Fast evaluation of radial basis functions : methods for generalised multiquadrics in ℝⁿ

Type of content
Discussion / Working Papers
Publisher's DOI/URI
Thesis discipline
Degree name
Publisher
University of Canterbury
Journal Title
Journal ISSN
Volume Title
Language
Date
2000
Authors
Cherrie, J. B.
Beatson, Richard Keith
Newsam, G.N.
Abstract

A generalised multiquadric radial basis function is a function of the form s(x) = ∑ᴺ𝑖₌₁ 𝑑𝑖 𝜙 (𝗅x-t𝑖𝗅), where 𝜙(r) = (r² + 𝝉²)ᵏ/², x ∈ ℝⁿ, and k ∈ Z is odd. The direct evaluation of an N centre generalised multiquadric radial basis function at m points requires 𝒪(mN) flops, which is prohibitive when m and N are large. Similar considerations apparently rule out fitting an interpolating N centre generalised multiquadric to N data points by either direct or iterative solution of the associated system of linear equations in realistic problems. In this paper we will develop far field expansions, recurrence relations for efficient formation of the expansions, error estimates, and translation formulas, for generalised multiquadric radial basis functions in n-variables. These pieces are combined in a hierarchical fast evaluator requiring only 𝒪((m + N) log N llog 𝜖lⁿ⁺¹) flops for evaluation of an N centre generalised multiquadric at m points. This flop count compares very favourably with the cost of the direct method. Moreover, used to compute matrix-vector products, the fast evaluator provides a basis for fast iterative fitting strategies.

Description
Citation
Keywords
Ngā upoko tukutuku/Māori subject headings
ANZSRC fields of research
Fields of Research::49 - Mathematical sciences::4904 - Pure mathematics::490408 - Operator algebras and functional analysis
Rights
All Rights Reserved