Fast algorithms for optimal control, anisotropic front propagation and multiple arrivals

TL;DR

This paper reviews recent fast algorithms for solving Hamilton-Jacobi equations related to optimal control, anisotropic front propagation, and multiple wave arrivals, emphasizing ordered methods for efficient computation.

Contribution

It introduces and discusses ordered algorithms for efficiently computing both viscosity and non-viscosity solutions to static Hamilton-Jacobi equations.

Findings

Ordered Upwind Methods effectively compute viscosity solutions.

Phase-space formulations enable solving multiple arrivals.

Algorithms improve accuracy and efficiency in wave propagation modeling.

Abstract

We review some recent work in fast, efficient and accurate methods to compute viscosity solutions and non-viscosity solutions to static Hamilton-Jacobi equations which arise in optimal control, anisotropic front propagation, and multiple arrivals in wave propagation. For viscosity solutions, the class of algorithms are known as ``Ordered Upwind Methods'', and rely on a systematic ordering inherent in the characteristic flow of information. For non-viscosity multiple arrivals, the techniques hinge on a static boundary value phase-space formulation which again can be solved through a systematic ordering.