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

TL;DR

The paper tackles fast computation for static Hamilton-Jacobi problems that arise in optimal control, anisotropic front propagation, and multi-arrival wave phenomena. It presents two complementary frameworks: Ordered Upwind Methods (OU) for viscosity solutions in physical space, and a phase-space boundary-value approach (Liouville formulation and Escape equations) for non-viscosity/multi-arrival problems. Key contributions include rigorous complexity and convergence results for one-pass OU schemes, isotropic and anisotropic extensions with $O(N log N)$ and $O((F_2/F_1)^2 M log M)$ bounds, and a phase-space, boundary-value formulation enabling simultaneous computation of all arrivals without time stepping. Together, these methods enable robust, real-time capable solutions for high-dimensional static HJ problems with broad applications in imaging, robotics, geophysics, and wave propagation.

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.

Figures (3)

• Figure 1: Viscosity solutions to static HJ equations
• Figure 2: Non-viscosity solutions
• Figure 3: Left: Anistropic front propagation Right: Arrival paths