Two-Scale Homogenization and Numerical Methods for Stationary Mean-Field Games Xianjin Yang, Ph.D. Student, Applied Mathematics and Computational Sciences 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.
On mean-field game price models Diogo Gomes, Program Chair, Applied Mathematics and Computational Sciences Nov 7, 12:00 - 13:00 B9 L2 H1 R2322 mean-field games price formation modeling linear-quadratic problem optimal transport with constraints supply and demand balance Abstract In this talk, we discuss a mean-field game price formation model. This model describes a large number of rational agents that can trade a commodity with an exogenous supply. The price is determined by a balance condition between supply and demand. We discuss the well-posedness of the model, the uniqueness and regularity of the price function. Then, we examine two explicit models - the linear-quadratic problem and a model with finitely many agents. Time permitting, we will examine the connections between this problem and optimal transport with constraints. Brief Biography Diogo Gomes
Stationary Mean-Field Games with Congestion David Evangelista, Ph.D. Student, Applied Mathematics and Computational Sciences May 28, 14:00 - 15:00 B2 L5 R5220 mean-field games Mean-field games MFG are models of large populations of rational agents who seek to optimize an objective function that takes into account their state variables and the distribution of the state variable of the remaining agents. MFG with congestion model problems where the agents’ motion is hampered in high-density regions. First, we study radial solutions for first- and second-order stationary MFG with congestion on R^d. Next, we consider second-order stationary MFG with congestion and prove the existence of stationary solutions. Additionally, we study first-order stationary MFG with congestion with quadratic or power-like Hamiltonians.
Asymptotic properties of composites Rita A. Ferreira, Research Scientist, Mean-field Games and Nonlinear PDE Mar 14, 12:00 - 13:00 B9 L2 R2322 H1 gamma-convergence Asymptotic Performance Analysis In this talk, I will start with an overview of my research to date. Then, I will address in more detail the study of the asymptotic behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. More precisely, we consider materials arranged into periodically alternated thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from our modeling assumptions are of integral form, featuring linear growth and non-standard differential constraints.
