mean-field games

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

-

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

-

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

-

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.

Stationary Mean-Field Games with Congestion

-

B2 L5 R5220

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.

On mean-field game price models

-

B9 L2 H1 R2322

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

Two-Scale Homogenization and Numerical Methods for Stationary Mean-Field Games

-

KAUST

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.

Mean Field Games: Numerical Methods and Applications

Mathieu Laurière, Postdoc, Department of Operations Research and Financial Engineering, Princeton University

-

KAUST

In this talk, several numerical methods will be presented and illustrated on examples. Borrowing tools from stochastic analysis, optimization, partial differential equations and machine learning, these methods enable us to solve mean field games with possibly complex sources of noise or high dimensional state variables.

Mean-Field Games on Networks and Wardrop Equilibria - Graduate Seminar

Ricardo De Lima Ribeiro, Research Specialist, CEMSE, KAUST

-

B9 L2 R2322

Models for flows on networks arise in the study of traffic and pedestrian crowds. These models encode congestion effects, the behavior and preferences of agents, such as aversion to crowds and their attempts to minimize travel time. We will present the Wardrop equilibrium model on networks with flow-dependent costs and its connection with stationary mean-field game.

Numerical approximations of Mean-Field-Games

-

B1 L2 R2202

In this thesis, we present three projects. First, we investigate the numerical approximation of Hamilton-Jacobi equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite-difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.

Mean Field Games price formation models

-

B1 L4 R4214

This thesis studies mean-field games (MFGs) models of price formation. The thesis focuses explicitly on a MFGs price formation model proposed by Gomes and Saúde. The thesis is divided into two parts. The first part examines the deterministic supply case, while the second part extends the model to incorporate a stochastic supply function. We explore different approaches, such as Aubry-Mather theory, to study the properties of the MFGs price formation model and alternative formulations using a convex variational problem with constraints. We propose machine-learning-based numerical methods to approximate the solution of the MFGs price formation model in the deterministic and stochastic setting.

Hessian-dependent Functionals: Mean Field Games, Free Boundaries and Improved Regularity

-

B2 L5 R5220

Hessian-dependent functionals play a pivotal role in a wide latitude of problems in mathematics. Arising in the context of differential geometry and probability theory, this class of problems find applications in the mechanics of deformable media (mostly in elasticity theory) and the modelling of slow viscous fluids. We study such functionals from three distinct perspectives.

KAUST Ph.D. students win best paper awards at mean-field games conferences

1 min read ·
By Sonia Turosienski KAUST Ph.D. students David Evangelista and Xianjin Yang, who are supervised by Diogo Gomes, professor of applied mathematics and computational science, won best paper awards at conferences this summer for their work in mean-field game (MFG) theory. Mean-field games model and seek to explain the behavior of rational agents in a competitive environment and have been used in diverse areas of research, including studies on non-renewable resources and mining models; opinion dynamics; price impact and order book modeling; and networks and energy management. Evangelista was

Peter Markowich elected Fellow of the European Academy of Sciences

1 min read ·
KAUST Professor Peter Markowich has been elected as a Fellow of the European Academy of Sciences (EurASc). Established in 2003, the EurASC is a fully independent international organization of distinguished scholars that aims to recognize and elect to its membership the best European researchers to strengthen European science and scientific cooperation.

Professor Markowich elected to the European Academy of Sciences and Arts

1 min read ·
KAUST Distinguished Professor of Applied Mathematics and Computational Science Dr. Peter Markowich has been named a member of the European Academy of Sciences and Arts (EASA). He will officially be inducted as a member at the organization's annual inauguration ceremony in early 2023.

Xianjin Yang

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

Peter Markowich

Al-Khawarzmi Distinguished Professor, Applied Mathematics and Computational Science
A distinguished professor of applied mathematics and computational science, Dr. Markowich is the author of over 200 publications in leading international journals and circa 14,000 citations.

Julian David Gutierrez Pineda

Julian received both B.S. in Mathematics and B.S. in Engineering from Escuela Colombiana de Ingenieria, Colombia. After receiving his M.Sc. in Mathematical Sciences from Universidad Nacional Autónoma de México, México, he joined Professor Diogo Gomes as a Ph.D. student in applied mathematics at King Abdullah University of Science and Technology, Saudi Arabia. His research interests are primarily optimal control, optimal transport, calculus of variations, partial differential equations, and mean-field games.

Diogo Gomes

Professor Diogo Gomes, a distinguished expert in Applied Mathematics and Computational Science at KAUST, leverages advanced mathematics – including partial differential equations, numerical methods and mean-field game models – to solve complex problems in social sciences, economics and finance

David Evangelista

David Evangelista is a Ph.D. Candidate at King Abdullah University of Science and Technology (KAUST), working in the areas of partial differential equations, mean-field games (MFG), and market microstructure. He is expect to receive a Ph.D. from KAUST in June 2019, under the direction of Professor Diogo A. Gomes. David's research interests are in optimal control theory, nonlinear partial differential equations, MFG, and market microstructure. MFG is a framework to study interactions of a large number of indistinguishable players that play differential games. MFG has become an active research