Numerical methods for the mean-field game and its inverse problems

Abstract

Mean-field games (MFG) study the behavior of a large number of rational agents in a non-cooperative game. An important task in mean-field games is to study the flow of all the agents in the state space and to understand the behavior of mean-field Nash equilibrium. It has wide applications in various fields. But it is not easy to solve the mean-field game numerically because of its complicated structure. And conventional studies on mean-field games focus on Euclidean spaces, which limits the application. In addition, while many real-life applications can be cast as an inverse mean-field game problem, there are not enough studies in this direction. In this thesis, we attempt to solve the aforementioned problems related to mean-field games. The theme is numerical methods for the mean-field game in Euclidean spaces, on manifolds, and for the inverse mean-field game.In the first chapter, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently. By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid becomes finer. Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies. We conduct comprehensive numerical experiments to confirm the convergence analysis of the proposed algorithm, to show its efficiency and mass preservation property by comparing it with state-of-the-art methods, and to illustrate its flexibility for handling various mean-field variational problems. In the second chapter, we explore the mean-field games on Riemannian manifolds. We formulate the mean-field game Nash Equilibrium on manifolds. We also establish the equivalence between the PDE system and the optimality conditions of the associated variational form on manifolds. Based on the triangular mesh representation of two-dimensional manifolds, we design a proximal gradient method for variational mean-field games. Our comprehensive numerical experiments on various manifolds illustrate the effectiveness and flexibility of the proposed model and numerical methods. In the third chapter, we propose a bilevel optimization formulation for inverse mean-field games and study the numerical methods for solving the bilevel problem. With the bilevel formulation, we preserve the convexity of the objective function and the linearity of the constraint in the forward problem. This formulation permits us to solve the problem with a gradient-based optimization algorithm and to have a nice convergence of the algorithm. We focus on inverse mean-field games with unknown obstacles and unknown metrics and implement our alternating gradient algorithm to solve the inverse problems. For the inverse mean-field game with unknown obstacles, we also establish the local unique identifiability result and verify the result with numerical experiments.