Parallel-in-Time Preconditioning for Time-Dependent Variational Mean Field Games

TL;DR

This paper introduces a parallel-in-time preconditioning method for efficiently solving time-dependent variational mean field game systems, improving scalability and robustness across different viscosities and geometries.

Contribution

It develops a novel parallel-in-time preconditioning technique using diagonalization and Fourier transforms, enhancing solver efficiency for mean field game systems.

Findings

Significant speedup in numerical experiments

Enhanced scalability with parallel computing

Robustness across various viscosities and geometries

Abstract

We study the numerical approximation of a time-dependent variational mean field game system with local couplings and either periodic or Neumann boundary conditions. Following a variational approach, we employ a finite difference discretization and solve the resulting finite-dimensional optimization problem using the Chambolle--Pock primal--dual algorithm. As this involves computing proximal operators and solving ill-conditioned linear systems at each iteration, we embed within our solver a general class of parallel-in-time preconditioners based on suitably-chosen diagonalization techniques, applied using discrete Fourier transforms. These enable efficient, scalable iterative solvers for each linear system, with robustness across a wide range of viscosities. We further develop fast solvers for the resulting ill-conditioned systems arising at each time step, using exact recursive schemes for structured grids while allowing for other geometries. Numerical experiments confirm the improved performance and parallel scalability of our approach.