On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local couplings

L Briceno-Arias (1) , D Kalise (1) , Z Kobeissi (2) , M Laurière (3, 4) , A Mateos Gonzalez (5, 6) , Francisco José Silva (7)

1 UTFSM - Universidad Tecnica Federico Santa Maria [Valparaiso]
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 ORFE - Department of Operations Research and Financial Engineering
4 Princeton University
5 IMAG - Institut Montpelliérain Alexander Grothendieck
6 UMR ISEM - Institut des Sciences de l'Evolution de Montpellier
7 XLIM-MATHIS - Mathématiques & Sécurité de l'information

We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [14] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [3]. In order to solve the finite dimensional variational problems, in [14] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [20], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [14] by suitable preconditioned iterative algorithms.

Domaines: Optimisation et contrôle [math.OC]

L Briceno-Arias, D Kalise, Z Kobeissi, M Laurière, A Mateos Gonzalez, et al.. On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local couplings. ESAIM: Proceedings and Surveys, 2019, 65, pp.330-348. ⟨10.1051/proc/201965330⟩. ⟨hal-01865488⟩