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IMA Journal of Numerical Analysis Advance Access originally published online on November 2, 2006
IMA Journal of Numerical Analysis 2007 27(3):427-450; doi:10.1093/imanum/drl027
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fast evaluation of polyharmonic splines in three dimensions

R. K. Beatson{dagger}

Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

M. J. D. Powell{ddagger}

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK

A. M. Tan

Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

{dagger} Email: rick.beatson{at}canterbury.ac.nz

{ddagger} Email: m.j.d.powell{at}damtp.cam.ac.uk

Received on 24 April 2006. Revised on 31 August 2006.


  Abstract

This paper concerns the fast evaluation of radial basis functions. It describes the mathematics of hierarchical and fast multipole methods for fast evaluation of splines of the form

Formula
where {nu} is a positive integer and p is a low-degree polynomial. Splines s of this form are polyharmonic splines in R3 and have been found to be very useful for providing solutions to scattered data interpolation problems in R3. As it is now well known, hierarchical methods reduce the incremental cost of a single extra evaluation from O(N) to O(log N) operations and reduce the cost of a matrix–vector product (evaluation of s at all the centres) from O(N2) to O(N log N) operations. We give appropriate far- and near-field expansions, together with error estimates, uniqueness theorems and translation formulae. A hierarchical code based on these formulae is detailed and some numerical results are given.

Key Words: fast evaluation; radial basis functions; polyharmonic splines in three dimensions


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