A discrete adapted hierarchical basis solver for radial basis function interpolation

A discrete adapted hierarchical basis solver for radial basis function interpolation

Julio E. Castrillon-Candas, Jun Li, Victor Eijkhout. "A discrete adapted hierarchical basis solver for radial basis function interpolation." BIT Numerical Mathematics, Volume 53, Issue 1 (2013), Page 57-86‚Äč
Julio E. Castrillon-Candas, Jun Li, Victor Eijkhout
Radial basis function, Interpolation, Hierarchical basis, Integral equations, Fast summation methods, Stable completion, Lifting, Generalized least squares, Best linear unbiased estimator
2013
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.
BIT Numerical Mathematics