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Monotone inclusion methods for a class of second-order non-potential mean-field games

Levon Nurbekyan, Siting Liu, Yat Tin Chow

TL;DR

The paper develops a monotone-splitting, PDHG-based method for second-order non-potential mean-field games by reformulating a semi-implicit finite-difference discretization as a pair of primal–dual monotone inclusions. A discretized Hamiltonian $H_h$ is crafted to preserve Lasry–Lions monotonicity, and a discrete energy $J$ built from the Legendre transform $L_h$ underpins a variational structure that yields first-order optimality conditions and KKT reformulations. By preconditioning and inner-product reweighting, the authors achieve grid-size independent time steps and establish convergence for the PDHG iterations, with the fourth-order space–time solve for the primal update rendered tractable via FFTs. Numerical experiments on a 2D torus congestion MFG demonstrate grid-independent performance and robustness to vanishing viscosity and small congestion, highlighting the method’s practical viability beyond potential MFGs. Overall, the work provides a scalable framework for solving non-potential MFGs with provable convergence guarantees and effective numerical behavior.

Abstract

We propose a monotone splitting algorithm for solving a class of second-order non-potential mean-field games. Following [Achdou, Capuzzo-Dolcetta, "Mean Field Games: Numerical Methods," SINUM (2010)], we introduce a finite-difference scheme and observe that the scheme represents first-order optimality conditions for a primal-dual pair of monotone inclusions. Based on this observation, we prove that the finite-difference system obtains a solution that can be provably recovered by an extension of the celebrated primal-dual hybrid gradient (PDHG) algorithm.

Monotone inclusion methods for a class of second-order non-potential mean-field games

TL;DR

The paper develops a monotone-splitting, PDHG-based method for second-order non-potential mean-field games by reformulating a semi-implicit finite-difference discretization as a pair of primal–dual monotone inclusions. A discretized Hamiltonian is crafted to preserve Lasry–Lions monotonicity, and a discrete energy built from the Legendre transform underpins a variational structure that yields first-order optimality conditions and KKT reformulations. By preconditioning and inner-product reweighting, the authors achieve grid-size independent time steps and establish convergence for the PDHG iterations, with the fourth-order space–time solve for the primal update rendered tractable via FFTs. Numerical experiments on a 2D torus congestion MFG demonstrate grid-independent performance and robustness to vanishing viscosity and small congestion, highlighting the method’s practical viability beyond potential MFGs. Overall, the work provides a scalable framework for solving non-potential MFGs with provable convergence guarantees and effective numerical behavior.

Abstract

We propose a monotone splitting algorithm for solving a class of second-order non-potential mean-field games. Following [Achdou, Capuzzo-Dolcetta, "Mean Field Games: Numerical Methods," SINUM (2010)], we introduce a finite-difference scheme and observe that the scheme represents first-order optimality conditions for a primal-dual pair of monotone inclusions. Based on this observation, we prove that the finite-difference system obtains a solution that can be provably recovered by an extension of the celebrated primal-dual hybrid gradient (PDHG) algorithm.
Paper Structure (14 sections, 8 theorems, 89 equations, 4 figures)

This paper contains 14 sections, 8 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Numerical analysis
  3. A finite-difference scheme
  4. A discrete energy
  5. A congestion model
  6. Properties of the discrete energy
  7. First-order optimality conditions
  8. A pair of primal-dual monotone inclusions
  9. Grid-size independent time-steps
  10. Further remarks on the algorithm
  11. Numerical Experiments
  12. Convergence of the algorithm
  13. Viscosity Effect
  14. Congestion Effect

Key Result

Lemma 2.2

\newlabellma:L_h0 Assume that $H_h$ is given by eq:H_h. Then for every $(t,x)\in \mathbb{R}\times \mathbb{T}^d$ and $\eta\geq 0$ we have that where $\beta'=\frac{\beta}{\beta-1}$. Furthermore, we have that where $w=(w_1,w_2,w_3,w_4)$.

Figures (4)

  • Figure 1: This figure shows the density evolution of $\rho(x,t)$ in the MFG model with congestion parameters $\alpha = 1, \beta = 2, \epsilon = 0.1$ and viscosity $\nu=0.1$, as detailed in \ref{['sec:num_eg1']}. Moving from left to right, at time points $t = 0, 0.25, 0.5, 0.75, 1$, the density starts in the form of $4$ Gaussians and eventually concentrates at points $(0.5, 0.25)$ and $(0.5, 0.75)$.
  • Figure 2: We record the number of iterations our algorithm requires to reach specific error thresholds, $\varepsilon$, across various mesh densities, along with the computational time needed (in seconds). Since the algorithm is not optimized for parallel execution, the computational time increases with the refinement of the mesh.
  • Figure 3: This figure shows the numerical results of density evolution of congestion model with $\nu = 0, 0.02, 0.1$ (from left to right): $\rho(0.5,x)$ (top) and $\rho(1,x)$ (bottom).
  • Figure 4: This figure shows the numerical results of density evolution of congestion model with $\epsilon = 0, 0.02, 0.1, 1, 5$ (from left to right): $\rho(0.5,x)$ (top) and $\rho(1,x)$ (bottom).

Theorems & Definitions (20)

  • Remark 2.1
  • Lemma 2.2
  • Proof 1
  • Lemma 2.3
  • Proof 2
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • Proof 3: Proof of \ref{['lma:J_min_eta']}
  • Lemma 2.7
  • ...and 10 more