Simone Cacace

Simone Cacace, Emiliano Cristiani, Maurizio Falcone et al.

In this paper we present a new algorithm for the solution of Hamilton-Jacobi-Bellman equations related to optimal control problems. The key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem. This can result in a rather complex geometrical subdivision, but it has the advantage that every subdomain is invariant with respect to the optimal dynamics, and then the solution can be computed independently in each subdomain. The features of this dynamics-dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization, since the classical transmission conditions at the boundaries of the subdomains can be avoided. For their properties, the subdomains are patches in the sense introduced by Ancona and Bressan [ESAIM Control Optim. Calc. Var., 4 (1999), pp. 445-471]. Several examples in two and three dimensions illustrate the properties of the new method.

Simone Cacace, Emiliano Cristiani, Maurizio Falcone

The use of local single-pass methods (like, e.g., the Fast Marching method) has become popular in the solution of some Hamilton-Jacobi equations. The prototype of these equations is the eikonal equation, for which the methods can be applied saving CPU time and possibly memory allocation. Then, some natural questions arise: can local single-pass methods solve any Hamilton-Jacobi equation? If not, where the limit should be set? This paper tries to answer these questions. In order to give a complete picture, we present an overview of some fast methods available in literature and we briefly analyze their main features. We also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods. We show that the construction of a local single-pass method for general Hamilton-Jacobi equations is very hard, if not impossible. Nevertheless, some special classes of problems can be actually solved, making local single-pass methods very useful from the practical point of view.

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.

Simone Cacace, Emiliano Cristiani, Maurizio Falcone

In this paper we propose a numerical method to obtain an approximation of Nash equilibria for multi-player non-cooperative games with a special structure. We consider the infinite horizon problem in a case which leads to a system of Hamilton-Jacobi equations. The numerical method is based on the Dynamic Programming Principle for every equation and on a global fixed point iteration. We present the numerical solutions of some two-player games in one and two dimensions. The paper has an experimental nature, but some features and properties of the approximation scheme are discussed.

Simone Cacace, Emiliano Cristiani, Leonardo Rocchi

3D printers based on the Fused Decomposition Modeling create objects layer-by-layer dropping fused material. As a consequence, strong overhangs cannot be printed because the new-come material does not find a suitable support over the last deposed layer. In these cases, one can add some support structures (scaffolds) which make the object printable, to be removed at the end. In this paper we propose a level set method to create object-dependent support structures, specifically conceived to reduce both the amount of additional material and the printing time. We also review some open problems about 3D printing which can be of interests for the mathematical community.

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.

Simone Cacace, Emiliano Cristiani, Roberto Ferretti

In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations and nonlinear conservation laws. By coupling a macroscopic (Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of multiscale numerical method.

Simone Cacace, Maurizio Falcone

We propose a parallel algorithm for the numerical solution of a class of second order semi-linear equations coming from stochastic optimal control problems, by means of a dynamic domain decomposition technique. The new method is an extension of the patchy domain decomposition method presented in a previous work for first order Hamilton-Jacobi-Bellman equations related to deterministic optimal control problems. The semi-Lagrangian scheme underlying the original method is modified in order to deal with (possibly degenerate) diffusion, by approximating the stochastic optimal control problem associated to the equation via discrete time Markov chains. We show that under suitable conditions on the discretization parameters and for sufficiently small values of the diffusion coefficient, the parallel computation on the proposed dynamic decomposition is faster than that on a static decomposition. To this end, we combine the parallelization with some well known techniques in the context of fast-marching-like methods for first order Hamilton-Jacobi equations. Several numerical tests in dimension two are presented, in order to show the features of the proposed method.

Alessio Oliviero, Simone Cacace, Giuseppe Visconti

We investigate the use of multi-agent systems to solve classical image processing tasks, such as colour quantization and segmentation. We frame the task as an optimal control problem, where the objective is to steer the multi-agent dynamics to obtain colour clusters that segment the image. To do so, we balance the total variation of the colour field and fidelity to the original image. The solution is obtained resorting to primal-dual splitting and the method of multipliers. Numerical experiments, implemented in parallel with CUDA, demonstrate the efficacy of the approach and its potential for high-dimensional data.

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.

Simone Cacace, Emiliano Cristiani, Francesca L. Ignoto

The wolf note is an acoustic instability that occurs in large bowed string instruments when a strong body resonance interacts with the vibrating string, producing amplitude modulation and loss of tonal control. Various wolf suppressors - tuned mass dampers attached to the string or to the instrument body - are used in practice to mitigate this effect. In this paper, we propose a mathematical model describing the coupled dynamics of a string and a two-dimensional body equipped with one or two wolf suppressors. Both string and body include elastic (second-order) and stiffness (fourth-order) contributions and can be excited either by plucking or bowing. Three performance indicators are introduced: The first one perceives the wolf-tone appearance, the second one quantifies the attenuation of the notes possibly caused by the wolf suppressor, and the third one measures the acoustic fidelity (in terms of spectrum) with respect to the original instrument. The proposed numerical tests give insights about optimal tuning and placement of one or two suppressors, achieving effective wolf-note suppression while preserving as much as possible the global tonal balance.

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.