Micro–macro modeling and scalable numerical methods for high-dimensional mean-field and control systems: structure-exploiting solvers and low-rank techniques

Résumé

In this thesis, we contribute to the study of micro-macro models in complex socio-economic systems, as well as to the investigation of the numerical challenges associated with solving mean-field systems. We then identify directions and propose strategies to address specific modeling and computational bottlenecks. In the first part, we develop a multi-class traffic framework that generalizes previous single-class micro-macro game approaches, linking mean-field game and finite-player differential game formulations. We introduce and study tailored numerical methods allowing large-scale simulations in space and time, including numerical linear algebra methods and high-performance computing techniques. We conduct extensive numerical experiments across diverse traffic scenarios, with a focus on autonomous vehicle behavior. We perform original computational comparisons between macroscopic and microscopic solutions, and validate the $\varepsilon$-Nash approximation with convergence rates better than theoretically expected. Finally, we analyze and discuss the numerical challenges and limitations encountered, and propose new directions for addressing them. In the second part, we focus on the most challenging of the two equations constituting the mean-field model, namely the Hamilton–Jacobi–Bellman (HJB) equation. We develop structure-exploiting methods, based on a Kronecker-product representation, to address the high-dimensional HJB equation. We show, using different linearization approaches, how the HJB equation can be reformulated to admit a separable structure. The well-posedness of the linearized HJB equation obtained via logarithmic transformation is proved. We construct two preliminary solvers: the first based on a block-in-time formulation and a parallel-in-time approach, and the second based on a low-rank reformulation and a fixed-point solver with adaptive damping. By evaluating the numerical performance and the rank evolution, we demonstrate that the proposed methods perform well for the one-dimensional linear-quadratic case. In the third part, we propose a new class of mean-field problems that we call mean-field optimal transport (MFOT), which represents a variation of mean-field control problems by drawing a connection with optimal transport theory. We propose and compare three neural network-based deep learning methods to solve high-dimensional MFOT problems. The first method relies on directly learning the control function that solves the MFOT problem. The second method solves a forward-backward PDE system, which is viewed as the optimality conditions for the MFOT problem. The third one uses an augmented Lagrangian approach to tackle the primal-dual formulation of the MFOT problem. We present and discuss numerical experiments to assess the effectiveness of the proposed approaches in dimensions up to five.

Citer ce document

Accès au document

Ce document est hébergé sur son dépôt institutionnel d'origine.

Auteur(s)

Statistiques

Consultations : 1

Téléchargements : 0