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Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions

Yohance A. P. Osborne, Iain Smears

TL;DR

The paper extends stationary Mean Field Games to nondifferentiable Hamiltonians by formulating a PDE inclusion using the Moreau–Rockafellar subdifferential $\partial_pH$ and analyzing weak solutions under a Lasry–Lions–type monotonicity framework. A Kakutani fixed-point argument yields existence, while strict monotonicity of the coupling $F$ with $G\ge0$ ensures uniqueness of the weak solution. A monotone continuous Galerkin finite element method with artificial diffusion on strictly acute meshes is developed, and rigorous convergence of the discrete solutions to the continuous weak solution is established, including strong $H^1$-convergence of the value function $u$ and strong/weak convergence of the density $m$ under additional regularity assumptions. Numerical experiments on problems with nonsmooth solutions and small viscosity demonstrate the method’s ability to achieve near-optimal convergence rates and robustness in challenging regimes. Overall, the work provides a theoretically sound and practically effective framework for solving MFGs with bang-bang controls and nondifferentiable Hamiltonians via a PDI formulation and a provably convergent monotone FEM discretization.

Abstract

The formulation of Mean Field Games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. We develop the analysis and numerical analysis of stationary MFG for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We establish existence of a weak solution to the MFG PDI system, and we further prove uniqueness under a similar monotonicity condition to the one considered by Lasry and Lions. We then propose a monotone finite element discretization of the problem, and we prove strong $H^1$-norm convergence of the approximations to the value function and strong $L^q$-norm convergence of the approximations of the density function. We illustrate the performance of the numerical method in numerical experiments featuring nonsmooth solutions.

Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions

TL;DR

The paper extends stationary Mean Field Games to nondifferentiable Hamiltonians by formulating a PDE inclusion using the Moreau–Rockafellar subdifferential and analyzing weak solutions under a Lasry–Lions–type monotonicity framework. A Kakutani fixed-point argument yields existence, while strict monotonicity of the coupling with ensures uniqueness of the weak solution. A monotone continuous Galerkin finite element method with artificial diffusion on strictly acute meshes is developed, and rigorous convergence of the discrete solutions to the continuous weak solution is established, including strong -convergence of the value function and strong/weak convergence of the density under additional regularity assumptions. Numerical experiments on problems with nonsmooth solutions and small viscosity demonstrate the method’s ability to achieve near-optimal convergence rates and robustness in challenging regimes. Overall, the work provides a theoretically sound and practically effective framework for solving MFGs with bang-bang controls and nondifferentiable Hamiltonians via a PDI formulation and a provably convergent monotone FEM discretization.

Abstract

The formulation of Mean Field Games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. We develop the analysis and numerical analysis of stationary MFG for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We establish existence of a weak solution to the MFG PDI system, and we further prove uniqueness under a similar monotonicity condition to the one considered by Lasry and Lions. We then propose a monotone finite element discretization of the problem, and we prove strong -norm convergence of the approximations to the value function and strong -norm convergence of the approximations of the density function. We illustrate the performance of the numerical method in numerical experiments featuring nonsmooth solutions.
Paper Structure (25 sections, 15 theorems, 101 equations, 4 figures)

This paper contains 25 sections, 15 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Continuous Problem and Main Results
  4. Problem Statement
  5. Main Results
  6. An Example
  7. Analysis of the Continuous Problem
  8. Preliminary Results
  9. Existence of Weak Solutions
  10. Uniqueness of Weak Solutions
  11. Monotone Continuous Galerkin Finite Element Scheme
  12. Notation
  13. A Monotone Finite Element Method
  14. Main Results
  15. Analysis of Monotone Finite Element Scheme
  16. ...and 10 more sections

Key Result

Theorem 1

There exists a pair $(u,m)\in H_0^1(\Omega)\times H_0^1(\Omega)$ that is a weak solution of sys in the sense of Definition weakdef satisfying for some constants $C^*,C^{**}\geq 0$ depending only on $n$, $\Omega$, $\nu$, $\|b\|_{C(\overline{\Omega}\times\mathcal{A};\mathbb{R}^n)}$, $\kappa$ and $c_1$.

Figures (4)

  • Figure 1: First experiment -- convergence plots for approximations of the value function, density function, and transport vector. The rate of convergence for $H^1$-norms of the errors of the approximations of the value function is close to the optimal value of $1/2$, and the rate of convergence in the $H^1$-norm for the density function is of order $1$.
  • Figure 2: Second experiment -- approximate contour plot of $u$ (left), $m$ (right) computed on a fine mesh.
  • Figure 3: Second experiment -- convergence plots for approximations of the value function, density function, and transport vector. We observe a first-order rate of convergence for the $H^1$-norms of the error of the approximations of the value function, and a rate of order approximately $1/2$ for the $H^1$-norm errors of the approximations of the density function.
  • Figure 4: Third experiment -- the error $\lVert m-m_k \rVert_\Omega$ and $\lVert u-u_k \rVert_{H^1(\Omega)}$ for various values of the diffusion coefficient $\nu$. In the coarse-mesh regime $\nu\ll h_k$, the effective rate of convergence is of order $1/2$ for both $\lVert m-m_k \rVert_\Omega$ and $\lVert u-u_k \rVert_{H^1(\Omega)}$ as a result of the strong boundary layer in $m$ that appears in the singularly perturbed limit.

Theorems & Definitions (36)

  • Example 1
  • Definition 1
  • Definition 2: Weak Solution of \ref{['sys']}
  • Remark 1
  • Theorem 1: Existence of Weak Solutions
  • Theorem 2: Uniqueness of Weak Solutions
  • Remark 2
  • Example 2
  • Lemma 1
  • Remark 3
  • ...and 26 more