3,308 research outputs found
A fully-discrete Semi-Lagrangian scheme for a first order mean field game problem
In this work we propose a fully-discrete Semi-Lagrangian scheme for a {\it
first order mean field game system}. We prove that the resulting discretization
admits at least one solution and, in the scalar case, we prove a convergence
result for the scheme. Numerical simulations and examples are also discussed.Comment: 28 pages,16 figure
A Semi-Lagrangian scheme for a degenerate second order Mean Field Game system
In this paper we study a fully discrete Semi-Lagrangian approximation of a
second order Mean Field Game system, which can be degenerate. We prove that the
resulting scheme is well posed and, if the state dimension is equals to one, we
prove a convergence result. Some numerical simulations are provided, evidencing
the convergence of the approximation and also the difference between the
numerical results for the degenerate and non-degenerate cases.Comment: 21 pages, 8 figure
A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow
In this paper we present a Semi-Lagrangian scheme for a regularized version
of the Hughes model for pedestrian flow. Hughes originally proposed a coupled
nonlinear PDE system describing the evolution of a large pedestrian group
trying to exit a domain as fast as possible. The original model corresponds to
a system of a conservation law for the pedestrian density and an Eikonal
equation to determine the weighted distance to the exit. We consider this model
in presence of small diffusion and discuss the numerical analysis of the
proposed Semi-Lagrangian scheme. Furthermore we illustrate the effect of small
diffusion on the exit time with various numerical experiments
A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations
We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the
dependence of the coefficients is nonlinear and nonlocal in time with respect
to the unknowns. We extend the numerical scheme proposed and studied recently
by the authors for a single FPK equation of this type. We analyse the
convergence of the scheme and we study its applicability in two examples. The
first one concerns a population model involving two interacting species and the
second one concerns two populations Mean Field Games
The Hughes model for pedestrian dynamics and congestion modelling
In this paper we present a numerical study of some variations of the Hughes
model for pedestrian flow under different types of congestion effects. The
general model consists of a coupled non-linear PDE system involving an eikonal
equation and a first order conservation law, and it intends to approximate the
flow of a large pedestrian group aiming to reach a target as fast as possible,
while taking into account the congestion of the crowd.
We propose an efficient semi-Lagrangian scheme (SL) to approximate the
solution of the PDE system and we investigate the macroscopic effects of
different penalization functions modelling the congestion phenomena.Comment: 6 page
Mean Field Games models of segregation
This paper introduces and analyses some models in the framework of Mean Field
Games describing interactions between two populations motivated by the studies
on urban settlements and residential choice by Thomas Schelling. For static
games, a large population limit is proved. For differential games with noise,
the existence of solutions is established for the systems of partial
differential equations of Mean Field Game theory, in the stationary and in the
evolutive case. Numerical methods are proposed, with several simulations. In
the examples and in the numerical results, particular emphasis is put on the
phenomenon of segregation between the populations.Comment: 35 pages, 10 figure
Semi-Lagrangian schemes for mean field game models
In this work we consider first and second order Mean Field Games (MFGs) systems, introduced in \cite{LasryLions06i,LasryLions06ii,LasryLions07}. For the first order case, we recall a fully-discrete Semi-Lagrangian (SL) scheme introduced in \cite{CS12} and its main properties. We propose the natural extension of this scheme for the second order case and we present some numerical simulations
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