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In this paper we develop a discrete Hierarchical Basis (HB) to efficientlysolve the Radial Basis Function (RBF) interpolation problem with variablepolynomial order. The HB forms an orthogonal set and is adapted to the kernelseed function and the placement of the interpolation nodes. Moreover, thisbasis is orthogonal to a set of polynomials up to a given order defined on theinterpolating nodes. We are thus able to decouple the RBF interpolation problemfor any order of the polynomial interpolation and solve it in two steps: (1)The polynomial orthogonal RBF interpolation problem is efficiently solved inthe transformed HB basis with a GMRES iteration and a diagonal, or block SSORpreconditioner. (2) The residual is then projected onto an orthonormalpolynomial basis. We apply our approach on several test cases to study itseffectiveness, including an application to the Best Linear Unbiased Estimatorregression problem.

A Discrete Adapted Hierarchical Basis Solver For Radial Basis Function Interpolation