Statistique, Analyse et Modélisation Multidisciplinaire

This SE aims at addressing a number of challenging mathematical problems related to stochastic interacting systems, with particular emphasis on the regularity properties of the solutions, their limiting behaviour and numerical computation. The equations we analyse arise from the modelisation of real-world phenomena in several fields of application, including spiking neural systems, hydrodynamics and financial/energy markets, and share nonlinearity as a common underlying trait. Critically, nonlinearity intertwines with other relevant features that include: low regularity of the coefficients, noise degeneracy, jump-diffusion dynamics, and high-dimensionality. The study of stochastic interacting systems is highly multidisciplinary from a two-fold perspective. On one hand, they have become a widespread modelling tool in a variety of applications. For example, they are used to model human neuron interfaces, particle systems, but also interacting agents in economics and finance, in relation to managing risk and decentralised production of renewable energy. On the other hand, the set of mathematical and computational tools needed to reach a holistic understanding of stochastic systems is very vast: ranging from stochastic (partial) differential equations, random measures, rough paths, gradient flows in metric measure spaces, numerical probability and computer simulation. We provide a team of experts that analyse stochastic systems integrating several approaches and techniques. The complementary expertise across the network, together with the consolidated experience and excellence of the key participants in their research areas, places our network in the privileged position to make relevant contributions across interconnected research fields, and to contribute to the training of the early career researchers involved in the project in an exciting field of pure and applied mathematics, with the possibility of boosting their careers in both academic and non-academic sectors.

Nous proposons un projet ANR dédié à l’étude des problématiques modernes en optimisation de forme, qui recèle à l’heure actuelle de multiples défis. L'objectif principal est de fédérer la communauté des chercheurs intéressés par cette discipline afin de développer de méthodes et outils originaux permettant d’aborder de nouveaux modèles ou considérer de nouvelles contraintes. Nous souhaitons concentrer nos efforts autour de quatre grandes thématiques : (1) l’étude de nouveaux modèles impliquant des EDP non-linéaires ainsi que des contraintes topologiques sur les domaines. (2) Le développement de méthodes permettant d’établir des inégalités géométriques quantitatives. (3) La prise en comte d’incertitudes dans les problèmes inverses impliquant des formes. (4) Le développement d’approches numériques originales permettant d’éviter l’utilisation de maillages coûteux. Le consortium a été composé de façon à mettre l’accent à la fois sur les aspects fondamentaux, applicatifs et numériques, qui sont totalement complémentaires et interconnectés dans ce cadre.