TY - JOUR
T1 - Stable computation of multiquadric interpolants for all values of the shape parameter
AU - Fornberg, B.
AU - Wright, G.
PY - 2004/9
Y1 - 2004/9
N2 - Abstract-Spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries. Since radial basis function (RBF) approxima-ions permit this even for multivariate scattered data, there has been much recent interest in practical algorithms to compute these approximations effectively. Several types of RBFs feature a free parameter (e.g., c in the multiquadric (MQ) case φ(r) = √r2 + c2). The limit of c → ∞ (increasingly flat basis functions) has not received much attention because it leads to a severely ill-conditioned problem. We present here an algorithm which avoids this difficulty, and which allows numerically stable computations of MQ RBF interpolants for all parameter values. We then find that the accuracy of the resulting approximations, in some cases, becomes orders of magnitude higher than was the case within the previously available parameter range. Our new method provides the first tool for the numerical exploration of MQ RBF interpolants in the limit of c → ∞. The method is in no way specific to MQ basis functions and can-without any change-be applied to many other cases as well.
AB - Abstract-Spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries. Since radial basis function (RBF) approxima-ions permit this even for multivariate scattered data, there has been much recent interest in practical algorithms to compute these approximations effectively. Several types of RBFs feature a free parameter (e.g., c in the multiquadric (MQ) case φ(r) = √r2 + c2). The limit of c → ∞ (increasingly flat basis functions) has not received much attention because it leads to a severely ill-conditioned problem. We present here an algorithm which avoids this difficulty, and which allows numerically stable computations of MQ RBF interpolants for all parameter values. We then find that the accuracy of the resulting approximations, in some cases, becomes orders of magnitude higher than was the case within the previously available parameter range. Our new method provides the first tool for the numerical exploration of MQ RBF interpolants in the limit of c → ∞. The method is in no way specific to MQ basis functions and can-without any change-be applied to many other cases as well.
KW - Ill-conditioning
KW - Multiquadrics
KW - Radial basis functions
KW - RBF
UR - http://www.scopus.com/inward/record.url?scp=14644431037&partnerID=8YFLogxK
U2 - 10.1016/j.camwa.2003.08.010
DO - 10.1016/j.camwa.2003.08.010
M3 - Article
AN - SCOPUS:14644431037
SN - 0898-1221
VL - 48
SP - 853
EP - 867
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
IS - 5-6
ER -