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IMA Journal of Numerical Analysis Advance Access originally published online on November 14, 2008
IMA Journal of Numerical Analysis 2009 29(3):806-813; doi:10.1093/imanum/drm028
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Remarks on the Formula implementation of the fast marching method

Christian Rasch{dagger} and Thomas Satzger{ddagger}

Zentrum Mathematik, Technische Universität München, 80290 München, Germany

{dagger} Corresponding author. Email: rasch{at}ma.tum.de

{ddagger} Email: satzger{at}ma.tum.de

Received on 15 March 2007. Revised on 1 August 2008.


  Abstract

The fast marching algorithm computes an approximate solution to the eikonal equation in Formula time, where the factor log N is due to the administration of a priority queue. Recently, Yatziv et al. (2006 J. Comput. Phys., 212, 393–399) have suggested using an untidy priority queue, reducing the overall complexity to Formula at the price of a small error in the computed solution. In this paper we give an explicit estimate of the error introduced, which is based on a discrete comparison principle. This estimate implies, in particular, that the choice of an accuracy level that is independent of the speed function F results in the complexity bound being Formula . A numerical experiment illustrates this robustness problem for large ratios F max /F min.

Key Words: fast marching method; eikonal equation; distance function; untidy priority queue


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