A patchy Dynamic Programming scheme for a class of Hamilton-Jacobi-Bellman equations

TL;DR

This paper introduces a novel patchy domain decomposition algorithm for solving Hamilton-Jacobi-Bellman equations, leveraging optimal dynamics to enable independent, parallelizable subdomain computations and improve efficiency.

Contribution

The paper presents a new patchy domain decomposition method that exploits optimal dynamics for efficient, parallel solution of Hamilton-Jacobi-Bellman equations, avoiding classical boundary transmission conditions.

Findings

Effective in 2D and 3D examples

Enables parallel computation without boundary transmission conditions

Accelerates solution of complex control problems

Abstract

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.