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On the convergence rates of generalized conditional gradient method for fully discretized Mean Field Games

Haruka Nakamura, Norikazu Saito

TL;DR

This work studies the convergence of a fully discrete generalized conditional gradient (GCG) method for Mean Field Games (MFG) by reformulating the problem via the Cole–Hopf transform into CH–GCG and applying finite-difference discretization. It proves a discrete maximum principle and derives explicit joint error bounds that couple discretization errors $( au, au_x)$ with iteration errors (in $k$), showing non-uniform iteration dependence but convergence as both mesh finiteness and iteration progress. Higher regularity of CH–GCG components yields improved discretization rates (up to first- or second-order in space-time under suitable conditions), while the iteration term decays polynomially in $k$, with the constants potentially growing with $k$. Numerical experiments in 1D and 2D corroborate the theoretical rates and demonstrate the practical impact of step-size rules on convergence, guiding parameter choices for efficient computation of fully discretized MFG solutions.

Abstract

We study convergence rates of the generalized conditional gradient (GCG) method applied to fully discretized Mean Field Games (MFG) systems. While explicit convergence rates of the GCG method have been established at the continuous PDE level, a rigorous analysis that simultaneously accounts for time-space discretization and iteration errors has been missing. In this work, we discretize the MFG system using finite difference method and analyze the resulting fully discrete GCG scheme. Under suitable structural assumptions on the Hamiltonian and coupling terms, we establish discrete maximum principles and derive explicit error estimates that quantify both discretization errors and iteration errors within a unified framework. Our estimates show how the convergence rates depend on the mesh sizes and the iteration number, and they reveal a non-uniform behavior with respect to the iteration. Moreover, we prove that higher convergence rates can be achieved under additional regularity assumptions on the solution. Numerical experiments are presented to illustrate the theoretical results and to confirm the predicted convergence behavior.

On the convergence rates of generalized conditional gradient method for fully discretized Mean Field Games

TL;DR

This work studies the convergence of a fully discrete generalized conditional gradient (GCG) method for Mean Field Games (MFG) by reformulating the problem via the Cole–Hopf transform into CH–GCG and applying finite-difference discretization. It proves a discrete maximum principle and derives explicit joint error bounds that couple discretization errors with iteration errors (in ), showing non-uniform iteration dependence but convergence as both mesh finiteness and iteration progress. Higher regularity of CH–GCG components yields improved discretization rates (up to first- or second-order in space-time under suitable conditions), while the iteration term decays polynomially in , with the constants potentially growing with . Numerical experiments in 1D and 2D corroborate the theoretical rates and demonstrate the practical impact of step-size rules on convergence, guiding parameter choices for efficient computation of fully discretized MFG solutions.

Abstract

We study convergence rates of the generalized conditional gradient (GCG) method applied to fully discretized Mean Field Games (MFG) systems. While explicit convergence rates of the GCG method have been established at the continuous PDE level, a rigorous analysis that simultaneously accounts for time-space discretization and iteration errors has been missing. In this work, we discretize the MFG system using finite difference method and analyze the resulting fully discrete GCG scheme. Under suitable structural assumptions on the Hamiltonian and coupling terms, we establish discrete maximum principles and derive explicit error estimates that quantify both discretization errors and iteration errors within a unified framework. Our estimates show how the convergence rates depend on the mesh sizes and the iteration number, and they reveal a non-uniform behavior with respect to the iteration. Moreover, we prove that higher convergence rates can be achieved under additional regularity assumptions on the solution. Numerical experiments are presented to illustrate the theoretical results and to confirm the predicted convergence behavior.
Paper Structure (10 sections, 17 theorems, 114 equations, 9 figures, 2 tables)

This paper contains 10 sections, 17 theorems, 114 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem settings
  3. The discrete GCG scheme and main results
  4. Preliminary results
  5. Proof of Theorem \ref{['I23_thm3.5']}
  6. Numerical Experiments
  7. One-dimensional Case
  8. Two-dimensional Case
  9. The case of the global coupling terms
  10. Concluding remarks

Key Result

Proposition 2.6

Suppose that (H), (f-B), (f-L), and (TIV) are satisfied. Then, the MFG system MFG_eq admits a unique solution $(\overline{u},\overline{m})\in W^{1,2,q}(Q)\times W^{1,2,q}(Q)$. Moreover, $\overline{\gamma}=f(\cdot,\cdot,\overline{m})\in\Gamma$. Furthermore, they are classical solutions in the sense t hold with some $\alpha\in (0,1)$.

Figures (9)

  • Figure 1: Left panel: density distribution; right panel: control input (Example \ref{['ex:1D']}).
  • Figure 2: Values of $I(\overline{M}^k, \overline{m})$ for different values of $\Delta x$ (red points) and the corresponding regression curve (Example \ref{['ex:1D']}).
  • Figure 3: Computed values of $I(\overline{M}^k, \overline{m})$ for different step sizes $\delta_k = k_2/(k + k_1)$ and iteration indices $k$ (red points), along with the corresponding regression curves (Example \ref{['ex:1D']}).
  • Figure 4: Left panel: density distribution; right panel: control input (Example \ref{['ex:1Db']}).
  • Figure 5: Values of $I(\overline{M}^k, \overline{m})$ for different values of $\Delta x$ (red points) and the corresponding regression curve (Example \ref{['ex:1Db']}).
  • ...and 4 more figures

Theorems & Definitions (45)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Remark 3.1
  • Remark 3.2
  • ...and 35 more