The Starting Grant ERC project HighLEAP (High-dimensional mathematical methods for LargE Agent and Particle systems) was awarded to Virginie Ehrlacher. The project will last five years from 1st December 2023.

Interacting particle or agent-based systems are ubiquitous in science. They arise in an extremely wide variety of applications including materials science, biology, economics and social sciences. Several mathematical models exist to account for the evolution of such systems at different scales, among which stand optimal transport problems, Fokker-Planck equations, mean-field games systems or stochastic differential equations. However, all of them suffer from severe limitations when it comes to the simulation of high-dimensional problems, the high-dimensionality character coming either from the large number of particles or agents in the system, the high amount of features of each agent or particle, or the huge quantity of parameters entering the model. The objective of this project is to provide a new mathematical framework for the development and analysis of efficient and accurate numerical methods for the simulation of high-dimensional particle or agent systems, stemming from applications in materials science and stochastic game theory.

The main challenges which will be addressed in this project are:

-sparse optimization problems for multi-marginal optimal transport problems, using moment constraints;

-numerical resolution of high-dimensional partial differential equations, with randomized iterative algorithms;

-efficient approximation of parametric stochastic differential equations, by means of reduced-order modeling approaches.

The potential impacts of the project are huge: making possible such extreme-scale simulations will enable to gain precious insights on the predictive power of agent- or particle-based models, with applications in various fields, such as quantum chemistry, molecular dynamics, crowd motion or urban traffic.

Molecular simulation is one of the most dynamic areas of scientific computing. Its field of application is very broad, ranging from theoretical chemistry and drug design to materials science and nanotechnology. Its importance in modern science has been acknowledged by two Nobel Prizes (Kohn & Pople in 1998; Karplus, Levitt & Warshel in 2013). It is also a gold mine of exciting problems for mathematicians and computer scientists.

EMC2 is an ERC Synergy project that aims to overcome some of the current limitations in the field of molecular simulation and aims to provide academic communities and industrial companies with new generation, dramatically faster and quantitatively reliable molecular simulation software. This will enable those communities to address major technological and societal challenges of the 21st century in health, energy and the environment for instance.

Les grandes thématiques de la Chaire ont été établies à partir des discussions qu’ont eues les enseignants-chercheurs et les cadres des Partenaires.

• Méthodes numériques pour les risques financiers
• Régulation, marchés
• Big data, smart data, détection d’anomalies et de ruptures

Elles tiennent compte des évolutions majeures ayant eu lieu dans les marchés financiers durant ces années de crise financières, tant que sur le plan de la modélisation que sur celui de l’implémentation numérique.

Les résultats de ces recherches doivent d’être d’un niveau académique d’excellence et donner lieu à des publications internationales dans les journaux de premier rang du domaine. Parallèlement, un effort pédagogique de valorisation de la recherche sera mis en oeuvre vers les praticiens et les étudiants.

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The operations research and machine learning chair is a long term academic-industrial partnership between Air France (the main French airline) and École des Ponts ParisTech (one of the leading French Grandes Écoles). Theoretical contributions and applications to industrial problems are in the scope of the chair. Research focuses on the following themes:

• Operations Research and Combinatorial Optimization.
• Machine Learning.
• Stochastic Combinatorial Optimization.

The coordinators of the chair are Pierre-Cyril Aubin and Axel Parmentier.

Scientific aim: Computing equilibrium properties in computational statistical physics can be performed efficiently using various variance reduction techniques. On the other hand, the simulation of nonequilibrium properties remains limited by the space-time scales which can be achieved by brute force numerical methods. In particular, the estimators of transport coefficients obtained from the linear response of steady-state averages of nonequilibrium stochastic dynamics are plagued by very large variances and/or biases. The aim of this project is to provide new tools for the mathematical and numerical analysis of nonequilibrium stochastic dynamics. The techniques employed lie at the interface of various subfields of mathematics, ranging from probability theory and the study of stochastic processes to functional analysis and the theory of partial differential equations; with an emphasis on the numerical analysis of Monte Carlo methods and time discretization of stochastic differential equations.

Keywords: probability, numerical analysis, partial differential equations, molecular dynamics, Monte Carlo methods, variance reduction methods.

Aims of the project: The main objectives of this project are to improve, implement and mathematically analyze sampling and optimisation methods based on interacting particle systems. The work we propose to undertake pertains to two particular classes of methods: consensus-based methods inspired by particle swarm optimisation, and ensemble Kalman-based methods, which were recently revealed to have a close connection to interacting Langevin diffusions. These methods have proven to be successful in variety of applications, including posterior sampling and maximum a posteriori estimation in the context of Bayesian inverse problems, as well as the training of large neural networks.