Advertisement

Anisotropic Weights for RBF-PU Interpolation with Subdomains of Variable Shapes

  • R. Cavoretto
  • A. De Rossi
  • G. E. Fasshauer
  • M. J. McCourt
  • E. PerracchioneEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

The partition of unity (PU) method, performed with local radial basis function (RBF) approximants, has already been proved to be an effective tool for solving interpolation or collocation problems when large data sets are considered. It decomposes the original domain into several subdomains or patches so that only linear systems of relatively small size need to be solved. In research on such partition of unity methods, such subdomains usually consist of spherical patches of a fixed radius. However, for particular data sets, such as track data, ellipsoidal patches seem to be more suitable. Therefore, in this paper, we propose a scheme based on a priori error estimates for selecting the sizes of such variable ellipsoidal subdomains. We jointly solve for both these domain decomposition parameters and the anisotropic RBF shape parameters on each subdomain to achieve superior accuracy in comparison to the standard partition of unity method.

Notes

Acknowledgements

This research has been accomplished within RITA (Rete ITaliana di Approssimazione) and partially supported by GNCS-INδAM. The first and second authors were partially supported by the 2016–2017 project Metodi numerici e computazionali per le scienze applicate of the Department of Mathematics of the University of Torino. The third author was partially supported by grant NSF-DMS #1522687. The last author is supported by the research project No. BIRD167404.

References

  1. 1.
    G. Allasia, R. Besenghi, R. Cavoretto, A. De Rossi, Scattered and track data interpolation using an efficient strip searching procedure. Appl. Math. Comput. 217, 5949–5966 (2011)MathSciNetzbMATHGoogle Scholar
  2. 2.
    R. Cavoretto, A. De Rossi, E. Perracchione, Efficient computation of partition of unity interpolants through a block-based searching technique. Comput. Math. Appl. 71, 2568–2584 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. Cavoretto, A. De Rossi, E. Perracchione, Optimal selection of local approximants in RBF-PU interpolation. J. Sci. Comput. 74, 1–22 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    G.E. Fasshauer, M.J. McCourt, Kernel-Based Approximation Methods Using Matlab (World Scientific, Singapore, 2015)CrossRefGoogle Scholar
  5. 5.
    J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optimiz. 9, 112–147 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Rippa, An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11, 193–210 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. Wendland, Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 17 (Cambridge University Press, Cambridge, 2005)Google Scholar
  8. 8.
    H. Wendland, Fast evaluation of radial basis functions: Methods based on partition of unity, in Approximation Theory X: Wavelets, Splines, and Applications, ed. by C.K. Chui et al. (Vanderbilt University Press, Nashville, 2002), pp. 473–483Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • R. Cavoretto
    • 1
  • A. De Rossi
    • 1
  • G. E. Fasshauer
    • 2
  • M. J. McCourt
    • 3
  • E. Perracchione
    • 4
    Email author
  1. 1.Department of Mathematics “Giuseppe Peano”University of TorinoTorinoItaly
  2. 2.Department of Applied Mathematics and StatisticsColorado School of MinesGoldenUSA
  3. 3.SigOpt, Inc.San FranciscoUSA
  4. 4.Department of Mathematics “Tullio Levi-Civita”University of PadovaPadovaItaly

Personalised recommendations