Approximation of stable solutions to a stationary MFG system

In this talk we consider stable solutions, in the sense of Briani-Cardaliaguet, to a stationary second order mean field game system with local coupling and quadratic Hamiltonian. We first provide sufficient conditions for the existence of such stable solutions and prove that they are isolated in some Banach space. After re-expressing solutions of the MFG system as zeros of a nonlinear map F we show that stable solutions have the property that the Fréchet differential of F at the solution is an isomorphism in a large class of Banach spaces. We then consider three applications of this fact. First we prove some error estimates for the finite element approximation of stable solutions. The finite element approximation of the stationary MFG system was studied by Osborne and Smears where the convergence of the method was proved without error estimates. We also obtain the convergence of Newton’s method both at continuous and discrete level. Finally we consider regular perturbations of the coupling and investigate the stability of solutions to the corresponding mean field game systems.

Conference on Numerical methods for optimal transport problems, mean field games, and multi-agent dynamics, Valparaiso, Chile

Universidad Técnica Federico Santa María