Preconditioning of Radial Basis Function Interpolation Systems via Accelerated Iterated Approximate Moving Least Squares Approximation

Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 11)

The standard approach to the solution of the radial basis function interpolation problem has been recognized as an ill-conditioned problem for many years. This is especially true when infinitely smooth basic functions such as multiquadrics or Gaussians are used with extreme values of their associated shape parameters. Various approaches have been described to deal with this phenomenon. These techniques include applying specialized preconditioners to the system matrix, changing the basis of the approximation space or using techniques from complex analysis. In this paper we present a preconditioning technique based on residual iteration of an approximate moving least squares quasi-interpolant that can be interpreted as a change of basis. In the limit our algorithm will produce the perfectly conditioned cardinal basis of the underlying radial basis function approximation space. Although our method is motivated by radial basis function interpolation problems, it can also be adapted for similar problems when the solution of a linear system is involved such as collocation methods for solving differential equations.

Keywords

Preconditioning methods Radial Basis Functions Accelerated Iterated least squares 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. F. Ashby, T. A. Manteuffel, and J. S. Otto, “A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems”, SIAM J. Sci. Statist. Comput. Vol. 13, pp. 1–29, 1992MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    B. J. C. Baxter, “Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices”, Comput. Math. Appl. Vol. 43, pp. 305–318, 2002MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. K. Beatson, J. B. Cherrie, and C. T. Mouat, “Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration”, Adv. Comput. Math. Vol. 11, pp. 253–270, 1999MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. K. Beatson, W. A. Light, and S. Billings, “Fast solution of the radial basis function interpolation equations: domain decomposition methods”, SIAM J. Sci. Comput. Vol. 22, pp. 1717–1740, 2000MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Benzi, “Preconditioning techniques for large linear systems: a survey”, J. Comput. Phys. Vol. 182, pp. 418–477, 2002MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. Brown, L. Ling, E. Kansa, and J. Levesley, “On approximate cardinal preconditioning methods for solving PDEs with radial basis functions”, Eng. Anal. Bound. Elem. Vol. 29, pp. 343–353, 2005CrossRefGoogle Scholar
  7. 7.
    M. D. Buhmann, “Radial Basis Functions”, Cambridge University Press, New York, 2003MATHGoogle Scholar
  8. 8.
    L. Cesari, “Sulla risoluzione dei sistemi di equazioni lineari per approssimazioni successive”, Ricerca Sci., Roma Vol. 2 8I, pp. 512–522, 1937Google Scholar
  9. 9.
    C. S. Chen, H. A. Cho and M. A. Golberg, “Some comments on the ill-conditioning of the method of fundamental solutions”, Eng. Anal. Bound. Elem. Vol. 30, pp. 405–410, 2006CrossRefGoogle Scholar
  10. 10.
    S. De Marchi and R. Schaback, “Stability of Kernel-Based Interpolation”, preprint, 2007Google Scholar
  11. 11.
    T. A. Driscoll and B. Fornberg, “Interpolation in the limit of increasingly flat radial basis functions”, Comput. Math. Appl., Vol. 43, pp. 413–422, 2002MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    T. A. Driscoll and A. R. H. Heryudono, “Adaptive residual subsampling methods for radial basis function interpolation and collocation problems”, Comput. Math. Appl., Vol. 53, pp. 927–939, 2007MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    N. Dyn, “Interpolation of scattered data by radial functions”, in Topics in Multivariate Approximation, C. K. Chui, L. L. Schumaker, and F. Utreras (eds.), Academic New York, pp. 47–61, 1987Google Scholar
  14. 14.
    N. Dyn, D. Levin, and S. Rippa, “Numerical procedures for surface fitting of scattered data by radial functions”, SIAM J. Sci. Statist. Comput. Vol. 7, pp. 639–659, 1986MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    G. E. Fasshauer, “Solving partial differential equations by collocation with radial basis functions”, in Surface Fitting and Multiresolution Methods, A. Le Mehaute, C. Rabut, and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, pp. 131–138, 1997Google Scholar
  16. 16.
    G. E. Fasshauer, “Meshfree approximation methods with Matlab”, Interdisciplinary Mathematical Sciences, Vol. 6, World Scientific Publishers, Singapore, 2007Google Scholar
  17. 17.
    G. E. Fasshauer and J. G. Zhang, “Scattered data approximation of noisy data via iterated moving least squares”, in Curve and Surface Fitting: Avignon 2006, T. Lyche, J. L. Merrien and L. L. Schumaker (eds.), Nashboro Press, Brentwood, TN, pp. 150–159, 2007Google Scholar
  18. 18.
    G. E. Fasshauer and J. G. Zhang, “Iterated approximate moving least squares approximation”, in Advances in Meshfree Techniques, V. M. A. Leitao, C. Alves and C. A. Duarte (eds.), Springer, New York, pp. 221–240, 2007CrossRefGoogle Scholar
  19. 19.
    G. E. Fasshauer and J. G. Zhang, “On choosing “optimal” shape parameters for RBF approximation”, Numer. Algorithms Vol. 45, pp. 345–368, 2007MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    B. Fornberg and C. Piret, “A stable algorithm for flat radial basis functions on a sphere”, SIAM J. Sci. Comp. Vol. 30, pp. 60–80, 2007CrossRefMathSciNetGoogle Scholar
  21. 21.
    B. Fornberg and G. Wright, “Stable computation of multiquadric interpolants for all values of the shape parameter”, Comput. Math. Appl. Vol. 47, pp. 497–523, 2004CrossRefMathSciNetGoogle Scholar
  22. 22.
    B. Fornberg and J. Zuev, “The Runge phenomenon and spatially variable shape parameters in RBF interpolation”, Comput. Math. Appl. Vol. 54, pp. 379–398, 2007MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    E. J. Kansa and R. E. Carlson. “Improved accuracy of multiquadric interpolation using variable shape parameters”, Comput. Math. Applic. Vol. 24, pp. 99–120, 1992MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    C.-F. Lee, L. Ling and R. Schaback, “On convergent numerical algorithms for unsymmetric collocation”, Adv. Comp. Math, to appearGoogle Scholar
  25. 25.
    L. Ling and E. J. Kansa, “Preconditioning for radial basis functions with domain decomposition methods”, Math. Comput. Model. Vol. 40, pp. 1413–1427, 2004MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    L. Ling and E. J. Kansa, “A least-squares preconditioner for radial basis functions collocation methods”, Adv. Comp. Math. Vol. 23, pp. 31–54, 2005MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    L. Ling and R. Schaback, “Stable and convergent unsymmetric meshless collocation methods”, SIAM J. Numer. Anal., to appearGoogle Scholar
  28. 28.
    V. Maz'ya, “A new approximation method and its applications to the calculation of volume potentials. Boundary point method”, in DFG-Kolloquium des DFG-Forschungsschwerpunktes “Randelementmethoden”, 1991Google Scholar
  29. 29.
    V. Maz'ya and G. Schmidt, “On quasi-interpolation with non-uniformly distributed centers on domains and manifolds”, J. Approx. Theory, Vol. 110, pp. 125–145, 2001MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    S. Rippa, “Algorithm for selecting a good value for the parameter c in radial basis function interpolation”, Adv. Comput. Math. Vol. 11, pp. 193–210, 1999MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005MATHGoogle Scholar
  32. 32.
    J. G. Zhang, “Iterated Approximate Moving Least-Squares: Theory and Applications”, Ph.D. Dissertation, Illinois Institute of Technology, 2007Google Scholar

Copyright information

© Springer Science + Business Media B.V 2009

Authors and Affiliations

  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA

Personalised recommendations