Lagrange Galerkin schemes for Mean Field Games

Speaker:  Elisabetta Carlini - Universit√† La Sapienza, Roma
  Thursday, April 27, 2023 at 3:30 PM Sala Verde (solo in presenza)
Abstract: We propose a numerical approximation of a mean-field game system with nonlocal couplings.
The system is characterized by a backward Hamilton-Jacobi-Bellman equation coupled with a forward Fokker-Planck equation. The approximation is constructed by combining Lagrange-Galerkin techniques, for the FP equation, with semi-Lagrangian techniques, for the HJB equation. The resulting discrete system is solved using fixed-point iterations.
We show that the scheme is conservative, consistent and stable for large time steps with respect to spatial steps. In the case of first-order MFG, we prove a convergence theorem for the exactly integrated Lagrange-Galerkin scheme in arbitrary spatial dimensions. In the case of second-order MFG, we construct an accurate high-order scheme. We propose an implementable version with inexact integration and finally show some numerical simulations.

Programme Director
Giacomo Albi

Publication date
April 19, 2023