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High order Tensor-Train-Based Schemes for High-Dimensional Mean Field Games

Elisabetta Carlini, Luca Saluzzi

TL;DR

The paper tackles solving high-dimensional mean field games by combining a second-order semi-Lagrangian policy-iteration framework with Tensor-Train representations to overcome the curse of dimensionality. It develops a fully discrete method that couples Crank–Nicolson type SL time stepping for the forward FP and backward HJB equations with TT-based low-rank approximations of the solution components, enabling polynomial-in-dimension storage and compute. Three quadrature-based expectations—SL1, SL2e, and a polynomial-growth SL2p—achieve second-order accuracy via moment matching, with a log-exp transform helping preserve positivity of densities. Numerical experiments up to fairly high dimensions demonstrate the method’s convergence, mass and moment preservation, and substantial efficiency gains over grid-based approaches, highlighting its practical viability for high-dimensional MFGs and related control problems.

Abstract

We introduce a fully discrete scheme to solve a class of high-dimensional Mean Field Games systems. Our approach couples semi-Lagrangian (SL) time discretizations with Tensor-Train (TT) decompositions to tame the curse of dimensionality. By reformulating the classical Hamilton-Jacobi-Bellman and Fokker-Planck equations as a sequence of advection-diffusion-reaction subproblems within a smoothed policy iteration, we construct both first and second order in time SL schemes. The TT format and appropriate quadrature rules reduce storage and computational cost from exponential to polynomial in the dimension. Numerical experiments demonstrate that our TT-accelerated SL methods achieve their theoretical convergence rates, exhibit modest growth in memory usage and runtime with dimension, and significantly outperform grid-based SL in accuracy per CPU second.

High order Tensor-Train-Based Schemes for High-Dimensional Mean Field Games

TL;DR

The paper tackles solving high-dimensional mean field games by combining a second-order semi-Lagrangian policy-iteration framework with Tensor-Train representations to overcome the curse of dimensionality. It develops a fully discrete method that couples Crank–Nicolson type SL time stepping for the forward FP and backward HJB equations with TT-based low-rank approximations of the solution components, enabling polynomial-in-dimension storage and compute. Three quadrature-based expectations—SL1, SL2e, and a polynomial-growth SL2p—achieve second-order accuracy via moment matching, with a log-exp transform helping preserve positivity of densities. Numerical experiments up to fairly high dimensions demonstrate the method’s convergence, mass and moment preservation, and substantial efficiency gains over grid-based approaches, highlighting its practical viability for high-dimensional MFGs and related control problems.

Abstract

We introduce a fully discrete scheme to solve a class of high-dimensional Mean Field Games systems. Our approach couples semi-Lagrangian (SL) time discretizations with Tensor-Train (TT) decompositions to tame the curse of dimensionality. By reformulating the classical Hamilton-Jacobi-Bellman and Fokker-Planck equations as a sequence of advection-diffusion-reaction subproblems within a smoothed policy iteration, we construct both first and second order in time SL schemes. The TT format and appropriate quadrature rules reduce storage and computational cost from exponential to polynomial in the dimension. Numerical experiments demonstrate that our TT-accelerated SL methods achieve their theoretical convergence rates, exhibit modest growth in memory usage and runtime with dimension, and significantly outperform grid-based SL in accuracy per CPU second.
Paper Structure (12 sections, 2 theorems, 61 equations, 17 tables)

This paper contains 12 sections, 2 theorems, 61 equations, 17 tables.

Table of Contents

  1. Introduction
  2. Semi-Lagrangian scheme based on Policy Iteration Method
  3. Second order Semi-Lagrangian approximation in time
  4. Approximation of the expectation
  5. Tensor-Train Approximation
  6. Tensor-Train-Based Semi-Lagrangian Method
  7. Numerical test
  8. Advection-Diffusion Equation with Exact Solution
  9. Local MFG with exact solution
  10. Non-local MFG with exact solution
  11. Proof of Proposition \ref{['prop1']}
  12. Derivation of the Moment Matching System

Key Result

Proposition 3.1

Assume $d=1$. Let $T>0$, let $m$ be a smooth solution of eq: ADeq in $\mathbb{R}\times[0,T]$ with $q\in C^2(\mathbb{R};\mathbb{R})$, let $\Delta t>0$ and $k\in \mathcal{I}_{\Delta t}^*$. Given the scheme eq:semi_m, define the corresponding one-step error $\mathcal{T}_k(x):= m(x,t_{k+1})-\sum_{\ell \newlabelprop10

Theorems & Definitions (4)

  • Remark 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3