Computational Trade-Offs Between Newton and Picard Solvers for Mean Field Game PDE Systems

Abstract: We study computational trade-offs between two solvers for the same semi-implicit finite-difference discretization of forward-backward partial differential equation (PDE) systems arising in mean field games (MFGs). The Picard method uses an outer fixed-point iteration that alternates a forward Fokker-Planck solve and a backward Hamilton-Jacobi-Bellman solve. The Newton method instead applies Newton's method directly to the coupled nonlinear space-time system. Across one- and two-dimensional MFG benchmarks, we find that the Picard method has much lower wall-clock cost when it converges, but may fail at sufficiently low viscosity and may require strong damping under temporal shocks. With parameter continuation, the Newton method is more robust in these regimes, at the cost of larger coupled linear systems. We relate these trade-offs to the residuals, Jacobian blocks, and sparsity structures produced by separable, local nonseparable, and nonlocal Hamiltonians. We also test a hybrid continuation strategy on a two-dimensional double-well MFG benchmark, using Picard iterations for the inexpensive part of the viscosity descent before switching to Newton continuation.