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Yves Achdou

Publications and source records attributed to Yves Achdou.

30 records · Page 2Linked to original sources
arXiv2017

Homogenization of a transmission problem with Hamilton-Jacobi equations and a two-scale interface. Effective transmission conditions

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of $ε$ while the period is of the order of $ε$ 2. As $ε$ $\rightarrow$ 0, the interfaces tend to a straight line $Γ$. We study the asymptotic behavior of the value function as $ε$ $\rightarrow$ 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $Γ$, with an effective transmission condition on $Γ$ keeping track of the oscillations.

math.AP↗
arXiv2017

Mean field games with congestion

We consider a class of systems of time dependent partial differential equations which arise in mean field type models with congestion. The systems couple a backward viscous Hamilton-Jacobi equation and a forward Kolmogorov equation both posed in $(0,T)\times (\mathbb{R}^N /\mathbb{Z}^N)$. Because of congestion and by contrast with simpler cases, the latter system can never be seen as the optimality conditions of an optimal control problem driven by a partial differential equation. The Hamiltonian vanishes as the density tends to $+\infty$ and may not even be defined in the regions where the density is zero. After giving a suitable definition of weak solutions, we prove the existence and uniqueness results of the latter under rather general assumptions. No restriction is made on the horizon $T$.

math.AP↗
arXiv2016

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.

math.AP↗
arXiv2016

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.

math.AP↗
arXiv2016

A transmission problem across a fractal self-similar interface

We consider a transmission problem in which the interior domain has infinitely ramified structures. Transmission between the interior and exterior domains occurs only at the fractal component of the interface between the interior and exterior domains. We also consider the sequence of the transmission problems in which the interior domain is obtained by stopping the self-similar construction after a finite number of steps; the transmission condition is then posed on a prefractal approximation of the fractal interface. We prove the convergence in the sense of Mosco of the energy forms associated with these problems to the energy form of the limit problem. In particular, this implies the convergence of the solutions of the approximated problems to the solution of the problem with fractal interface. The proof relies in particular on an extension property. Emphasis is put on the geometry of the ramified domain. The convergence result is obtained when the fractal interface has no self-contact, and in a particular geometry with self-contacts, for which an extension result is proved.

math.AP↗
arXiv2015

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 $Γ_ε$. The oscillations of the interface have small period and amplitude, both of the order of $ε$, and the interfaces $Γ_ε$ tend to a straight line $Γ$. We study the asymptotic behavior as $ε\to 0$. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $Γ$, with an effective transmission condition on $Γ$ keeping track of the oscillations of $Γ_ε$.

math.AP↗
arXiv2015

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 +$\infty$, 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.

math.NA↗
arXiv2015

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.

math.AP↗
arXiv2015

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.

math.AP↗
arXiv2012

Mean field games: convergence of a finite difference method

Mean field type models describing the limiting behavior, as the number of players tends to $+\infty$, 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 assumptions

math.NA↗
arXiv2009

Homogenization of first order equations with $u/ε$-periodic Hamiltonian: Rate of convergence as $ε\to 0$ and numerical approximation of the effective Hamiltonian

We consider homogenization problems for first order Hamilton-Jacobi equations with $u^ε/ε$ periodic dependence, recently introduced by C. Imbert and R. Monneau, and also studied by G. Barles: this unusual dependence leads to a nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter $ε$ tends to 0. We obtain the same rates as those obtained by I. Capuzzo Dolcetta and H. Ishii for the more usual homogenization problems without the dependence in $u^ε/ε$. In a second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.

math.AP↗