Fast evaluation of radial basis functions : moment based methods (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) operations and an incremental cost per evaluation of s(x) of 𝜙(⃒log𝜖⃒) operations. It is based on a hierarchical subdivision of the unit interval, the adaptive construction of a corresponding hierarchy of polynomial approximations, and the fast accumulation of moments. It can be applied in any case where the basic function 𝜙 smooth on (0, 1], and on any grid of centres ｛Xn｝. The algorithm does not require that 𝜙 be analytic at infinity, nor that the user specify new polynomial approximations or modify the data structures for each new 𝜙, nor that the points Xn form any sort of regular array. Furthermore the algorithm can be extended to problems in higher dimensions.
ANZSRC Fields of Research01 - Mathematical Sciences
RightsCopyright Richard Keith Beatson
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Cherrie, J. B.; Beatson, R. K.; Ragozin, D. L. (University of Canterbury. Department of Mathematics & Statistics, 2000)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 ...
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 ...
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 ...