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A discrete adapted hierarchical basis solver for radial basis function interpolation

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. King Abdullah University of Science and Technology, Building #1 (UN 1550), Office No 4204, Level 4, Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Julio E. Castrillón-Candás

  2. Schlumberger, 5599 San Felipe, Ste 100, Houston, TX, 77056, USA

    Jun Li

  3. Texas Advanced Computing Center, University of Texas at Austin, Austin, USA

    Victor Eijkhout

Authors
  1. Julio E. Castrillón-Candás
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  2. Jun Li
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Corresponding author

Correspondence to Julio E. Castrillón-Candás.

Additional information

Communicated by Michiel Hochstenbach.

J.E. Castrillón-Candás was supported in part by the Institute for Computational Engineering and Sciences Postdoctoral Fellowship at the University of Texas at Austin.

J. Li was supported in part by grant # NIH R01 GM074258.

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Castrillón-Candás, J.E., Li, J. & Eijkhout, V. A discrete adapted hierarchical basis solver for radial basis function interpolation. Bit Numer Math 53, 57–86 (2013). https://doi.org/10.1007/s10543-012-0397-x

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