106 research outputs found
Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games
Mean field type models describing the limiting behavior of stochastic
differential games as the number of players tends to +, have been
recently introduced by J-M. Lasry and P-L. Lions. Under suitable assumptions,
they lead to a system of two coupled partial differential equations, a forward
Bellman equation and a backward Fokker-Planck equations. Finite difference
schemes for the approximation of such systems have been proposed in previous
works. Here, we prove the convergence of these schemes towards a weak solution
of the system of partial differential equations
Mean field type control with congestion
We analyze some systems of partial differential equations arising in the
theory of mean field type control with congestion effects. We look for weak
solutions. Our main result is the existence and uniqueness of suitably defined
weak solutions, which are characterized as the optima of two optimal control
problems in duality
On the system of partial differential equations arising in mean field type control
We discuss the system of Fokker-Planck and Hamilton-Jacobi-Bellman equations
arising from the finite horizon control of McKean-Vlasov dynamics. We give
examples of existence and uniqueness results. Finally, we propose some simple
models for the motion of pedestrians and report about numerical simulations in
which we compare mean filed games and mean field type control
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
Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface
We consider a family of optimal control problems in the plane with dynamics
and running costs possibly discontinuous across an oscillatory interface
. The oscillations of the interface have small period and
amplitude, both of the order of , and the interfaces
tend to a straight line . We study the asymptotic
behavior as . We prove that the value function tends to the
solution of Hamilton-Jacobi equations in the two half-planes limited by
, with an effective transmission condition on keeping track of
the oscillations of
Mean field games: convergence of a finite difference method
Mean field type models describing the limiting behavior, as the number of
players tends to , of stochastic differential game problems, have been
recently introduced by J-M. Lasry and P-L. Lions. Numerical methods for the
approximation of the stationary and evolutive versions of such models have been
proposed by the authors in previous works . Convergence theorems for these
methods are proved under various assumption
Neumann conditions on fractal boundaries
We consider some elliptic boundary value problems in a self-similar ramified domain of ℝ2 with a fractal boundary with Laplace's equation and nonhomogeneous Neumann boundary conditions. The Hausdorff dimension of the fractal boundary is greater than one. The goal is twofold: first rigorously define the boundary value problems, second approximate the restriction of the solutions to subdomains obtained by stopping the geometric construction after a finite number of steps. For the first task, a key step is the definition of a trace operator. For the second task, a multiscale strategy based on transparent boundary conditions and on a wavelet expansion of the Neumann datum is proposed, following an idea contained in a previous work by the same authors. Error estimates are given and numerical results are presented
Mean Field Type Control with Congestion (II): An Augmented Lagrangian Method
This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential equations arising from the model were discussed and existence and uniqueness were proved. Here, the focus is put on numerical methods: a monotone finite difference scheme is proposed and shown to have a variational interpretation. Then an Alternating Direction Method of Multipliers for solving the variational problem is addressed. It is based on an augmented Lagrangian. Two kinds of boundary conditions are considered: periodic conditions and more realistic boundary conditions associated to state constrained problems. Various test cases and numerical results are presented
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