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.

翻译：本文提出了一种全离散格式用于求解一类高维平均场博弈系统。该方法将半拉格朗日时间离散化与张量列分解相结合，以克服维度灾难问题。通过将经典的Hamilton-Jacobi-Bellman方程和Fokker-Planck方程重构为平滑策略迭代框架内的一系列对流-扩散-反应子问题，我们构建了时间上一阶和二阶精度的半拉格朗日格式。张量列格式配合适当的数值积分规则，将存储和计算成本从维度的指数级降低至多项式级。数值实验表明：我们的张量列加速半拉格朗日方法达到了理论收敛阶；内存占用和运行时间随维度增长保持适度；在单位CPU时间精度方面显著优于基于网格的半拉格朗日方法。