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Error Analysis of Splines for Periodic Problems Using Lattice Designs

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Monte Carlo and Quasi-Monte Carlo Methods 2004

Summary

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.

This research was supported by a Hong Kong RGC grant HKBU2007/03P.

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Author information

Author notes

  1. Xiaoyan Zeng & Fred J. Hickernell

    Present address: Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

    Xiaoyan Zeng, King-Tai Leung & Fred J. Hickernell

Authors
  1. Xiaoyan Zeng
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  2. King-Tai Leung
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  3. Fred J. Hickernell
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Editor information

Editors and Affiliations

  1. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

  2. INRIA Sophia Antipolis, route des lucioles 2004, 06902, Sophia Antipolis Cedex, France

    Denis Talay

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© 2006 Springer-Verlag Berlin Heidelberg

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Zeng, X., Leung, KT., Hickernell, F.J. (2006). Error Analysis of Splines for Periodic Problems Using Lattice Designs. In: Niederreiter, H., Talay, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31186-6_31

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