Fast evaluation of radial basis functions : methods for four-dimensional polyharmonic splines (2000)
Type of ContentDiscussion / Working Papers
PublisherUniversity of Canterbury. Department of Mathematics & Statistics
- Engineering: Reports 
As is now well known for some basic functions ϕ, 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 [FORM] p a low degree polynomial and certain choices of ⏀, the cost of a single extra evaluation can be reduced from O(N) to O(log N), or even O(1), operations and the cost of a matrix-vector product (i.e., evaluation at all centres) can be decreased from O(N²) to O(N log N), or even O(N), operations. This paper develops the mathematics required by methods of these types for polyharmonic splines in R⁴. That is for splines s built from a basic function from the list ⏀(r) = r⁻² or ⏀(r) = r²n ln(r), n = 0, 1, .... We give appropriate far and near field expansions, together with corresponding error estimates, uniqueness theorems, and translation formulae. A significant new feature of the current work is the use of arguments based on the action of the group of non-zero quaternions, realised as 2 x 2 complex matrices
acting on C² = R⁴. Use of this perspective allows us to give a relatively efficient development of the relevant spherical harmonics and their properties.
ANZSRC Fields of Research49 - Mathematical sciences::4904 - Pure mathematics::490406 - Lie groups, harmonic and Fourier analysis
RightsAll Rights Reserved
Showing items related by title, author, creator and subject.
Cherrie, J. B.; Beatson, Richard Keith; Newsam, G.N. (University of Canterbury, 2000)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 ...
Beatson, Richard Keith; Newsam, G.N. (University of Canterbury. Dept. of Mathematics, 1995)In this paper we introduce a new algorithm for fast evaluation of univariate radial basis functions of the form s(x) = Σᶰn₌₁ dn𝜙(⃒x - xn⃒) to within accuracy 𝜖. The algorithm has a setup cost of 𝜙(N⃒log𝜖⃒log⃒log𝜖⃒) ...
The performance of available methods for computing the polynomial coefficients of the quadratic function approximation is evaluated. By comparing the numerical results to those obtained by symbolic methods, for a variety of functions, the direct solution of the matrix equation and a variety of recursive algorithms are all shown to be numerically unstable. AMS classification Balakrishnan, Easwaran; McInnes, A.W. (University of Canterbury. Mathematics, 1991)The performance of available methods for computing the polynomial coefficients of the quadratic function approximation is evaluated. By comparing the numerical results to those obtained by symbolic methods, for a variety ...