Fast evaluation of radial basis functions: I

• Published in: Computers & mathematics with applications (1987) Vol. 24; no. 12; pp. 7 - 19
• Main Authors: Beatson, R.K., Newsam, G.N.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.1992
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/0898-1221(92)90167-G

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s( x) = Σ j=1 N d j φ( x− x j ), where φ( u) = | u| 2log| u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O( N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s( x) to within precision ϵ, can be cut to O(1+|log ϵ|) operations. In particular, if A is the interpolation matrix, a i, j = φ( x i − x j , the technique allows computation of the matrix-vector product Ad in O( N), rather than the previously required O( N 2) operations, and using only O( N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.