On the Use of Discrete Laplace Operator for Preconditioning Kernel Matrices

This paper studies 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 technique is to differentiate the kernel 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 ridding 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 respect to kernel matrices.