TITLE

# A New Error Bound for Shifted Surface Spline Interpolation

AUTHOR(S)
Lin-Tian Luh
PUB. DATE
November 2010
SOURCE
Studies in Mathematical Sciences;Nov2010, Vol. 1 Issue 1, p1
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
Shifted surface spline is a frequently used radial function for scattered data interpolation. The most frequently used error bounds for this radial function are the one raised by Wu and Schaback in  and the one raised by Madych and Nelson in . Both are O(dl) as d â†’ 0, where l is a positive integer and d is the well-known fill-distance which roughly speaking measures the spacing of the data points. Then RBF people found that there should be an error bound of the form O(Ï‰1/d ) because shifted surface spline is smooth and every smooth function shares this property. The only problem was that the value of the cucial constant Ï‰ was unknown. Recently Luh raised an exponential-type error bound with convergence rate O(Ï‰1/d) as d â†’ 0 where 0 < Ï‰ < 1 is a fixed constant which can be accurately computed . Although the exponential-type error bound converges much faster than the algebraic-type error bound, the constant Ï‰ is intensely influenced by the dimension n in the sense Ï‰ â†’ 1 rapidly as n â†’ âˆž. Here the variable x of both the interpolated and interpolating functions lies in Rn. In this paper we present an error bound which is O(âˆšdÏ‰'1/d) where 0 < Ï‰' < 1 is a fixed constant for any fixed n, and is only mildly influenced by n. In other words, Ï‰' â†’ 1 very slowly as n â†’ âˆž, and Ï‰' << Ï‰, especially for high dimensions. Moreover, Ï‰' can be accurately computed without slight difficulty. This provides a good error estimate for high-dimensional problems which are of growing importance.
ACCESSION #
60435447

## Related Articles

• Multivariate interpolation with increasingly flat radial basis functions of finite smoothness. Song, Guohui; Riddle, John; Fasshauer, Gregory; Hickernell, Fred // Advances in Computational Mathematics;Apr2012, Vol. 36 Issue 3, p485

In this paper, we consider multivariate interpolation with radial basis functions of finite smoothness. In particular, we show that interpolants by radial basis functions in â„ with finite smoothness of even order converge to a polyharmonic spline interpolant as the scale parameter of the...

• Blending moving least squares techniques with NURBS basis functions for nonlinear isogeometric analysis. Cardoso, Rui; Cesar de Sa, J. // Computational Mechanics;Jun2014, Vol. 53 Issue 6, p1327

IsoGeometric Analysis (IGA) is increasing its popularity as a new numerical tool for the analysis of structures. IGA provides: (i) the possibility of using higher order polynomials for the basis functions; (ii) the smoothness for contact analysis; (iii) the possibility to operate directly on CAD...

• Interpolating point spread function anisotropy. Gentile, M.; Courbin, F.; Meylan, G. // Astronomy & Astrophysics / Astronomie et Astrophysique;Jan2013, Vol. 549, p1

Planned wide-field weak lensing surveys are expected to reduce the statistical errors on the shear field to unprecedented levels. In contrast, systematic errors like those induced by the convolution with the point spread function (PSF) will not benefit from that scaling effect and will require...

• Wendland functions with increasing smoothness converge to a Gaussian. Chernih, A.; Sloan, I.; Womersley, R. // Advances in Computational Mathematics;Feb2014, Vol. 40 Issue 1, p185

The Wendland functions are a class of compactly supported radial basis functions with a user-specified smoothness parameter. We prove that with an appropriate rescaling of the variables, both the original and the 'missing' Wendland functions converge uniformly to a Gaussian as the smoothness...

• Designing Font using Radial Basis Function. Ahmad, Azhar; Ali, Jamaludin Md; Ramli, Ahmad // AIP Conference Proceedings;2015, Vol. 1691, p1

A new way of designing font is developed for the font using Radial Basis Function. The approach using Radial Basis Function is used on the interpolation of point. The data point will be used to generate the curve which will then form the shape of the font. To generate the curve, we will use the...

• Designing Font using Radial Basis Function. Ahmad, Azhar; Ali, Jamaludin Md; Ramli, Ahmad // AIP Conference Proceedings;2015, Vol. 1691, p1

A new way of designing font is developed for the font using Radial Basis Function. The approach using Radial Basis Function is used on the interpolation of point. The data point will be used to generate the curve which will then form the shape of the font. To generate the curve, we will use the...

• APPROXIMATE IMPLICITIZATION BASED ON RBF NETWORKS AND MQ QUASI-INTERPOLATION. Renhong Wang; Jinming Wu // Journal of Computational Mathematics;Jan2007, Vol. 25 Issue 1, p97

In this paper, we propose a new approach to solve the approximate implicitization problem based on RBF networks and MQ quasi-interpolation. This approach possesses the advantages of shape preserving, better smoothness, good approximation behavior and relatively less data etc. Several numerical...

• A note on radial basis function interpolant limits. BUHMANN, MARTIN D.; DINEW, SŁAWOMIR; LARSSON, ELISABETH // IMA Journal of Numerical Analysis;Apr2010, Vol. 30 Issue 2, p543

Radial basis functions (RBFs) are very useful in multivariate interpolation because of their ability to produce highly accurate results for scattered data. Many of them, especially the Gaussian RBF and the multiquadric RBF, contain parameters that need to be adjusted in order to improve the...

• Approximation on the sphere using radial basis functions plus polynomials. Ian Sloan; Alvise Sommariva // Advances in Computational Mathematics;Aug2008, Vol. 29 Issue 2, p147

AbstractÂ Â In this paper we analyse a hybrid approximation of functions on the sphere by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at...

Share