A Fast Treecode for Multiquadric Interpolation with Varying Shape Parameters
Abstract
A treecode algorithm is presented for the fast evaluation of multiquadric radial basis function (RBF) approximations. The method is a dual approach to one presented by Krasny and Wang, which applies far-field expansions to clusters of RBF centers (source points). The new approach clusters evaluation points instead and is therefore easily able to cope with basis functions that have different multiquadric shape parameters. The new treecode is able to evaluate an approximation on N centers at M points in $O((N+M) \log M)$ time in the ideal case when evaluation points are uniformly distributed. When coupled with a two-level restricted additive Schwarz preconditioner for GMRES iterations, the treecode is well suited for use within an adaptive RBF iteration, previously described by Driscoll and Heryudono, as is demonstrated by experiments on test functions.
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Published In
SIAM Journal on Scientific Computing
Pages: A1126 - A1140
DOI: 10.1137/110836225
ISSN (online): 1095-7197
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Copyright © 2012 Society for Industrial and Applied Mathematics.
History
Submitted: 2 June 2011
Accepted: 30 January 2012
Published online: 10 April 2012
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