Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation

Xue, Fei; Elman, Howard C.

doi:10.1016/j.laa.2010.06.021

Cited by 25 publications

(18 citation statements)

References 24 publications

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“…In this work, we show that if A is diagonalizable the convergence theory developed in [28] yields an insightful explanation for the GMRES behavior in inverse iteration with different types of preconditioned inner solves. Moreover, we explain why the GMRES residual often decreases sharply in the first iteration [31]. A more detailed description of this phenomenon is given in Section 2.2.…”

Section: Algorithm 1: Inexact Inverse Iterationmentioning

confidence: 89%

GMRES Convergence Bounds for Eigenvalue Problems

Freitag

Kürschner

Pestana

2017

Computational Methods in Applied Mathematics

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The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g. tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.

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Section: Algorithm 1: Inexact Inverse Iterationmentioning

confidence: 89%

GMRES Convergence Bounds for Eigenvalue Problems

Freitag

Kürschner

Pestana

2017

Computational Methods in Applied Mathematics

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“…An important consideration independent of the convergence rate of the outer iteration is the choice of the preconditioner for the solution to the linear systems. For standard (one‐sided) eigenproblems, a ‘tuned’ preconditioner, a rank‐1 modification of the standard preconditioner, reduces the number of iterations for the inner solve considerably , a result that has been extended to inverse subspace iteration in . As our main novel contribution of this article, we extend the result to two‐sided inverse iteration and RQI, where, due to the structure and the simultaneous solution of a forward and adjoint linear system, a rank‐2 modification of the standard preconditioner is necessary for an efficient tuning strategy.…”

Section: Motivationmentioning

confidence: 94%

“…The following lemma explains the role the right‐hand side plays in the solution of the linear system C x = b when Krylov subspace methods are used. It follows directly from , Theorem 3.7], , Lemma 3.1]. Lemma Suppose the field of values

W (C_{2}) = ({}, \frac{z^{*} C_{2} z}{z^{*} z} : z \in {double-struckC}^{n - 1}, z \neq 0)

or the ε ‐pseudospectrum

{normalΛ}_{ε} = {z \in {double-struckC}^{n - 1} : ∥ {(zI - C_{2})}^{- 1} ∥ > ε}

is contained in a convex closed bounded set E in the convex plane with 0 ∉ E .…”

Section: The Inner Iteration and Tuned Preconditionersmentioning

confidence: 99%

“…For details about D a and D b , we refer the reader to , for our analysis these constants have very little significance. We have the following theorem for the number of inner iterations of the unpreconditioned algorithm (see [, Theorem 3.2]) applied to the forward linear system. Theorem Assume that (un)preconditioned GMRES is used to solve the linear system C u k + 1 = M u k , where C = A − θ M to the prescribed tolerance in , and

ξ_{k}^{R} = η^{R} ∥ r_{u_{k}} ∥

from .…”

Section: The Inner Iteration and Tuned Preconditionersmentioning

confidence: 99%

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Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration

Freitag

Kürschner

2014

Numer. Linear Algebra Appl.

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SUMMARYConvergence results are provided for inexact two‐sided inverse and Rayleigh quotient iteration, which extend the previously established results to the generalized non‐Hermitian eigenproblem and inexact solves with a decreasing solve tolerance. Moreover, the simultaneous solution of the forward and adjoint problem arising in two‐sided methods is considered, and the successful tuning strategy for preconditioners is extended to two‐sided methods, creating a novel way of preconditioning two‐sided algorithms. Furthermore, it is shown that inexact two‐sided Rayleigh quotient iteration and the inexact two‐sided Jacobi‐Davidson method (without subspace expansion) applied to the generalized preconditioned eigenvalue problem are equivalent when a certain number of steps of a Petrov–Galerkin–Krylov method is used and when this specific tuning strategy is applied. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Belos provides two single-vector recycling solvers: Recycling GM-RES (GCRO-DR) and Recycling CG. Sandia National Laboratories and Temple University are currently collaborating on a Block Recycling GMRES algorithm to be deployed in Belos [53,61].…”

Section: Recycling Solversmentioning

confidence: 99%

Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

Bavier¹,

Hoemmen

Rajamanickam

et al. 2012

Scientific Programming

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Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.

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Fast inexact subspace iteration for generalized eigenvalue problems with spectral transformation

Cited by 25 publications

References 24 publications

GMRES Convergence Bounds for Eigenvalue Problems

GMRES Convergence Bounds for Eigenvalue Problems

Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration

Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

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