A discrete adapted hierarchical basis solver for radial basis function interpolation |
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Authors: | Julio E Castrillón-Candás Jun Li Victor Eijkhout |
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Institution: | 1. King Abdullah University of Science and Technology, Building #1 (UN 1550), Office No 4204, Level 4, Thuwal, 23955-6900, Kingdom of Saudi Arabia 2. Schlumberger, 5599 San Felipe, Ste 100, Houston, TX, 77056, USA 3. Texas Advanced Computing Center, University of Texas at Austin, Austin, USA
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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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