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1. Machado Velho, Roberto. Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods.

Degree: 2017, King Abdullah University of Science and Technology

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).

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…

Advisors/Committee Members: Gomes, Diogo A., Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, Tempone, Raul Fidel, Sun, Shuyu, Falcone, Maurizio.

Subjects/Keywords: Mean-field games; Crowd motion; Fokker-Planck Equation; Numerical Methods; Hamilton-Jacobi equations

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APA (6th Edition):

Machado Velho, R. (2017). Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods. (Thesis). King Abdullah University of Science and Technology. Retrieved from http://hdl.handle.net/10754/625444

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Machado Velho, Roberto. “Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods.” 2017. Thesis, King Abdullah University of Science and Technology. Accessed November 23, 2017. http://hdl.handle.net/10754/625444.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Machado Velho, Roberto. “Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods.” 2017. Web. 23 Nov 2017.

Vancouver:

Machado Velho R. Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods. [Internet] [Thesis]. King Abdullah University of Science and Technology; 2017. [cited 2017 Nov 23]. Available from: http://hdl.handle.net/10754/625444.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Machado Velho R. Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods. [Thesis]. King Abdullah University of Science and Technology; 2017. Available from: http://hdl.handle.net/10754/625444

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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