A fast-marching algorithm for non-monotonically evolving fronts

TL;DR

This paper introduces a fast-marching algorithm for non-monotonically evolving fronts based on solving Dirichlet problems, achieving first-order accuracy and comparable complexity to existing methods.

Contribution

A novel fast-marching algorithm that handles non-monotonic front propagation without restrictions on speed sign, with proven convergence and stability.

Findings

Algorithm is globally first-order accurate.

Computational complexity is comparable to Fast Marching Method.

Performance is independent of front monotonicity.

Abstract

The non-monotonic propagation of fronts is considered. When the speed function $F:\mathbb{R}^{n} \times [0,T]\rightarrow \mathbb{R}$ is prescribed, the non-linear advection equation $\phi_{t}+F|\nabla \phi|=0$ is a Hamilton-Jacobi equation known as the level-set equation. It is argued that a small enough neighbourhood of the zero-level-set $\mathcal{M}$ of the solution $\phi: \mathbb{R}^{n} \times [0,T] \rightarrow \mathbb{R}$ is the graph of $\psi:\mathbb{R}^{n} \rightarrow \mathbb{R}$ where $\psi$ solves a Dirichlet problem of the form $H(\vec{u},\psi(\vec{u}),\nabla \psi(\vec{u}))=0$. A fast-marching algorithm is presented where each point is computed using a discretization of such a Dirichlet problem, with no restrictions on the sign of $F$. The output is a directed graph whose vertices evenly sample $\mathcal{M}$. The convergence, consistency and stability of the scheme are addressed. Bounds on the computational complexity are estimated, and experimentally shown to be on par with the Fast Marching Method. Examples are presented where the algorithm is shown to be globally first-order accurate. The complexities and accuracies observed are independent of the monotonicity of the evolution.