Publications (83)54.56 Total impact
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ABSTRACT: We propose a numerical method for stationary Mean Field Games defined on a network. In this framework a correct approximation of the transition conditions at the vertices plays a crucial role. We prove existence, uniqueness and convergence of the scheme and we also propose a least squares method for the solution of the discrete system. Numerical experiments are carried out. 
Research: On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators
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DESCRIPTION: We present a f�nite di�fference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators. The principal eigenvalue is computed by solving a �finitedimensional nonlinear minmax optimization problem. We prove the convergence of the method and we discussits implementation. Some examples where the exact solution is explicitly known show the eff�ectiveness of the method. 
Article: On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators
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ABSTRACT: We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators. The principal eigenvalue is computed by solving a finitedimensional nonlinear minmax optimization problem. We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method.  [Show abstract] [Hide abstract]
ABSTRACT: We consider a stationary Mean Field Games system defined on a network. In this framework, the transition conditions at the vertices play a crucial role: the ones here considered are based on the optimal control interpretation of the problem. We prove separately the wellposedness for each of the two equations composing the system. Finally, we prove existence and uniqueness of the solution of the Mean Field Games system.  [Show abstract] [Hide abstract]
ABSTRACT: We study feedback interconnections of two nonlinear systems, that are asymptotically stable at a fixed point. It is shown that if the subsystems are inputtostate stable and the corresponding gains satisfy a small gain condition, then estimates for the domain of attraction of the whole system may be obtained by calculating robust Lyapunov functions for the subsystems. The latter task can be solved using available Zubov techniques. In total this approach makes numerical computations feasible, as high cost computations only have to be performed in lower dimensions. This comes at the price, that in general only lower approximations of the domain of attraction are obtained and that the system has to be brought into a form, where a small gain condition holds.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the Cauchy problem\[\partial_t u+H(x,Du)=0 \quad (x,t)\in\Gamma \times (0,T),\quad u(x,0)=u_0(x)\; x\in\Gamma \]where $\Gamma$ is a network and $H$ is a convex and positive homogeneous Hamiltonian. We introduce a definition of viscosity solution and we prove that the unique viscosity solution of the problem is given by a HopfLax type formula. In the second part of the paper we study flame propagation in a network and we seek an optimal strategy to block a fire breaking up in some part of a pipeline.  [Show abstract] [Hide abstract]
ABSTRACT: We study the eikonal equation on the Sierpinski gasket in the spirit of the construction of the Laplacian in Kigami [8]: we consider graph eikonal equations on the prefractals and we show that the solutions of these problems converge to a function defined on the fractal set. We characterize this limit function as the unique metric viscosity solution to the eikonal equation on the Sierpinski gasket according to the definition introduced in [3].Mathematische Annalen 04/2014; DOI:10.1007/s0020801512517 · 1.13 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In [14], Gueant, Lasry and Lions considered the model problem ``What time does meeting start?'' as a prototype for a general class of optimization problems with a continuum of players, called Mean Field Games problems. In this paper we consider a similar model, but with the dynamics of the agents defined on a network. We discuss appropriate transition conditions at the vertices which give a well posed problem and we present some numerical results.Discrete and Continuous Dynamical Systems 02/2014; 35(9). DOI:10.3934/dcds.2015.35.4173 · 0.83 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The equivalence between logarithmic Sobolev inequalities and hypercontractivity of solutions of HamiltonJacobi equations has been proved in [5]. We consider a semiLagrangian approximation scheme for the HamiltonJacobi equation and we prove that the solution of the discrete problem satisfies a hypercontractivity estimate. We apply this property to obtain an error estimate of the set where the truncation error is concentrated.  [Show abstract] [Hide abstract]
ABSTRACT: We consider continuousstate and continuoustime control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We define a notion of constrained viscosity solution of Hamilton–Jacobi equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained viscosity solution of the Hamilton–Jacobi equation on the network.Nonlinear Differential Equations and Applications NoDEA 06/2013; 20(3). DOI:10.1007/s0003001201581 · 0.90 Impact Factor 
Article: A comparison among various notions of viscosity solution for HamiltonJacobi equations on networks
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ABSTRACT: Three definitions of viscosity solutions for HamiltonJacobi equations on networks recently appeared in literature ([1,4,6]). Being motivated by various applications, they appear to be considerably different. Aim of this note is to establish their equivalence.Journal of Mathematical Analysis and Applications 01/2013; 407(1). DOI:10.1016/j.jmaa.2013.05.015 · 1.12 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: For a HamiltonJacobi equation defined on a network, we introduce its vanishing viscosity approximation. The elliptic equation is given on the edges and coupled with Kirchhofftype conditions at the transition vertices. We prove that there exists exactly one solution of this elliptic approximation and mainly that, as the viscosity vanishes, it converges to the unique solution of the original problem.Journal of Differential Equations 07/2012; 254(10). DOI:10.1016/j.jde.2013.02.013 · 1.68 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 JM. Lasry and PL. 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 assumptionsSIAM Journal on Numerical Analysis 07/2012; 51(5). DOI:10.1137/120882421 · 1.79 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider a model first order mean field game problem, introduced by J. M. Lasry and P. L. Lions [C. R., Math., Acad. Sci. Paris 343, No. 9, 619–625 (2006; Zbl 1153.91009)]. Its solution (v,m) can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [loc. cit.]). We propose a semidiscrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to (v,m) when the discretization parameter tends to zero.Networks and Heterogeneous Media 06/2012; 2(2). DOI:10.3934/nhm.2012.7.263 · 0.65 Impact Factor  Networks and Heterogeneous Media 06/2012; 7(2):III. DOI:10.3934/nhm.2012.7.2i · 0.65 Impact Factor

Article: Eikonal equations on ramified spaces
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ABSTRACT: We generalize the results in [16] to higher dimensional ramified spaces. For this purpose we introduce ramified manifolds and, as special cases, locally elementary polygonal ramified spaces (LEP spaces). On LEP spaces we develop a theory of viscosity solutions for HamiltonJacobi equations, providing existence and uniqueness results.Interfaces and Free Boundaries 01/2012; 15(1). DOI:10.4171/IFB/297 · 0.65 Impact Factor 
Article: A semidiscrete in time approximation for a model 1 st orderfinite horizon mean field game problem
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ABSTRACT: In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [17]. Its solution (v, m) can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [17]). We propose a semidiscrete in time approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to (v, m) when the discretization parameter tends to zero. Keywords. First order mean field game, semidiscrete in time approximation.  [Show abstract] [Hide abstract]
ABSTRACT: Mean field games describe the asymptotic behavior of differential games in which the number of players tends to +infinity. Here we focus on the optimal planning problem, i.e., the problem in which the positions of a very large number of identical rational agents, with a common value function, evolve from a given initial spatial density to a desired target density at the final horizon time. We propose a finite difference semiimplicit scheme for the optimal planning problem, which has an optimal control formulation. The latter leads to existence and uniqueness of the discrete control problem. We also study a penalized version of the semiimplicit scheme. For solving the resulting system of equations, we propose a strategy based on Newton iterations. We describe some numerical experiments.SIAM Journal on Control and Optimization 01/2012; 50(1). DOI:10.1137/100790069 · 1.46 Impact Factor  Networks and Heterogeneous Media 01/2012; 2(2). · 0.65 Impact Factor
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ABSTRACT: We consider approximation schemes for monotone systems of fully nonlinear second order partial differential equations. We first prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of BarlesSouganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approximation schemes for weakly coupled systems of HamiltonJacobiBellman equations. Examples including finite difference schemes and SemiLagrangian schemes are discussed.01/2012; 4(2). DOI:10.7153/dea0418
Publication Stats
704  Citations  
54.56  Total Impact Points  
Top Journals
Institutions

19962013

Sapienza University of Rome
 • Department of Basic and Applied Sciences for Engineering
 • Department of Computer Science
Roma, Latium, Italy


2012

University of Rome Tor Vergata
Roma, Latium, Italy


20022009

Università degli Studi dell'Aquila
 Department of Chemistry, Chemical Engineering and Materials
Aquila, Abruzzo, Italy


2004

University of Bayreuth
 Institute of Mathematics
Bayreuth, Bavaria, Germany


2003

Universität Bremen
Bremen, Bremen, Germany


1999

Università degli Studi di Torino
Torino, Piedmont, Italy
