Localized spectrum slicing

Abstract

Given a sparse Hermitian matrix $A$ and a real number $\mu$, we construct a set of sparse vectors, each approximately spanned only by eigenvectors of $A$ corresponding to eigenvalues near $\mu$. This set of vectors spans the column space of a localized spectrum slicing (LSS) operator, and is called an LSS basis set. The sparsity of the LSS basis set is related to the decay properties of matrix Gaussian functions. We present a divide-and-conquer strategy with controllable error to construct the LSS basis set. This is a purely algebraic process using only submatrices of $A$, and can therefore be applied to general sparse Hermitian matrices. The LSS basis set leads to sparse projected matrices with reduced sizes, which allows the projected problems to be solved efficiently with techniques using sparse linear algebra. As an example, we demonstrate that the LSS basis set can be used to solve interior eigenvalue problems for a discretized second order partial differential operator in one-dimensional and two-dimensional domains, as well as for a matrix of general sparsity pattern.