Simone Cacace

A Patchy Dynamic Programming Scheme for a Class of Hamilton--Jacobi--Bellman Equations

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.

A Local Ordered Upwind Method for Hamilton-Jacobi and Isaacs Equations

Abstract We present a generalization of the Fast Marching (FM) method for the numerical solution of a class of Hamilton-Jacobi equations, including Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations. The method is able to compute an approximation of the viscosity solution concentrating the computations only in a small evolving trial region, as the original FM method. The main novelty is that the size of the trial region does not depend on the dynamics. We compare the new method with the standard iterative algorithm and the FM method, in terms of accuracy and order of computations on the grid nodes.

CAN LOCAL SINGLE-PASS METHODS SOLVE ANY STATIONARY HAMILTON-JACOBI-BELLMAN EQUATION? ∗

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 questions naturally arise: Can local single-pass methods solve any Hamilton--Jacobi equation? If not, where should the limit 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 the literature and 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 actually be solved, making local single-pass methods very useful from a p...

A Generalized Newton Method for Homogenization of Hamilton--Jacobi Equations

We propose a new approach to the numerical solution of cell problems arising in the homogenization of Hamilton--Jacobi equations. It is based on a Newton-like method for solving inconsistent systems of nonlinear equations, coming from the discretization of the corresponding cell problems. 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 dimensions one and two shows the performance of the proposed method, both in terms of accuracy and computational time.

Blended numerical schemes for the advection equation and conservation laws

In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating a new scheme which inherits 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.

Two Semi-Lagrangian Fast Methods for Hamilton-Jacobi-Bellman Equations

In this paper we apply the Fast Iterative Method (FIM) for solving general Hamilton–Jacobi–Bellman (HJB) equations and we compare the results with an accelerated version of the Fast Sweeping Method (FSM). We find that FIM can be indeed used to solve HJB equations with no relevant modifications with respect to the original algorithm proposed for the eikonal equation, and that it overcomes FSM in many cases. Observing the evolution of the active list of nodes for FIM, we recover another numerical validation of the arguments recently discussed in [1] about the impossibility of creating local single-pass methods for HJB equations.

Myrmedrome: Simulating the Life of an Ant Colony

Ants are a Family of insects that includes about 12,200 different species, counting only those studied so far. Although they all appear similar, these social insects show an enormous biological diversity: from their anatomical characteristics to their reproductive behaviour, from the population densities of colonies to the types of nests, from what they eat to the many kinds of interactions that many species of ants establish with different species of living organisms. Even though it is difficult to generalise, let us describe the ‘typical’ biology of these extremely evolved creatures.

A Differential Model for Growing Sandpiles on Networks

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 introduced in [P.-L. Lions and P. E. Souganidis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), pp. 535--545], we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.