BIT Numerical Mathematics

, Volume 53, Issue 1, pp 57–86 | Cite as

A discrete adapted hierarchical basis solver for radial basis function interpolation

  • Julio E. Castrillón-CandásEmail author
  • Jun Li
  • Victor Eijkhout


In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal (or block SSOR) preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness.


Radial basis function Interpolation Hierarchical basis Integral equations Fast summation methods Stable completion Lifting Generalized least squares Best linear unbiased estimator 

Mathematics Subject Classification (2010)

65D05 65D07 65F25 65F10 62J05 41A15 



We are grateful to Lexing Ying for providing a single processor version of the KIFMM3d code. We also appreciate the discussions, assistance and feedback from Raul Tempone, Robert Van De Gein, Vinay Siddavanahalli and the members of the Computational Visualization Center (Institute for Computational Engineering and Sciences) at the University of Texas at Austin. In addition, we appreciate the invaluable feedback from the reviewers of this paper.


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Copyright information

© Springer Science + Business Media B.V. 2012

Authors and Affiliations

  • Julio E. Castrillón-Candás
    • 1
    Email author
  • Jun Li
    • 2
  • Victor Eijkhout
    • 3
  1. 1.King Abdullah University of Science and TechnologyThuwalKingdom of Saudi Arabia
  2. 2.SchlumbergerHoustonUSA
  3. 3.Texas Advanced Computing CenterUniversity of Texas at AustinAustinUSA

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