High order Tensor-Train-Based Schemes for High-Dimensional Mean Field Games
This work addresses computational challenges in high-dimensional Mean Field Games, which model large-scale agent interactions in fields like economics and robotics, though it appears incremental as it builds on existing semi-Lagrangian and Tensor-Train techniques.
The authors tackled high-dimensional Mean Field Games by developing a fully discrete scheme that combines semi-Lagrangian time discretizations with Tensor-Train decompositions to overcome the curse of dimensionality. Their method achieved theoretical convergence rates, showed modest growth in memory and runtime with dimension, and significantly outperformed grid-based approaches in accuracy per CPU second.
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.