A Discrete Adapted Hierarchical Basis Solver For Radial Basis Function Interpolation

TL;DR

This paper introduces a discrete hierarchical basis method for efficiently solving variable polynomial order RBF interpolation problems, improving computational efficiency and flexibility.

Contribution

The paper develops a novel orthogonal hierarchical basis tailored to RBF kernels and node placement, enabling decoupled, efficient polynomial and RBF interpolation solutions.

Findings

Effective in solving variable polynomial order RBF interpolation

Reduces computational complexity with orthogonal basis and preconditioning

Demonstrated success in test cases including regression applications

Abstract

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial order. 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 order defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any order 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, including an application to the Best Linear Unbiased Estimator regression problem.