Fast evaluation of polyharmonic splines in three dimensions |
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Authors: | Beatson, R. K. Powell, M. J. D. Tan, A. M. |
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Affiliation: | Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand |
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Abstract: | M. J. D. Powell This paper concerns the fast evaluation of radial basis functions.It describes the mathematics of hierarchical and fast multipolemethods for fast evaluation of splines of the form where is a positive integer andp is a low-degree polynomial. Splines s of this form are polyharmonicsplines in 3 and have been found to be very useful for providingsolutions to scattered data interpolation problems in 3. Asit is now well known, hierarchical methods reduce the incrementalcost of a single extra evaluation from O(N) to O(log N) operationsand reduce the cost of a matrixvector product (evaluationof s at all the centres) from O(N2) to O(N log N) operations.We give appropriate far- and near-field expansions, togetherwith error estimates, uniqueness theorems and translation formulae.A hierarchical code based on these formulae is detailed andsome numerical results are given. |
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Keywords: | fast evaluation radial basis functions polyharmonic splines in three dimensions |
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