Rational invariant subspace approximations with applications

Includes bibliographical references.Subspace methods such as MUSIC, Minimum Norm, and ESPRIT have gained considerable attention due to their superior performance in sinusoidal and direction-of-arrival (DOA) estimation, but they are also known to be of high computational cost. In this paper, new fast algorithms for approximating signal and noise subspaces and that do not require exact eigendecomposition are presented. These algorithms approximate the required subspace using rational and power-like methods applied to the direct data or the sample covariance matrix. Several ESPRIT- as well as MUSIC-type methods are developed based on these approximations. A substantial computational saving can be gained comparing with those associated with the eigendecomposition-based methods. These methods are demonstrated to have performance comparable to that of MUSIC yet will require fewer computation to obtain the signal subspace matrix

Rational approximations from power series of vector-valued meromorphic functions

Let F(z) be a vector-valued function, F: C yields C(sup N), which is analytic at z = 0 and meromorphic in a neighborhood of z = 0, and let its Maclaurin series be given. In this work we developed vector-valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyzed some of the algebraic and analytic properties of the rational approximations thus obtained, and showed that they were akin to Pade approximations. In particular, we proved a Koenig type theorem concerning their poles and a de Montessus type theorem concerning their uniform convergence. We showed how optical approximations to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces was also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a forthcoming paper we exploited the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to approximate simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods

Spectral discretization errors in filtered subspace iteration

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates

Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential

The overlap operator in lattice QCD requires the computation of the sign function of a matrix, which is non-Hermitian in the presence of a quark chemical potential. In previous work we introduced an Arnoldi-based Krylov subspace approximation, which uses long recurrences. Even after the deflation of critical eigenvalues, the low efficiency of the method restricts its application to small lattices. Here we propose new short-recurrence methods which strongly enhance the efficiency of the computational method. Using rational approximations to the sign function we introduce two variants, based on the restarted Arnoldi process and on the two-sided Lanczos method, respectively, which become very efficient when combined with multishift solvers. Alternatively, in the variant based on the two-sided Lanczos method the sign function can be evaluated directly. We present numerical results which compare the efficiencies of a restarted Arnoldi-based method and the direct two-sided Lanczos approximation for various lattice sizes. We also show that our new methods gain substantially when combined with deflation.Comment: 14 pages, 4 figures; as published in Comput. Phys. Commun., modified data in Figs. 2,3 and 4 for improved implementation of FOM algorithm, extended discussion of the algorithmic cos

An Entropy Stable Discontinuous Galerkin Finite-Element Moment Method for the Boltzmann Equation

This paper presents a numerical approximation technique for the Boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous Galerkin finite-element approximation in position dependence. The closure relation for the moment systems derives from minimization of a suitable {\phi}-divergence. This divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The resulting combined discontinuous Galerkin moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. We present a new class of numerical flux functions, based on the underlying renormalized Boltzmann equation, that ensure entropy dissipation of the approximation scheme. Numerical results are presented for a one-dimensional test case.Comment: arXiv admin note: substantial text overlap with arXiv:1503.0518