Neighbor-Gradient Single-Pass Method for solving anisotropic eikonal equation

TL;DR

This paper introduces a fast, single-pass method for solving anisotropic eikonal equations by replacing the search along the accepted front with neighbor gradient information, reducing computational cost.

Contribution

The paper presents the Neighbor-Gradient Single-Pass Method, a novel approach that avoids the traditional SAAF in anisotropic eikonal equation solutions, improving efficiency.

Findings

Method reduces computational cost significantly.

Works well on various anisotropic eikonal equations.

Avoids the need for search along the accepted front.

Abstract

We develop a single pass method for approximating the solution to an anisotropic eikonal equation related to the anisotropic min-time optimal trajectory problem. Ordered Upwind Method (OUM) solves this equation, which is a single-pass method with an asymptotic complexity. OUM uses the search along the accepted front (SAAF) to update the value at considered nodes. Our technique, which we refer to as "Neighbor-Gradient Single-Pass Method", uses the minimizer of the Hamiltonian, in which the gradient is substituted with neighbor gradient information, to avoid SAAF. Our technique is considered in the context of control-theoretic problem. We begin by discussing SAAF of OUM. We then prove some properties of the value function and its gradient, which provide the key motivation for constructing our method. Based on these discussions, we present a new single-pass method, which is fast since it does not require SAAF. We test this method with several anisotropic eikonal equations to observe that it works well while significantly reducing the computational cost.