2025/07/02 by Heidi W. Ljósheim, Dante Kalise, Ljósheim, Heidi Wolles +5
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems
paper · pdf · doi:10.48550/arxiv.2507.01958
openalex publication_date 2025/07/02 · openalex created_date 2025/09/26 · openalex updated_date 2026/07/28
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.