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Fast computation of orthonormal basis for RBF spaces through Krylov space methods

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Abstract

In recent years, in the setting of radial basis function, the study of approximation algorithms has particularly focused on the construction of (stable) bases for the associated Hilbert spaces. One of the ways of describing such spaces and their properties is the study of a particular integral operator and its spectrum. We proposed in a recent work the so-called WSVD basis, which is strictly connected to the eigen-decomposition of this operator and allows to overcome some problems related to the stability of the computation of the approximant for a wide class of radial kernels. Although effective, this basis is computationally expensive to compute. In this paper we discuss a method to improve and compute in a fast way the basis using methods related to Krylov subspaces. After reviewing the connections between the two bases, we concentrate on the properties of the new one, describing its behavior by numerical tests.

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Acknowledgments

We would like to thank the anonymous referees who, thanks to their suggestions, allow to significantly improve the paper and the results here presented. The authors have been supported by the funds 2012 of the University of Padua, project CPDA124755 “Multivariate approximation with application to image reconstruction”. Both authors are members of GNCS-INdAM.

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  1. Department of Mathematics, University of Padova, via Trieste 63, Padova, Italy

    Stefano De Marchi & Gabriele Santin

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  1. Stefano De Marchi
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  2. Gabriele Santin
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Correspondence to Gabriele Santin.

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Communicated by Michiel Hochstenbach.

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De Marchi, S., Santin, G. Fast computation of orthonormal basis for RBF spaces through Krylov space methods. Bit Numer Math 55, 949–966 (2015). https://doi.org/10.1007/s10543-014-0537-6

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  • DOI: https://doi.org/10.1007/s10543-014-0537-6

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