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 proceeding › 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	1st
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

Access to Document

10.1007/3-540-31186-6_31

Cite this

@inproceedings{e804c583d4a1414a981d9d3006d41152,

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 King-Tai Leung 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",

year = "2005",

month = dec,

day = "6",

doi = "10.1007/3-540-31186-6_31",

language = "English",

isbn = "9783540255413",

pages = "501--514",

editor = "Harald Niederreiter and Denis Talay",

booktitle = "Monte Carlo and Quasi-Monte Carlo Methods 2004",

publisher = "Springer Berlin Heidelberg",

edition = "1st",

}

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. 1st. ed. Springer Berlin Heidelberg, 2005. p. 501-514.

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

TY - GEN

T1 - Error Analysis of Splines for Periodic Problems Using Lattice Designs

AU - Zeng, Xiaoyan

AU - Leung, King-Tai

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.

KW - spline

KW - lattice designs

KW - circulant matrix

UR - https://link.springer.com/book/10.1007/3-540-31186-6?page=2#bibliographic-information

U2 - 10.1007/3-540-31186-6_31

DO - 10.1007/3-540-31186-6_31

M3 - Conference proceeding

SN - 9783540255413

SP - 501

EP - 514

BT - Monte Carlo and Quasi-Monte Carlo Methods 2004

A2 - Niederreiter, Harald

A2 - Talay, Denis

PB - Springer Berlin Heidelberg

ER -

Error Analysis of Splines for Periodic Problems Using Lattice Designs

Abstract

User-Defined Keywords

Access to Document

Other files and links

Fingerprint

Cite this