Two semi-Lagrangian fast methods for Hamilton-Jacobi-Bellman equations

TL;DR

This paper compares two fast numerical methods, FIM and FSM, for solving Hamilton-Jacobi-Bellman equations, demonstrating FIM's effectiveness and discussing fundamental limitations of local single-pass methods.

Contribution

It applies the Fast Iterative Method to HJB equations and shows its advantages over the Fast Sweeping Method, highlighting limitations of local single-pass approaches.

Findings

FIM can solve HJB equations with minimal modifications.

FIM outperforms FSM in many cases.

The study supports the argument against local single-pass methods for HJB equations.

Abstract

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 [Cacace et al., SISC 36 (2014), A570-A587] about the impossibility of creating local single-pass methods for HJB equations.