About Xianjin Yang Xianjin Yang Ph.D. Student, Applied Mathematics and Computational Science differential game theories mean-field games Xianjin Yang's research interests fall on mean-field games, which is an active research area in differential game theories. In his PhD period, Xianjin focuses on the homogenization and numerical methods for stationary mean-field games. Education and Early Career Xianjin Yang got his bachelor's degree in Software Engineering from Chongqing University of China in 2011. Then, he obtained his Master with majors in Computer Science from Zhejiang University of China in 2014. Later on, Xianjin Yang received his second Master's degree in Applied Mathematics from KAUST in 2016. He continues to study at Events Presented Events Jun 28 - Jul 4, 2020 Two-Scale Homogenization and Numerical Methods for Stationary Mean-Field Games Xianjin Yang, Ph.D. Student, Applied Mathematics and Computational Science Jul 1, 16:00 - 18:00 KAUST mean-field games 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 dissertation, we study the homogenization of a stationary first-order MFG and seek to find 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. Our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems. Moreover, we prove the existence and uniqueness of the solution to these limit problems. Next, we notice 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 develop algorithms to solve the homogenized problem, effective Hamiltonians, and Mather measures.
Two-Scale Homogenization and Numerical Methods for Stationary Mean-Field Games Xianjin Yang, Ph.D. Student, Applied Mathematics and Computational Science Jul 1, 16:00 - 18:00 KAUST mean-field games 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 dissertation, we study the homogenization of a stationary first-order MFG and seek to find 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. Our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems. Moreover, we prove the existence and uniqueness of the solution to these limit problems. Next, we notice 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 develop algorithms to solve the homogenized problem, effective Hamiltonians, and Mather measures.
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