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

In: Numerical Mathematics and Advanced Applications 2011

Author

Listed:
  • E. H. Georgoulis

    (University of Leicester, Department of Mathematics)

  • J. Levesley

    (University of Leicester, Department of Mathematics)

  • F. Subhan

    (University of Leicester, Department of Mathematics)

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.

Suggested Citation

  • E. H. Georgoulis & J. Levesley & F. Subhan, 2013. "Multilevel Sparse Kernel-Based Interpolation Using Conditionally Positive Definite Radial Basis Functions," Springer Books, in: Andrea Cangiani & Ruslan L. Davidchack & Emmanuil Georgoulis & Alexander N. Gorban & Jeremy Levesley (ed.), Numerical Mathematics and Advanced Applications 2011, edition 127, pages 157-164, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-33134-3_17
    DOI: 10.1007/978-3-642-33134-3_17
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