[Paper Review] Splitting methods for a class of non-potential mean field games

Siting Liu, Levon Nurbekyan|arXiv (Cornell University)|Jun 30, 2020

Numerical methods in inverse problems22 references4 citations

TL;DR

This paper introduces a novel primal-dual splitting method for non-potential mean field games (MFGs) with mixed local and nonlocal couplings by formulating the system as a monotone inclusion problem. By representing nonlocal couplings in Fourier or feature spaces via orthogonal projections, the method achieves dimension reduction, enables modular and parallelizable algorithm design, and ensures convergence via convex analysis, extending PDHG-type solvers beyond potential MFGs.

ABSTRACT

We extend the methods from Nurbekyan, Saude "Fourier approximation methods for first-order nonlocal mean-field games" [Port. Math. 75 (2018), no. 3-4] and Liu, Jacobs, Li, Nurbekyan, Osher "Computational methods for nonlocal mean field games with applications" [arXiv:2004.12210] to a class of non-potential mean-field game (MFG) systems with mixed couplings. Up to now, splitting methods have been applied to potential MFG systems that can be cast as convex-concave saddle-point problems. Here, we show that a class of non-potential MFG can be cast as primal-dual pairs of monotone inclusions and solved via extensions of convex optimization algorithms such as the primal-dual hybrid gradient (PDHG) algorithm. A critical feature of our approach is in considering dual variables of nonlocal couplings in Fourier or feature spaces.

Motivation & Objective

• To develop a computational framework for non-potential mean field game systems with both local and nonlocal interactions, which generalizes existing splitting methods beyond potential MFGs.
• To overcome high memory and computational costs of traditional methods by projecting nonlocal kernels onto low-dimensional feature spaces using orthogonal bases.
• To enable efficient solution of MFG systems via monotone inclusion formulations with dual variables in Fourier space, ensuring algorithmic modularity and parallelism.
• To provide convergence guarantees through convex duality and monotone operator theory, extending the applicability of PDHG-type algorithms to non-potential settings.

Proposed method

• The method formulates the non-potential MFG system as a primal-dual monotone inclusion problem by introducing dual variables for each interaction term, including local and nonlocal couplings.
• Nonlocal couplings are represented via orthogonal projections of interaction kernels $K$ and $S$ onto a finite-dimensional subspace spanned by basis functions $\{\zeta_i\}$, reducing the dimensionality of the problem.
• Dual variables for nonlocal terms are defined in Fourier or feature space, enabling efficient computation via fast transforms and avoiding full matrix-vector products on fine grids.
• The algorithm uses a primal-dual hybrid gradient (PDHG) scheme applied to the monotone inclusion, with updates decoupled across different interaction types for parallelization.
• The formulation leverages convex duality and Legendre transforms to derive equivalent inclusion conditions, ensuring convergence under monotonicity assumptions.
• The method is modular: adding new local or nonlocal interactions requires only minimal modifications to the algorithmic structure.

Experimental results

Research questions

• RQ1 Can splitting methods like PDHG be extended to non-potential MFG systems with mixed local and nonlocal couplings?
• RQ2 How can nonlocal interactions be represented efficiently to reduce memory and computational costs in MFG solvers?
• RQ3 What is the role of dual variables in Fourier or feature space in enabling scalable and modular solution schemes for non-potential MFGs?
• RQ4 Can convergence guarantees be preserved when extending PDHG to non-potential settings via monotone inclusion formulations?
• RQ5 How does the use of orthogonal projections of kernels $K$ and $S$ affect the accuracy and stability of the resulting MFG solution?

Key findings

• The proposed method successfully extends PDHG-type solvers to non-potential MFG systems by formulating them as monotone inclusion problems with dual variables in Fourier space.
• Nonlocal couplings are approximated via projections onto a low-dimensional basis, achieving significant dimension reduction without sacrificing algorithmic modularity.
• The algorithm is highly modular and parallelizable, as updates for dual variables corresponding to different interaction terms are decoupled.
• Convergence is guaranteed by the maximal monotonicity of the operators involved, derived from convex duality and Legendre transform properties.
• The method enables efficient computation of nonlocal terms via fast transforms, avoiding expensive matrix-vector products on fine grids.
• The approach is general and can be applied to a wide class of non-potential MFGs with arbitrary combinations of local and nonlocal interactions.