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.