Université de Montréal

Part 3: Variational structures and Fourier approximation techniques for mean-field game systems

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B1 L3 R3119

Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems.

Part 1: Variational structures and Fourier approximation techniques for mean-field game systems

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B1 L3 R3119

Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems. Day 1: I will show how to transform suitable mean-field game (MFG) systems into infinite-dimensional convex optimization problems. Furthermore, I will present Uzawa’s algorithm and augmented Lagrangian approach for solving convex optimization problems. Finally, I will demonstrate how to apply these methods to approximate solutions of corresponding MFG systems.

Part 2: Variational structures and Fourier approximation techniques for mean-field game systems

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B1 L3 R3119

Mean-field game (MFG) systems of partial differential equations (PDE) arise in modeling huge populations of indistinguishable rational agents that play non-cooperative differential games. Mathematically, an MFG system comprises of a Hamilton-Jacobi-Bellman PDE coupled with a Kolmogorov-Fokker-Planck PDE in a highly nonlinear fashion. Hence, theoretical and numerical treatments of MFG systems are highly challenging problems. Day 2: I will present new Fourier approximation techniques for nonlocal MFG systems and discuss how these techniques fit the variational framework presented on Day 1. Then I will introduce the Chambolle-Pock primal-dual hybrid gradient optimization method and apply a variant of this method to approximate nonlocal MFG models via Fourier approximation techniques above.