A simple multiscale method for mean field games

On the Quadratic Convergence of Newton’s Method for Mean Field Games with Non-separable Hamiltonian

“…In this regard, a significant part of it is dedicated to the study of numerical methods and algorithms for the computation of the solution to the MFG model, both in the formulation as a PDEs system and as an optimal control problem of a PDE. Such approaches, just to mention a few, include finite differences, semi-Lagrangian methods and Fourier expansions with regard to the approximation methods (see [1,2,6,7,11,13,14,28,29,31,33,34]). Many of these methods exploit the variational structure of the problem, concerns the case in which the coupling term involving the distribution of the population is separated from the Hamiltonian, while relatively few works have been dedicated to the so-called non-separable case…”

Deep learning for Mean Field Games with non-separable Hamiltonians

“…In this regard, a significant part of it is dedicated to the study of numerical methods and algorithms for the computation of the solution to the MFG model, both in the formulation as a PDEs system and as an optimal control problem of a PDE. Such approaches, just to mention a few, include finite differences, semi-Lagrangian methods and Fourier expansions with regard to the approximation methods and policy iteration, Newton method, fictitious play, convex programming for the algorithms (see [1,2,6,5,9,11,12,26,27,29,31,32]). Most of the convergence results for numerical methods, which often exploit the variational structure of the problem, concerns the case in which the coupling term involving the distribution of the population is separated from the Hamiltonian, while relatively few works have been dedicated to the so-called non-separable case…”

“…In MFG theory, the Newton method has been applied, after a standard finite differences discretization developed in [1,2], to solve the resulting finite dimensional problem (see [6,3,26]). A distinction should be made between two applications of Newton method to MFGs: to compute an approximation of the solution to the fully nonlinear coupled system, as in our approach and [6,26], or only to solve the Hamilton-Jacobi equation at each iteration while using a fixed point iterations for the MFG system [26,29]. The numerical implementation of the Newton algorithm, as discussed in our paper, has been considered in [3,6,26].…”

Random Features for High-Dimensional Nonlocal Mean-Field Games

“…For related work on numerical methods for nonlocal MFG we refer to [40,41,42,43] for game theoretic approach, [44,45,46,47] for semi-Lagrangian schemes, [48] for deep learning approach, and [49] for a multiscale method. In all of these methods the nonlocal terms are discretized directly in the statespace.…”