Fabio Camilli

Yves Achdou, Fabio Camilli, Italo Capuzzo Dolcetta

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

Fabio Camilli, Adriano Festa, Dirk Schieborn

In this paper we study an approximation scheme for an Hamilton-Jacobi 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 semi-Lagrangian type converges to the unique solution of the problem.

Yves Achdou, Fabio Camilli, Lucilla Corrias

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.

Simone Cacace, Fabio Camilli, Claudio Marchi

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.

Fabio Camilli, Adriano Festa, Silvia Tozza

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law while the minimization principle is described by a graph eikonal equation. We show that the model is well posed and we implement some numerical examples to demonstrate the validity of the proposed model.

Simone Cacace, Fabio Camilli

We propose a new approach to the numerical solution of ergodic problems arising in the homogenization of Hamilton-Jacobi (HJ) equations. It is based on a Newton-like method for solving inconsistent systems of nonlinear equations, coming from the discretization of the corresponding ergodic HJ equations. We show that our method is able to solve efficiently cell problems in very general contexts, e.g., for first and second order scalar convex and nonconvex Hamiltonians, weakly coupled systems, dislocation dynamics and mean field games, also in the case of more competing populations. A large collection of numerical tests in dimension one and two shows the performance of the proposed method, both in terms of accuracy and computational time.

Fabio Camilli, Elisabetta Carlini, Claudio Marchi

We consider the Cauchy problem \[\partial_t u+H(x,Du)=0 \quad (x,t)\inΓ\times (0,T),\quad u(x,0)=u_0(x) \quad x\inΓ\] where $Γ$ is a network and $H$ is a convex and positive homogeneous Hamiltonian which may change from edge to edge. In the former part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.

Fabio Camilli, Raul De Maio, Elisa Iacomini

We prove a representation formula of Hopf-Lax type for the solution of a Hamilton-Jacobi equation involving Caputo time-fractional derivative. Equations of these type are associated with optimal control problems where the controlled dynamics is replaced by a time-changed stochastic process describing the trajectory of a particle subject to random trapping effects.

Isabeau Birindelli, Fabio Camilli, Italo Capuzzo Dolcetta

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 finite-dimensional nonlinear min-max 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.

Simone Cacace, Fabio Camilli, Lucilla Corrias

We consider a system of differential equations of Monge-Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton-Jacobi equations on networks, recently introduced by P.-L. Lions and P. E. Souganidis, we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.

Fabio Camilli, Espen R. Jakobsen

We construct a finite element like scheme for fully non-linear integro-partial differential equations arising in optimal control of jump-processes. 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 jump-models 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.

Fabio Ancona, Alessio Basti, Fabio Camilli

Building on the information-theoretic perspective of P.~D.~Lax [\textit{Proc.\ Sympos., Math.\ Res.\ Center, Univ.\ Wisconsin}, 1978], we establish a two-sided quantitative compactness estimate for numerical solutions of scalar conservation laws with a uniformly convex flux, expressed in terms of Kolmogorov $\varepsilon$-entropy. We prove that, under specific grid constraints, conservative, monotone finite-difference schemes satisfying a discrete one-sided Lipschitz condition (OSLC) preserve the $1/\varepsilon$ Kolmogorov entropy scaling of the corresponding exact entropy solution set, matching the bounds obtained by De~Lellis and Golse [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{58} (2005)] and by Ancona, Glass, and Nguyen [\textit{Comm.\ Pure Appl.\ Math.}\ \textbf{65} (2012)]. Specifically, the upper bound follows from the discrete OSLC, while the lower bound relies on a uniform approximation argument on a bounded-variation precursor class. Our results show that prototypical first-order methods are high-resolution in Lax's sense. Finally, we abstract the lower bound mechanism into a general transfer principle, discuss implications for information recovery via post-processing, and indicate directions for future work.

Alessio Basti, Fabio Camilli, Adriano Festa

We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Moving beyond static or heuristic approaches, we formulate the problem as a population dynamics game governed by a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system. Driven by a variational cost functional rather than predefined statistical shapes, this continuous-time formulation provides a flexible basis for non-parametric cluster evolution. To validate the framework, we analyze the setting of time-dependent Gaussian mixtures, showing that the MFG dynamics recover the trajectories of the classical Expectation-Maximization (EM) algorithm while ensuring mass conservation. Furthermore, we introduce time-averaged log-likelihood functionals to regularize temporal fluctuations. Numerical experiments illustrate the stability of our approach and suggest a path toward more general non-parametric clustering applications where traditional EM methods may face limitations.

Laura Aquilanti, Simone Cacace, Fabio Camilli et al.

Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.