Generic preconditioning based on the inverse with respect to the semi-inner product for iterative solvers for radial basis function interpolation

Abstract

The inverse matrix of radial basis function (RBF) interpolation systems can be stated concisely in terms of an inverse with respect to the semi-inner product induced by the interpolation kernel. Based on this representation, a separation of the solution process is justified and consequently splitting methods and an orthogonal projection method based on the semi-inner norm induced by the RBF are established. The requirements for preconditioning operators are derived and exemplary domain decomposition method preconditioning operators are presented. The introduced representation using the inverse with respect to the semi-inner product clarifies the coherence with well-known concepts from numerical linear algebra. The generic formulation of the preconditioned orthogonal projection method and the requirements for suitable preconditioners serve as building blocks to create solvers tailored for the specific assets of available hardware. Exemplary, design variants of the established subspace projection method and the respective preconditioners are tested on replicable data up to $2^{19}$ interpolation centers.