Publications (66)16.45 Total impact
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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].04/2014;  [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.02/2014;  [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.12/2013; 
Article: A comparison among various notions of viscosity solutions 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.01/2013;  [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 01/2013; 20(3). · 0.67 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).  [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 assumptions07/2012; 
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.01/2012;  [Show abstract] [Hide abstract]
ABSTRACT: Mean fields games describe the asymptotic behavior of differential games in which the number of players tends to +∞. Here we consider a numerical method for the optimal planning problem, i.e. the problem in which the positions of a very large number of identical rational agents, with common value function, evolve from a given initial spatial density to a desired target density at the final horizon time.01/2012;  [Show abstract] [Hide abstract]
ABSTRACT: Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) We study approximation strategies for the limit problem arising in the homogenization of HamiltonJacobi equations. They involve first an approximation of the effective Hamiltonian then a discretization of the HamiltonJacobi equation with the approximate effective Hamiltonian. We give a global error estimate which takes into account all the parameters involved in the approximation.Mathematical Models and Methods in Applied Sciences 11/2011; 18(07). · 1.87 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In the present article, we study the numerical approximation of a system of HamiltonJacobi and transport equations arising in geometrical optics. We consider a semiLagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscositymeasure valued solution of the exact problem.10/2011;  [Show abstract] [Hide abstract]
ABSTRACT: Aim of this paper is to extend the continuous dependence estimates proved in \cite{JK1} to quasimonotone systems of fully nonlinear secondorder parabolic equations. As byproduct of these estimates, we get an H\"older estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation. In the second part of the paper we employ similar techniques to study the periodic homogenization of quasimonotone systems of fully nonlinear secondorder uniformly parabolic equations. Finally, some examples are discussed.09/2011;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we study an approximation scheme for an HamiltonJacobi equation of Eikonal type defined on a network. We introduce an appropriate notion of viscosity solution for this class of equations (see \cite{sc}) and we prove that an approximation scheme of semiLagrangian type converges to the unique solution of the problem.05/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We show a large time behavior result for class of weakly coupled systems of firstorder HamiltonJacobi equations in the periodic setting. We use a PDE approach to extend the convergence result proved by Namah and Roquejoffre (1999) in the scalar case. Our proof is based on new comparison, existence and regularity results for systems. An interpretation of the solution of the system in terms of an optimal control problem with switching is given.Nonlinear Differential Equations and Applications NoDEA 04/2011; · 0.67 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we introduce a notion of viscosity solutions for Eikonal equations defined on topological networks. Existence of a solution for the Dirichlet problem is obtained via representation formulas involving a distance function associated to the Hamiltonian. A comparison theorem based on Ishii's classical argument yields the uniqueness of the solution.Calculus of Variations 03/2011; · 1.24 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: An important problem in graph theory is to detect the shortest paths connecting the vertices of a graph to a prescribed target vertex. Here we study a generalization of the previous problem: finding the shortest path connecting any point of a graph (and not only a vertex) to the target. Our approach is based on the study of Eikonal equations and the corresponding theory of viscosity solutions on topological graphs.01/2011;  [Show abstract] [Hide abstract]
ABSTRACT: This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form u ε + H ε x, x ε , . . . , x ε k , Du ε , D 2 u ε = 0. The operators H ε are a regular perturbations of some uniformly elliptic, convex operator H. As ε → 0, the solutions u ε converge locally uniformly to the solution u of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed. MSC 2000: 35B27, 35J60, 49L25.11/2009; 
Article: A Finite Element Like Scheme for IntegroPartial Differential HamiltonJacobiBellman Equations.
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ABSTRACT: We construct a finite element like scheme for fully nonlinear integropartial differential equations arising in optimal control of jumpprocesses. Special cases of these equations include optimal portfolio and option pricing equations in Finance. The schemes are monotone and robust. We prove that they converge in very general situations, including degenerate equations, multiple dimensions, relatively low regularity of the data, and for most (if not all) types of jumpmodels used in Finance. In all cases we provide (probably optimal) error bounds. These bounds apply when grids are unstructured and integral terms are very singular, two features that are new or highly unusual in this setting.SIAM J. Numerical Analysis. 01/2009; 47:24072431.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we study singular perturbations of weakly coupled systems of elliptic equations. A model problem is given by small random perturbations of random evolution processes. In this setting we give a PDE proof of large deviation results, analogous to those studied in Ann. Probab.374. An essential tool in our approach is the weak KAM theory introduced in Calc. Var. Partial Differential Equations 22 (2005), 185–228.Asymptotic Analysis Stochastics Stochastics Rep. Probab. Theory Related Fields. 01/2009; 65(94):646658.  [Show abstract] [Hide abstract]
ABSTRACT: We consider periodic homogenization of the fully nonlinear uniformly elliptic equation . We give an estimate of the rate of convergence of ue to the solution u of the homogenized problem . Moreover we describe a numerical scheme for the approximation of the effective nonlinearity and we estimate the corresponding rate of convergence.Nonlinearity 01/2009; · 1.60 Impact Factor
Publication Stats
336  Citations  
16.45  Total Impact Points  
Top Journals
Institutions

1996–2013

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


2011

Paris Diderot University
Lutetia Parisorum, ÎledeFrance, France


2009

University of Padova
Padua, Veneto, Italy


2003–2009

Università degli Studi dell'Aquila
Aquila, Abruzzo, Italy


2006

Centro di Ricerca in Matematica Pura ed Applicata
Fisciano, Campania, Italy


2004–2006

University of Bayreuth
 Institute of Mathematics
Bayreuth, Bavaria, Germany
