On the Use of Discrete Laplace Operator for Preconditioning Kernel Matrices

Abstract

This paper presents a preconditioning strategy applied to certain types of kernel matrices that are increasingly ill-conditioned. The ill-conditioning of these matrices is tied to the unbounded variation of the Fourier transform of the kernel function. Hence, the basic idea is to differentiate the kernel in order to suppress the variation. The idea resembles some existing preconditioning methods for Toeplitz matrices, where the theory heavily relies on the underlying fixed generating function. The theory does not apply to the case of a fixed domain with increasingly fine discretizations because the generating function depends on the grid size. For this case, we prove equal distribution results on the spectrum of the resulting matrices. Furthermore, the proposed preconditioning technique also applies to non-Toeplitz matrices, thus eliminating the reliance on a regular grid structure of the points. The preconditioning strategy can be used to accelerate an iterative solver for solving linear systems with res...