Preconditioned Conjugate Gradients, Radial Basis Functions and Toeplitz Matrices

TL;DR

This paper introduces an efficient preconditioner for conjugate gradient methods solving Toeplitz matrix systems arising from radial basis function interpolation, significantly reducing iteration counts regardless of problem size.

Contribution

The paper develops a novel preconditioning technique tailored for Toeplitz matrices in radial basis function interpolation, enhancing computational efficiency.

Findings

Iteration count is independent of the number of variables.

Preconditioning significantly accelerates convergence.

Method applies to multivariate PDE solutions.

Abstract

Radial basis functions provide highly useful and flexible interpolants to multivariate functions. Further, they are beginning to be used in the numerical solution of partial differential equations. Unfortunately, their construction requires the solution of a dense linear system. Therefore much attention has been given to iterative methods. In this paper, we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data. Thus our method applies to the corresponding Toeplitz matrices. The number of iterations required to achieve a given tolerance is independent of the number of variables.