High-dimensional mathematical methods for LargE Agent and Particle systems

Introduction

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 stochastic differential equations, optimal transport problems, Fokker-Planck equations, or mean-field games systems.

Limitations of Current Models

However, all of them suffer from severe limitations when it comes to the simulation of high-dimensional problems. The high-dimensionality character comes either from:

1. The large number of particles or agents in the system.
2. The high amount of features of each agent or particle.
3. The huge quantity of parameters entering the model.

Project Objective

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.

Main Challenges

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 stochastic iterative algorithms.
• Efficient approximation of parametric stochastic differential equations, by means of reduced-order modeling approaches.

Potential Impacts

The potential impacts of the project are huge. Making possible such extreme-scale simulations will enable gaining 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
• Urban traffic