On Use of Discrete Laplace Operator for Preconditioning Kernel Matrices

TitleOn Use of Discrete Laplace Operator for Preconditioning Kernel Matrices
Publication TypeJournal Article
Year of Publication2012
AuthorsChen, J
JournalSIAM Journal on Scientific Computing
Date Published04/2013
Other NumbersANL/MCS-P2076-0412

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 dierentiate the kernel to suppress the variation. The idea resembles some existing preconditioning methods for Toeplitz matrices, where the theory heavily relies on the underlying xed generating function. The theory does not apply to the case of a xed domain with increasingly ne discretiza-tions, 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 precondi-tioning 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.