Asymptotic Convergence of Conjugate Gradient Methods for the Partial Symmetric Eigenproblem

Bergamaschi, Luca; Gambolati, Giuseppe; Pini, Giorgio

doi:10.1002/(sici)1099-1506(199703/04)4:2<69::aid-nla98>3.0.co;2-f

Cited by 40 publications

(45 citation statements)

References 18 publications

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“…The vector is then used to enlarge the search space from which a new approximation is obtained through the Rayleigh-Ritz ALGORITHM 2.1. The Generalized Davidson algorithm for one eigenpair (1) start with v 0 starting vector (2) t (0) = v 0 , m = 0, nmv = 0 (3) while nmv < max num matvecs (5) Orthonormalize t (m) against v i , i = 1, . .…”

Section: The Generalized-davidson As An Outer Iteration Modelmentioning

confidence: 99%

“…The appropriate forms of the constrained version for the NLCG and for the case of many required eigenvalues are given in [15]. Researchers have been using similar type of recurrences quite successfully for many decades; see Bradbury and Fletcher's seminal work [8], a long list of references in [15], as well as work in [20,5].…”

Section: The Conjugate Gradients Viewmentioning

confidence: 99%

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Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems under Limited Memory. Part I: Seeking One Eigenvalue

Stathopoulos¹

2007

SIAM J. Sci. Comput.

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Abstract. Large, sparse, Hermitian eigenvalue problems are still some of the most computationally challenging tasks. Despite the need for a robust, nearly optimal preconditioned iterative method that can operate under severe memory limitations, no such method has surfaced as a clear winner. In this research we approach the eigenproblem from the nonlinear perspective that helps us develop two nearly optimal methods. The first extends the recent Jacobi-Davidson-Conjugate-Gradient (JDCG) method to JDQMR, improving robustness and efficiency. The second method, Generalized-Davidson+1 (GD+1), utilizes the locally optimal Conjugate Gradient recurrence as a restarting technique to achieve almost optimal convergence. We describe both methods within a unifying framework, and provide theoretical justification for their near optimality. A choice between the most efficient of the two can be made at runtime. Our extensive experiments confirm the robustness, the near optimality, and the efficiency of our multimethod over other state-of-the-art methods.1. Introduction. The numerical solution of large, sparse, Hermitian and real symmetric eigenvalue problems is central to many applications in science and engineering. It is also one of the most time consuming tasks. Recently, electronic structure calculations, with eigenproblems at their core, have displaced Quantum Chromodynamics as the top supercomputer cycle user. The symmetric eigenvalue problem seems deceptively simple to solve, given well conditioned eigenvalues and a wealth of theoretical knowledge. Yet, these advantages have enabled researchers to push modeling accuracy to unprecedented levels, routinely solving for a few extreme eigenvalues of matrices of size more than a million, while an order of a billion has also been tried [48].The sheer size of these problems can only be addressed by iterative methods. At the same time, the size imposes limits on the memory available to these methods. Moreover, preconditioning becomes imperative to reduce the total number of iterations. While many eigenvalue iterative methods have been proposed, there is little consensus on which method is the best and in what situations. The unrestarted Lanczos method is known to be optimal for solving Hermitian eigenvalue problems but, unlike the Conjugate Gradient (CG) method for linear systems, it requires unlimited storage of its iteration vectors. With preconditioning, or under limited storage, the question of optimality remains open. Furthermore, there is a noticeable scarcity of high quality, general purpose software for preconditioned eigensolvers. In this research we seek an optimal, or nearly optimal, method that can utilize preconditioning and that can be implemented in a robust software that is also flexible.In the particular case of seeking one eigenpair, if the eigenvalue were known, solving a linear system using CG to obtain the corresponding eigenvector would yield the optimal Lanczos convergence. In practice both the eigenvalue and the eigenvector are unknown, so the appropriate w...

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Section: The Generalized-davidson As An Outer Iteration Modelmentioning

confidence: 99%

Section: The Conjugate Gradients Viewmentioning

confidence: 99%

Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems under Limited Memory. Part I: Seeking One Eigenvalue

Stathopoulos¹

2007

SIAM J. Sci. Comput.

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“…Concerning the choice of the Krylov solver, experiments with GMRES, CGS [32], and Bi-CGSTAB [38] were performed by our group [6]. Further experiments with TFQMR [15] were also carried out.…”

Section: Sequential Jdmentioning

confidence: 99%

“…Several techniques for solving this problem have been proposed: Subspace iteration [3,31], Lanczos method [14,23,30]; more recently, the restarted ArnoldiLanczos algorithm [24], the Jacobi-Davidson method [34], hereafter labeled JD, and optimization methods by Conjugate Gradient schemes [5,21,33].…”

Section: Introductionmentioning

confidence: 99%

Computational experience with sequential and parallel, preconditioned Jacobi–Davidson for large, sparse symmetric matrices

Bergamaschi

Pini

Sartoretto³

2003

Journal of Computational Physics

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The Jacobi-Davidson (JD) algorithm was recently proposed for evaluating a number of the eigenvalues of a matrix. JD goes beyond pure Krylov-space techniques; it cleverly expands its search space, by solving the so-called correction equation, thus in principle providing a more powerful method. Preconditioning the Jacobi-Davidson correction equation is mandatory when large, sparse matrices are analyzed. We considered several preconditioners: Classical block-Jacobi, and IC(0), together with approximate inverse (AINV or FSAI) preconditioners. The rationale for using approximate inverse preconditioners is their high parallelization potential, combined with their efficiency in accelerating the iterative solution of the correction equation. Analysis was carried on the sequential performance of preconditioned JD for the spectral decomposition of large, sparse matrices, which originate in the numerical integration of partial differential equations arising in physical and engineering problems. It was found that JD is highly sensitive to preconditioning, and it can display an irregular convergence behavior. We parallelized JD by data-splitting techniques, combining them with techniques to reduce the amount of communication data. Our own parallel, preconditioned code was executed on a dedicated parallel machine, and we present the results of our experiments. Our JD code provides an appreciable parallel degree of computation. Its performance was also compared with those of PARPACK and parallel DACG.

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“…The setup time often increases due to the increasing complexity of the preconditioner but it can be kept under control mainly by proper choice of the prefiltration parameters. We also successfully tried RFSAI to accelerate a PCG-like iterative eigensolver (DACG, see [4]) to compute the leftmost eigenpairs of our SPD test matrices.…”

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confidence: 99%

Banded target matrices and recursive FSAI for parallel preconditioning

Bergamaschi

Martínez

2012

Numer Algor

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Abstract. In this paper we propose a parallel preconditioner for the CG solver based on successive applications of the FSAI preconditioner. We first compute an FSAI factor Gout for coefficient matrix A, and then another FSAI preconditioner is computed for either the preconditioned matrix S = GoutAG T out or a sparse approximation of S. This process can be iterated to obtain a sequence of triangular factors whose product forms the final preconditioner. Numerical results onto large SPD matrices arising from geomechanical models account for the efficiency of the proposed preconditioner which provides a reduction of the iteration number and of the CPU time of the iterative phase with respect to the original FSAI preconditioner.

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Asymptotic Convergence of Conjugate Gradient Methods for the Partial Symmetric Eigenproblem

Cited by 40 publications

References 18 publications

Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems under Limited Memory. Part I: Seeking One Eigenvalue

Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems under Limited Memory. Part I: Seeking One Eigenvalue

Computational experience with sequential and parallel, preconditioned Jacobi–Davidson for large, sparse symmetric matrices

Banded target matrices and recursive FSAI for parallel preconditioning

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