Fast Evaluation of Radial Basis Functions: Methods for Four-Dimensional Polyharmonic Splines
| Autor: | David L. Ragozin, J. B. Cherrie, Rick Beatson |
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| Rok vydání: | 2001 |
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| Zdroj: | SIAM Journal on Mathematical Analysis. 32:1272-1310 |
| ISSN: | 1095-7154 0036-1410 |
| DOI: | 10.1137/s0036141099361767 |
| Popis: | As is now well known for some basic functions $\phi$, hierarchical and fast multipole-like methods can greatly reduce the storage and operation counts for fitting and evaluating radial basis functions. In particular, for spline functions of the form \begin{equation*} s(x) = p(x) + \sum_{k=1}^{N} d_k \phi(|x - x_k|), \end{equation*} where p is a low degree polynomial and with certain choices of $\phi$, the cost of a single extra evaluation can be reduced from \Order{N} to \Order{\log N}, or even \Order{1}, operations and the cost of a matrix-vector product (i.e., evaluation at all centers) can be decreased from \Order{N^2} to \Order{N\log N}, or even \Order{N}, operations. This paper develops the mathematics required by methods of these types for polyharmonic splines in $\mathbb{R}^4$. That is, for splines s built from a basic function from the list $\phi(r) = r^{-2}$ or $\phi(r)= r^{2n}\ln(r)$, $n = 0,1,\ldots$. We give appropriate far and near field expansions, together with corresponding error estimates... |
| Databáze: | OpenAIRE |
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