176 research outputs found
Two harmonic Jacobi--Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair
Two harmonic extraction based Jacobi--Davidson (JD) type algorithms are
proposed to compute a partial generalized singular value decomposition (GSVD)
of a large regular matrix pair. They are called cross product-free (CPF) and
inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and
IF-HJDGSVD, respectively. Compared with the standard extraction based JDGSVD
algorithm, the harmonic extraction based algorithms converge more regularly and
suit better for computing GSVD components corresponding to interior generalized
singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some
deflation and purgation techniques are developed to compute more than one GSVD
components. Numerical experiments confirm the superiority of CPF-HJDGSVD and
IF-HJDGSVD to the standard extraction based JDGSVD algorithm.Comment: 24 pages, 5 figure
Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector
We consider algebraic iterative reconstruction methods with applications in
image reconstruction. In particular, we are concerned with methods based on an
unmatched projector/backprojector pair; i.e., the backprojector is not the
exact adjoint or transpose of the forward projector. Such situations are common
in large-scale computed tomography, and we consider the common situation where
the method does not converge due to the nonsymmetry of the iteration matrix. We
propose a modified algorithm that incorporates a small shift parameter, and we
give the conditions that guarantee convergence of this method to a fixed point
of a slightly perturbed problem. We also give perturbation bounds for this
fixed point. Moreover, we discuss how to use Krylov subspace methods to
efficiently estimate the leftmost eigenvalue of a certain matrix to select a
proper shift parameter. The modified algorithm is illustrated with test
problems from computed tomography
The joint bidiagonalization of a matrix pair with inaccurate inner iterations
The joint bidiagonalization (JBD) process iteratively reduces a matrix pair
to two bidiagonal forms simultaneously, which can be used for
computing a partial generalized singular value decomposition (GSVD) of
. The process has a nested inner-outer iteration structure, where the
inner iteration usually can not be computed exactly. In this paper, we study
the inaccurately computed inner iterations of JBD by first investigating
influence of computational error of the inner iteration on the outer iteration,
and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality
of a part of Lanczos vectors. An error analysis of the rJBD is carried out to
build up connections with Lanczos bidiagonalizations. The results are then used
to investigate convergence and accuracy of the rJBD based GSVD computation. It
is shown that the accuracy of computed GSVD components depend on the computing
accuracy of inner iterations and condition number of while the
convergence rate is not affected very much. For practical JBD based GSVD
computations, our results can provide a guideline for choosing a proper
computing accuracy of inner iterations in order to obtain approximate GSVD
components with a desired accuracy. Numerical experiments are made to confirm
our theoretical results
Theoretical and Computable Optimal Subspace Expansions for Matrix Eigenvalue Problems
Consider the optimal subspace expansion problem for the matrix eigenvalue
problem : {\em Which vector in the current subspace
, after multiplied by , provides an optimal subspace expansion
for approximating a desired eigenvector in the sense that has the
smallest angle with the expanded subspace , i.e.,
}? This problem
is important as many iterative methods construct nested subspaces that
successively expand to . Ye ({\em Linear Algebra
Appl.}, 428 (2008), p. 911--918) derives an expression of for
general, but it could not be exploited to construct a computable (nearly)
optimally expanded subspace. He turns to deriving a maximization
characterization of for a {\em given} when is Hermitian, but his proof and analysis cannot extend to
the non-Hermitian case. We generalize Ye's maximization characterization to the
general case and find its maximizer. Our main contributions consist of explicit
expressions of , and the optimally expanded subspace
for general, where is the orthogonal
projector onto . These results can be fully exploited to obtain
computable optimally expanded subspaces
within the framework of the standard, harmonic, refined, and refined harmonic
Rayleigh--Ritz methods.Comment: 20 pages, 3 figure
A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc
In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. 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A Jacobi–Davidson type method for the generalized singular value problem
AbstractWe discuss a new method for the iterative computation of some of the generalized singular values and vectors of a large sparse matrix. Our starting point is the augmented matrix formulation of the GSVD. The subspace expansion is performed by (approximately) solving a Jacobi–Davidson type correction equation, while we give several alternatives for the subspace extraction. Numerical experiments illustrate the performance of the method
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