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 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 in 1999. Several examples in dimension two and three illustrate the properties of the new method.

We study, by means of Γ-convergence, the asymptotic behavior of a variational model for dislocations moving on a slip plane. The variational functional is a two-dimensional multi-phase transition-type energy given by a nonlocal term and a nonlinear potential which penalizes noninteger values for the components of the phase. In the limit we obtain an anisotropic sharp interfaces model. The relevant feature of this problem is that optimal sequences in general are not given by a onedimensional profile, but they can create microstructure.

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 HamiltonJacobi 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 practical point of view.1. Introduction. The study of Hamilton-Jacobi (HJ) equations arises in several applied contexts, including classical mechanics, front propagation, control problems, and differential games, and has had a great impact in many areas, such as robotics, aeronautics, and electrical and aerospace engineering. In particular, for control/game problems, an approximation of the value function allows for the synthesis of optimal control laws in feedback form, and then for the computation of optimal trajectories. The value function for a control problem (resp., differential game) can be characterized as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation (resp., Hamilton-Jacobi-Isaacs (HJI) equation), and it is obtained by passing to the limit in Bellman's well-known dynamic programming (DP) principle. The DP approach can be rather expensive from a computational point of view, but in some situations it gives a real advantage over methods based on Pontryagin's maximum principle, because the latter approach allows one to compute only open-loop controls and locally optimal trajectories. Moreover, weak solutions to HJ equations are nowadays well understood in the framework of viscosity solutions, which offers the correct notion of solution for many applied problems.The above remarks have motivated the research of efficient and accurate numerical methods. Indeed, an increasing number of techniques have been proposed for

We propose a new approach to the numerical solution of cell 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 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 dimension one and two shows the performance of the proposed method, both in terms of accuracy and computational time.

We study an implicit and discontinuous scheme for a non-local Hamilton-Jacobi equation modelling dislocation dynamics. For the evolution problem, we prove an a posteriori estimate of Crandall-Lions type for the error between continuous and discrete solutions. We deduce an a posteriori error estimate for the effective Hamiltonian associated to a stationary cell problem. In dimension one and under suitable assumptions, we also give improved a posteriori estimates. Numerical simulations are provided.