JKO) scheme provides a time-discrete approximation of Wasserstein gradient flows. A natural question is whether the quantitative properties established for the continuous gradient flow can also be recovered at the discrete level. In this talk, we focus on the dissipation rate of an internal functional G along the JKO scheme. The computation is easy to obtain in the gradient flow level, but if one tries to perform a similar computation along the iterations of the JKO scheme, the so-called flow interchange technique is needed, and this requires geodesic convexity of the functional G. We will show an approach that removes this assumption by approximating the functional G with \lambda-geodesically convex functional. Under some regularity conditions, the same estimate is built up to an error term that vanishes as the JKO scheme converges to the continuous gradient flow. This is a joint work with Filippo Santambrogio.

Titre : Persistance et vitesse d'une impulsion de coopération lorsque les coopérateurs sont plus mobiles que les tricheurs.

Résumé : On considère la dynamique spatiale d'une population structurée par un trait régissant à la fois la vitesse de diffusion et le niveau de coopération, et soumise à la compétition. Le taux de reproduction croit avec le niveau local moyen de coopération, tandis qu'il décroît avec le niveau individuel de coopération, de telle sorte qu'une population spatialement homogène ne peut croître du fait du caractère rapidement prépondérant des tricheurs. Des simulations numériques indiquent qu'une impulsion («pulse», onde localisée) peut se propager, avec une population de coopérateurs croissant à l'avant, tandis que les tricheurs ne deviennent dominant et n'entraîne la décroissance de la population à l'arrière. Afin de prédire pour quelles valeurs des paramètres ce phénomène peut produire, ainsi que la vitesse de propagation, nous étudions, dans le cadre d'un modèle déterministe, la relation de dispersion et la limite faible diffusion, qui est une équation d'Hamilton-Jacobi avec un Hamiltonien non convexe. Les impulsions observées ne peuvent correspondre à des solutions de viscosité de l'équation limite. Cela nous amène à proposer une nouvelle heuristique, qui nous permet de trouver la vitesse effectivement observée. Afin de la justifier, nous étudions l'effet d'une perturbation sur une solution de type exponentielle pour le modèle sans compétition.

Titre : A kernel based approximation for constrained optimal smoothing problem

Résumé : If the constrained optimal smoothing problem is posed in a Reproducing Kernel Hilbert Space, there is a new way of approaching it by defining a sequence of finite dimensional Reproducing Kernel Hilbert Spaces and a particular manner to discretize the problem. We present this method of approximation and prove its convergence to the exact solution. This problem can be reformulated as a Bayesian estimation problem involving a Gaussian process related to the kernel of the RKHS. Our approximation corresponds to its Maximum A Posteriori. We propose an error bounds for the convergence of a piecewise linear approximation of the constrained optimal smoothing problem which depends of the grid size, the regularity of the kernel, and the distance from the kernel interpolant of the approximation to the set of constraints.

References :

[1] Aronszajn, N. (1950), Theory of Reproducing Kernels, Transactions of the American Mathematical Society.
[2] Bay, X., Grammont, L., and Maatouk, H. (2017), A New Method for Interpolating in a Convex Subset of a Hilbert Space, Computational Optimization and Applications.
[3] Berlinet, A., and Thomas-Agnan, C. (2011), Reproducing Kernel Hilbert Spaces in Probability and Statistics, Springer Science and Business Media.
[4] Grammont, L. , Maatouk, H. and Bay, X. (2024) Equivalence between constrained optimal smoothing and Bayesian estimation Journal of nonparametric statistics.
[5] Grammont, L. Bachoc, F. and Lopez Lopera, A. (2025) Error Bounds for a Kernel-Based Constrained Optimal Smoothing Approximation, SIAM/ASA Journal on Uncertainty Quantification.
[6] Schwartz, L. (1964) Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associ´es (Noyaux reproduisants), Journal d’Analyse Mathématique.