# Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods

Handle URI:
http://hdl.handle.net/10754/625444
Title:
Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods
Authors:
Machado Velho, Roberto ( 0000-0003-0801-7934 )
Abstract:
In this dissertation, we present two research projects, namely finite-state mean-field games and the Hughes model for the motion of crowds. In the first part, we describe finite-state mean-field games and some applications to socio-economic sciences. Examples include paradigm shifts in the scientific community and the consumer choice behavior in a free market. The corresponding finite-state mean-field game models are hyperbolic systems of partial differential equations, for which we propose and validate a new numerical method. Next, we consider the dual formulation to two-state mean-field games, and we discuss numerical methods for these problems. We then depict different computational experiments, exhibiting a variety of behaviors, including shock formation, lack of invertibility, and monotonicity loss. We conclude the first part of this dissertation with an investigation of the shock structure for two-state problems. In the second part, we consider a model for the movement of crowds proposed by R. Hughes in [56] and describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an Eikonal equation with Dirichlet or Neumann data. We first establish a priori estimates for the solutions. Next, we consider radial solutions, and we identify a shock formation mechanism. Subsequently, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. We also propose a new numerical method for the solution of Fokker-Planck equations and then to systems of PDEs composed by a Fokker-Planck equation and a potential type equation. Finally, we illustrate the use of the numerical method both to the Hughes model and mean-field games. We also depict cases such as the evacuation of a room and the movement of persons around Kaaba (Saudi Arabia).
Gomes, Diogo A. ( 0000-0002-3129-3956 )
Committee Member:
Tempone, Raul Fidel ( 0000-0003-1967-4446 ) ; Sun, Shuyu ( 0000-0002-3078-864X ) ; Falcone, Maurizio
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Applied Mathematics and Computational Science
Issue Date:
10-Sep-2017
Type:
Dissertation
Appears in Collections:
Dissertations

DC FieldValue Language
dc.date.accessioned2017-09-11T12:50:16Z-
dc.date.available2017-09-11T12:50:16Z-
dc.date.issued2017-09-10-
dc.identifier.urihttp://hdl.handle.net/10754/625444-
dc.description.abstractIn this dissertation, we present two research projects, namely finite-state mean-field games and the Hughes model for the motion of crowds. In the first part, we describe finite-state mean-field games and some applications to socio-economic sciences. Examples include paradigm shifts in the scientific community and the consumer choice behavior in a free market. The corresponding finite-state mean-field game models are hyperbolic systems of partial differential equations, for which we propose and validate a new numerical method. Next, we consider the dual formulation to two-state mean-field games, and we discuss numerical methods for these problems. We then depict different computational experiments, exhibiting a variety of behaviors, including shock formation, lack of invertibility, and monotonicity loss. We conclude the first part of this dissertation with an investigation of the shock structure for two-state problems. In the second part, we consider a model for the movement of crowds proposed by R. Hughes in [56] and describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an Eikonal equation with Dirichlet or Neumann data. We first establish a priori estimates for the solutions. Next, we consider radial solutions, and we identify a shock formation mechanism. Subsequently, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. We also propose a new numerical method for the solution of Fokker-Planck equations and then to systems of PDEs composed by a Fokker-Planck equation and a potential type equation. Finally, we illustrate the use of the numerical method both to the Hughes model and mean-field games. We also depict cases such as the evacuation of a room and the movement of persons around Kaaba (Saudi Arabia).en
dc.language.isoenen
dc.subjectMean-field gamesen
dc.subjectCrowd motionen
dc.subjectFokker-Planck Equationen
dc.subjectNumerical Methodsen
dc.subjectHamilton-Jacobi equationsen
dc.titleFinite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methodsen
dc.typeDissertationen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen
dc.contributor.committeememberTempone, Raul Fidelen
dc.contributor.committeememberSun, Shuyuen
dc.contributor.committeememberFalcone, Maurizioen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameDoctor of Philosophyen
dc.person.id129109en