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Optimal Selection of Local Approximants in RBF-PU Interpolation

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Abstract

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has been proved to be an effective tool for solving large scattered data interpolation problems. However, in order to achieve a good accuracy, the question about how many points we have to consider on each local subdomain, i.e. how large can be the local data sets, needs to be answered. Moreover, it is well-known that also the shape parameter affects the accuracy of the local RBF approximants and, as a consequence, of the PU interpolant. Thus here, both the shape parameter used to fit the local problems and the size of the associated linear systems are supposed to vary among the subdomains. They are selected by minimizing an a priori error estimate. As evident from extensive numerical experiments and applications provided in the paper, the proposed method turns out to be extremely accurate also when data with non-homogeneous density are considered.

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Acknowledgements

We sincerely thank the two anonymous referees for helping us to significantly improve our paper. This work was partially supported by the project “Metodi e modelli numerici per le scienze applicate” of the Department of Mathematics of the University of Turin.

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Authors and Affiliations

  1. Department of Mathematics “G. Peano”, University of Turin, via Carlo Alberto 10, 10123, Turin, Italy

    Roberto Cavoretto & Alessandra De Rossi

  2. Department of Mathematics “T. Levi-Civita”, University of Padua, via Trieste 63, 35131, Padua, Italy

    Emma Perracchione

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  1. Roberto Cavoretto
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  2. Alessandra De Rossi
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  3. Emma Perracchione
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Correspondence to Emma Perracchione.

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Cavoretto, R., De Rossi, A. & Perracchione, E. Optimal Selection of Local Approximants in RBF-PU Interpolation. J Sci Comput 74, 1–22 (2018). https://doi.org/10.1007/s10915-017-0418-7

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  • DOI: https://doi.org/10.1007/s10915-017-0418-7

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