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

  • Gregory E. Fasshauer
  • Jack G. Zhang
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.


Preconditioning methods Radial Basis Functions Accelerated Iterated least squares 


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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

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