Stable Computation of Multiquadric Interpolants for All Values of the Shape Parameter

K e y w o r d s R a d i a l basis functions, RBF, Multiquadrics, Ill-conditioning. 1. I N T R O D U C T I O N Linear combinations of radial basis functions (RBFs) can provide very good interpolants for multivariate data. Multiquadric (MQ) basis functions, generated by ¢(r) = ~ (or in the notation used in this paper, ¢(r) = V/1 + (er) 2 with E = l/c), have proven to be particularly successful [1]. However, there have been three main difficulties with this approach: severe numerical ill-conditioning for a fixed N (the number of data points) and small E, similar ill-conditioning *This work was supported by NSF Grants DMS-9810751 (VIGRE), DMS-0073048, and a Faculty Fellowship from the University of Colorado at Boulder. tThis work was supported by an NSF VIGRE Graduate Traineeship under Grant DMS-9810751. 0898-1221/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by .AAdS-TEX doi:10.1016/j.camwa.2003.08.010 854 B. FORNBERO AND G. WRIGHT problems for a fixed E and large N, and high computational cost. This study shows how the first of these three problems can be resolved. Large values of parameter E are well known to produce very inaccurate results (approaching linear interpolation in the case of l-D). Decreasing e usually improves the accuracy significantly [2]. However, the direct way of computing the RBF interpolant suffers from severe ill-conditioning as E is decreased [3]. Several numerical methods have been developed for selecting the "optimal" value of e (e.g., [4-6]). However, because of the ill-conditioning problem, they have all been limited in the range of values that could be considered, having to resort to high-precision arithmetic, for which the cost of computing the interpolant increases to infinity as e --* 0 (timing illustrations for this will be given later). In this study, we present the first algorithm which not only can compute the interpolant for the full range e > 0, but it does so entirely without a progressive cost increase as e -~ 0. In the highly special case of MQ RBF interpolation on an infinite equispaced Cartesian grid, Buhmann and Dyn [7] showed that the interpolants obtain spectral convergence for smooth functions as the grid spacing goes to zero (see, for example, [8] for the spectral convergence properties of MQ and other RBF interpolants for scattered finite data sets). Additionally, for an infinite equispaced grid, but with a fixed grid spacing, Baxter [9] showed the MQ RBF interpolant in the limit of ~ --+ 0 to cardinal data (equal to one at one data point and zero at all others) exists and goes to the multidimensional sinc function--just as the case would be for a Fourier spectral method. Limiting (6 --+ 0) interpolants on scattered finite data sets were studied by Driscoll and Fornberg [10]. They noted that, although the limit usually exists, it can fail to do so in exceptional cases. The present numerical algorithm handles both of these situations. It also applies--without any change--t0 many other types of basis functions. The cases we will give computational examples for are listed in Table 1. Note that for all these cases, the limits of flat basis functions correspond to e --* 0. Table 1. Name of RBF Abbreviation Definition Multiquadrics Inverse Quadratics