Error Analysis of Splines for Periodic Problems Using Lattice Designs

Xiaoyan Zeng; King Tai Leung; Fred J. Hickernell

doi:10.1007/3-540-31186-6_31

Error Analysis of Splines for Periodic Problems Using Lattice Designs

Xiaoyan Zeng, King Tai Leung, Fred J. Hickernell

Department of Mathematics

Research output: Chapter in book/report/conference proceeding › Conference contribution › peer-review

Abstract

Splines are minimum-norm approximations to functions that interpolate the given data, (x_i, f(x_i)). Examples of multidimensional splines are those based on radial basis functions. This paper considers splines of a similar form but where the kernels are not necessarily radially symmetric. The designs, {x_i}, considered here are node-sets of integration lattices. Choosing the kernels to be periodic facilitates the computation of the spline and the error analysis. The error of the spline is shown to depend on the smoothness of the function being approximated and the quality of the lattice design. The quality measure for the lattice design is similar, but not equivalent to, traditional quality measures for lattice integration rules.

Original language	English
Title of host publication	Monte Carlo and Quasi-Monte Carlo Methods 2004
Editors	Harald Niederreiter, Denis Talay
Publisher	Springer Berlin Heidelberg
Pages	501-514
Number of pages	14
Edition	1
ISBN (Electronic)	9783540311867
ISBN (Print)	9783540255413
DOIs	https://doi.org/10.1007/3-540-31186-6_31
Publication status	Published - 6 Dec 2005

User-Defined Keywords

spline
lattice designs
circulant matrix

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10.1007/3-540-31186-6_31

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title = "Error Analysis of Splines for Periodic Problems Using Lattice Designs",

abstract = "Splines are minimum-norm approximations to functions that interpolate the given data, (xi, f(xi)). Examples of multidimensional splines are those based on radial basis functions. This paper considers splines of a similar form but where the kernels are not necessarily radially symmetric. The designs, {xi}, considered here are node-sets of integration lattices. Choosing the kernels to be periodic facilitates the computation of the spline and the error analysis. The error of the spline is shown to depend on the smoothness of the function being approximated and the quality of the lattice design. The quality measure for the lattice design is similar, but not equivalent to, traditional quality measures for lattice integration rules.",

keywords = "spline, lattice designs, circulant matrix",

author = "Xiaoyan Zeng and Leung, {King Tai} and Hickernell, {Fred J.}",

note = "Funding information: This research was supported by a Hong Kong RGC grant HKBU2007/03P. Publisher copyright: {\textcopyright} 2006 Springer-Verlag Berlin Heidelberg",

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Error Analysis of Splines for Periodic Problems Using Lattice Designs. / Zeng, Xiaoyan; Leung, King Tai; Hickernell, Fred J.

Monte Carlo and Quasi-Monte Carlo Methods 2004. ed. / Harald Niederreiter; Denis Talay. 1. ed. Springer Berlin Heidelberg, 2005. p. 501-514.

Research output: Chapter in book/report/conference proceeding › Conference contribution › peer-review

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AU - Hickernell, Fred J.

PY - 2005/12/6

Y1 - 2005/12/6

N2 - Splines are minimum-norm approximations to functions that interpolate the given data, (xi, f(xi)). Examples of multidimensional splines are those based on radial basis functions. This paper considers splines of a similar form but where the kernels are not necessarily radially symmetric. The designs, {xi}, considered here are node-sets of integration lattices. Choosing the kernels to be periodic facilitates the computation of the spline and the error analysis. The error of the spline is shown to depend on the smoothness of the function being approximated and the quality of the lattice design. The quality measure for the lattice design is similar, but not equivalent to, traditional quality measures for lattice integration rules.

AB - Splines are minimum-norm approximations to functions that interpolate the given data, (xi, f(xi)). Examples of multidimensional splines are those based on radial basis functions. This paper considers splines of a similar form but where the kernels are not necessarily radially symmetric. The designs, {xi}, considered here are node-sets of integration lattices. Choosing the kernels to be periodic facilitates the computation of the spline and the error analysis. The error of the spline is shown to depend on the smoothness of the function being approximated and the quality of the lattice design. The quality measure for the lattice design is similar, but not equivalent to, traditional quality measures for lattice integration rules.

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KW - lattice designs

KW - circulant matrix

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A2 - Niederreiter, Harald

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PB - Springer Berlin Heidelberg

ER -

Error Analysis of Splines for Periodic Problems Using Lattice Designs

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