Fabio Camilli

Sapienza University of Rome, Roma, Latium, Italy

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Publications (75)51.97 Total impact

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    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 Hopf-Lax 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.
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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].
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    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.
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    Fabio Camilli, Paola Loreti, Cristina Pocci
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    ABSTRACT: The equivalence between logarithmic Sobolev inequalities and hypercontractivity of solutions of Hamilton-Jacobi equations has been proved in [5]. We consider a semi-Lagrangian approximation scheme for the Hamilton-Jacobi 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.
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    Fabio Camilli, Claudio Marchi
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    ABSTRACT: Three definitions of viscosity solutions for Hamilton-Jacobi 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
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    ABSTRACT: We consider continuous-state and continuous-time 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). DOI:10.1007/s00030-012-0158-1 · 0.97 Impact Factor
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    Fabio Camilli, Claudio Marchi, Dirk Schieborn
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    ABSTRACT: For a Hamilton-Jacobi equation defined on a network, we introduce its vanishing viscosity approximation. The elliptic equation is given on the edges and coupled with Kirchhoff-type 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 · 1.57 Impact Factor
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    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 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
    SIAM Journal on Numerical Analysis 07/2012; 51(5). DOI:10.1137/120882421 · 1.69 Impact Factor
  • Fabio Camilli, Francisco Silva
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    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 semi-discrete 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 · 0.95 Impact Factor
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    Fabio Camilli, Dirk Schieborn, Claudio Marchi
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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 Hamilton-Jacobi equations, providing existence and uniqueness results.
    Interfaces and Free Boundaries 01/2012; 15(1). DOI:10.4171/IFB/297 · 0.57 Impact Factor
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    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 semi-implicit 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 semi-implicit 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 · 1.39 Impact Factor
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    ABSTRACT: The theory of Mean Field Games (MFG, in short) is a branch of the theory of Differential Games which aims at modeling and analyzing complex decision processes involving a large number of indistinguishable rational agents who have individually a very small influence on the overall system and are, on the other hand, influenced by the mass of the other agents. The name comes from particle physics where it is common to consider interactions among particles as an external mean field which influences the particles. In spite of the optimization made by rational agents, playing the role of particles in such models, appropriate mean field equations can be derived to replace the many particles interactions by a single problem with an appropriately chosen external mean field which takes into account the global behavior of the individuals.
    Networks and Heterogeneous Media 01/2012; 2(2). · 0.95 Impact Factor
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    Ariela Briani, Fabio Camilli, Hasnaa Zidani
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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 Barles-Souganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approximation schemes for weakly coupled systems of Hamilton-Jacobi-Bellman equations. Examples including finite difference schemes and Semi-Lagrangian schemes are discussed.
    01/2012; 4(2). DOI:10.7153/dea-04-18
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    ABSTRACT: Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) We study approximation strategies for the limit problem arising in the homogeniza-tion of Hamilton-Jacobi equations. They involve first an approximation of the effective Hamiltonian then a discretization of the Hamilton-Jacobi 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). DOI:10.1142/S0218202508002978 · 2.35 Impact Factor
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    Yves Achdou, Fabio Camilli, Lucilla Corrias
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    ABSTRACT: In the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. We prove the well posedness of the discrete problem and the convergence of the approximated solution toward the viscosity-measure valued solution of the exact problem.
    Discrete and Continuous Dynamical Systems - Series B 10/2011; 19(3). DOI:10.3934/dcdsb.2014.19.629 · 0.63 Impact Factor
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    Fabio Camilli, Claudio Marchi
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    ABSTRACT: Aim of this paper is to extend the continuous dependence estimates proved in \cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic equations. As by-product 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 quasi-monotone systems of fully nonlinear second-order uniformly parabolic equations. Finally, some examples are discussed.
    Nonlinear Analysis 09/2011; 75(13). DOI:10.1016/j.na.2012.04.026 · 1.61 Impact Factor
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    Fabio Camilli, Adriano Festa, Dirk Schieborn
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    ABSTRACT: In this paper we study approximation of Hamilton-Jacobi equations defined on a network. We introduce an appropriate notion of viscosity solution on networks which satisfies existence, uniqueness and stability properties. Then we define an approximation scheme of semi-Lagrangian type by discretizing in time the representation formula for the solution of Hamilton-Jacobi equations and we prove that the discrete problem admits a unique solution. Moreover we prove that the solution of the approximation scheme converges to the solution of the continuous problem uniformly on the network. In the second part of the paper we study a fully discrete scheme obtained via a finite elements discretization of the semi-discrete problem. Also for fully discrete scheme we prove the well posedness and the convergence to the viscosity solution of the Hamilton-Jacobi equation. We also discuss some issues concerning the implementation of the algorithm and we present some numerical examples.
    Applied Numerical Mathematics 05/2011; 73. DOI:10.1016/j.apnum.2013.05.003 · 1.04 Impact Factor
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    ABSTRACT: We show a large time behavior result for class of weakly coupled systems of first-order Hamilton-Jacobi 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; 19(6). DOI:10.1007/s00030-011-0149-7 · 0.97 Impact Factor
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    Dirk Schieborn, Fabio Camilli
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    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; 46(3-4). DOI:10.1007/s00526-012-0498-z · 1.53 Impact Factor
  • Fabio Camilli, Adriano Festa, Dirk Schieborn
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    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.

Publication Stats

583 Citations
51.97 Total Impact Points

Institutions

  • 1996–2013
    • Sapienza University of Rome
      • • Department of Basic and Applied Sciences for Engineering
      • • Department of Computer Science
      Roma, Latium, Italy
  • 2012
    • The American University of Rome
      Roma, Latium, Italy
  • 2011
    • Paris Diderot University
      Lutetia Parisorum, Île-de-France, France
  • 2004–2009
    • Università degli Studi dell'Aquila
      Aquila, Abruzzo, Italy
    • University of Bayreuth
      • Institute of Mathematics
      Bayreuth, Bavaria, Germany
  • 2003
    • Universität Bremen
      Bremen, Bremen, Germany