Two-scale Homogenization and Numerical Methods for Stationary Mean-field Games
数学专题报告
报告题目(Title)：Two-scale Homogenization and Numerical Methods for Stationary Mean-field Games
报告人(Speaker)：杨先津 博士 （清华大学）
地点(Place)：后主楼 1124
时间(Time)：2021 年 09 月 17 日（周五） 15:00--16:00
邀请人(Inviter)：苏喜锋
报告摘要
Mean-field games (MFGs) study the behavior of rational and indistinguishable agents in a large population. Agents seek to minimize their cost based upon statistical information on the population’s distribution. In this talk, we first give a brief introduction of the theory of MFGs. Then, we introduce the homogenization of a stationary first-order MFG and a numerical method to solve the homogenized problem. More precisely, we characterize the asymptotic behavior of a first-order stationary MFG with a periodically oscillating potential. The main tool is the two-scale convergence. Using this convergence, we show the two-scale homogenized and the homogenized MFG problems. Moreover, we give the proof of the existence and uniqueness of the solution to these limit problems. Next, we show that the homogenized problem resembles the problem involving effective Hamiltonians and Mather measures, which arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, and Aubry–Mather theory.
Thus, we present algorithms to solve the homogenized problem, the effective Hamiltonians, and Mather measures. We introduce the Hessian Riemannian flow and give the convergence proof of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.
主讲人简介
Xianjin Yang is currently a PostDoc researcher in Tsinghua University and BIMSA (beijing institute of mathematical sciences and applications). He received his Ph.D. degree from KAUST (King Abdullah University of Science and Technology) under the supervision of Diogo A. Gomes. His research interests are in Mean-field Games and Optimization.
