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Finite element approximation of time-dependent mean field games with nondifferentiable Hamiltonians

Yohance A. P. Osborne, Iain Smears

TL;DR

This work develops a time-dependent mean field game framework with nondifferentiable Hamiltonians by formulating the system as a partial differential inclusion using the Hamiltonian subdifferential. It proposes a monotone finite element method with mass lumping and a novel volume-based stabilization that enforces a discrete maximum principle, enabling stable and convergent space-time discretizations without restrictive time-step constraints. The authors prove existence and uniqueness of weak solutions for the MFG-PDI and establish convergence of the discrete solutions to the continuous weak solution, with strong convergence for the value function and density under standard monotonicity assumptions. A numerical experiment corroborates the theoretical results, displaying expected convergence rates and strong transport-field convergence, highlighting practical applicability to MFGs with nondifferentiable Hamiltonians.

Abstract

The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonian. However in many cases, the structure of the underlying optimal problem leads to a convex but nondifferentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusion (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniqueness of weak solutions to the resulting MFG PDI system under standard assumptions in the literature. We propose a monotone stabilized finite element discretization of the problem, using conforming affine elements in space and an implicit Euler discretization in time with mass-lumping. We prove the strong convergence in $L^2(H^1)$ of the value function approximations, and strong convergence in $L^p(L^2)$ of the density function approximations, together with strong $L^2$-convergence of the value function approximations at the initial time.

Finite element approximation of time-dependent mean field games with nondifferentiable Hamiltonians

TL;DR

This work develops a time-dependent mean field game framework with nondifferentiable Hamiltonians by formulating the system as a partial differential inclusion using the Hamiltonian subdifferential. It proposes a monotone finite element method with mass lumping and a novel volume-based stabilization that enforces a discrete maximum principle, enabling stable and convergent space-time discretizations without restrictive time-step constraints. The authors prove existence and uniqueness of weak solutions for the MFG-PDI and establish convergence of the discrete solutions to the continuous weak solution, with strong convergence for the value function and density under standard monotonicity assumptions. A numerical experiment corroborates the theoretical results, displaying expected convergence rates and strong transport-field convergence, highlighting practical applicability to MFGs with nondifferentiable Hamiltonians.

Abstract

The standard formulation of the PDE system of Mean Field Games (MFG) requires the differentiability of the Hamiltonian. However in many cases, the structure of the underlying optimal problem leads to a convex but nondifferentiable Hamiltonian. For time-dependent MFG systems, we introduce a generalization of the problem as a Partial Differential Inclusion (PDI) by interpreting the derivative of the Hamiltonian in terms of the subdifferential set. In particular, we prove the existence and uniqueness of weak solutions to the resulting MFG PDI system under standard assumptions in the literature. We propose a monotone stabilized finite element discretization of the problem, using conforming affine elements in space and an implicit Euler discretization in time with mass-lumping. We prove the strong convergence in of the value function approximations, and strong convergence in of the density function approximations, together with strong -convergence of the value function approximations at the initial time.
Paper Structure (34 sections, 25 theorems, 133 equations, 1 figure)

This paper contains 34 sections, 25 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and notation
  3. Basic notation
  4. Subdifferential of the Hamiltonian
  5. Continuous Problem
  6. Notation and setting for discretization
  7. Meshes
  8. Notation for inequalities
  9. Sets of vertices and edges
  10. Spatial finite element spaces
  11. Mass lumping
  12. Spatial stabilization
  13. Space-time finite element spaces
  14. Reconstruction of time derivatives
  15. Discrete norms
  16. ...and 19 more sections

Key Result

Lemma 2.1

For every $v\in L^1(0,T;W^{1,1}(\Omega))$, the set $\mathcal{D}_p H[v]$ is a nonempty subset of $L^\infty(Q_T;\mathbb{R}^d)$ and

Figures (1)

  • Figure 1: Convergence plots for approximations of the value function, density function, and transport vector. Optimal rates of convergence are observed for the value function and density function errors in the $H^1$-norm.

Theorems & Definitions (46)

  • Example 2.1
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Definition 3.1: Weak Solution
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1: Existence of Solutions
  • Theorem 3.2: Uniqueness of Solutions
  • ...and 36 more