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Numerical Mathematics and Advanced Applications 2011 pp 157–164Cite as

Multilevel Sparse Kernel-Based Interpolation Using Conditionally Positive Definite Radial Basis Functions

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Abstract

A multilevel sparse kernel-based interpolation (MLSKI) method, suitable for moderately high-dimensional function interpolation problems has been recently proposed in (Georgoulis et al. Multilevel sparse kernel-based interpolation, submitted for publication). The method uses both level-wise and direction-wise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites, whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level. On each level, anisotropic radial basis functions (RBFs) are used for solving a number of small interpolation problems, which are subsequently linearly combined to produce the interpolant. Here, we investigate the use of conditionally positive definite RBFs within the MLSKI setting, thus extending the results from (Georgoulis et al. Multilevel sparse kernel-based interpolation, submitted for publication), where (strictly) positive definite RBFs are used only.

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK

    E. H. Georgoulis, J. Levesley & F. Subhan

Authors
  1. E. H. Georgoulis
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  2. J. Levesley
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  3. F. Subhan
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Corresponding author

Correspondence to E. H. Georgoulis .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

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Georgoulis, E.H., Levesley, J., Subhan, F. (2013). Multilevel Sparse Kernel-Based Interpolation Using Conditionally Positive Definite Radial Basis Functions. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_17

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