Sharp Convergence Estimates for the Preconditioned Steepest Descent Method for Hermitian Eigenvalue Problems

Ovtchinnikov, Evgueni

doi:10.1137/040620643

Cited by 21 publications

(22 citation statements)

References 11 publications

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“…The estimate of the previous section (which is asymptotically sharp owing to the sharpness of (2.20); cf. also [18]) indicates that the convergence of steepest descent iterations for such problems may be very slow, and, indeed, this is the case in practical computation, the same being true for a closely related CG method. In this section we briefly discuss a technique, known as preconditioning, that can be employed to accelerate the convergence, and apply the results of section 2.2 to the convergence analysis of the preconditioned steepest descent.…”

Section: Jacobi Correction Operator and Preconditioningmentioning

confidence: 82%

“…The derivation of the above convergence result for the preconditioned steepest descent for the Rayleigh quotient minimization is remarkably simple, compared to much more complicated proofs of similar results that can be found in the literature; see, e.g., [8,13,12] and the relevant results in [16,17,18]. 2 Admittedly, (2.24) is given in different terms.…”

Section: Jacobi Correction Operator and Preconditioningmentioning

confidence: 93%

“…It should be noted, however, that[13,12] and[18] present sharp estimates, whereas (2.24) is only asymptotically sharp. Downloaded 12/30/12 to 129.173.72.87.…”

mentioning

confidence: 87%

“…All of the results of this paper apply to the generalized eigenvalue problem after simple modifications (cf. the introduction in [18] for the relevant arguments; cf. also Remark 1 in section 2.1).…”

Section: Introductionmentioning

confidence: 99%

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Jacobi Correction Equation, Line Search, and Conjugate Gradients in Hermitian Eigenvalue Computation I: Computing an Extreme Eigenvalue

Ovtchinnikov¹

2008

SIAM J. Numer. Anal.

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This paper is concerned with the convergence properties of iterative algorithms of conjugate gradient type applied to the computation of an extreme eigenvalue of a Hermitian operator via the optimization of the Rayleigh quotient functional. The algorithms in focus employ the line search for the extremum of the Rayleigh quotient functional in the direction that is a linear combination of the gradient and the previous search direction. An asymptotically explicit equation for the reduction in the eigenvalue error after the line search step as a function of the search direction is derived and applied to the analysis of the local convergence properties (i.e., the error reduction after an iteration) of the algorithms in question. The local (stepwise) asymptotic equivalence of various conjugate gradient algorithms and the generalized Davidson method is proved, and a new convergence estimate for conjugate gradient iterations is derived, showing the reduction in the eigenvalue error after any two consecutive iterations. The paper's analysis extensively employs remarkable properties of the operator of the so-called Jacobi orthogonal complement correction equation.

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Section: Jacobi Correction Operator and Preconditioningmentioning

confidence: 82%

Section: Jacobi Correction Operator and Preconditioningmentioning

confidence: 93%

“…It should be noted, however, that[13,12] and[18] present sharp estimates, whereas (2.24) is only asymptotically sharp. Downloaded 12/30/12 to 129.173.72.87.…”

mentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

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Jacobi Correction Equation, Line Search, and Conjugate Gradients in Hermitian Eigenvalue Computation I: Computing an Extreme Eigenvalue

Ovtchinnikov¹

2008

SIAM J. Numer. Anal.

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“…The reader is referred to [29,49] for details. Theorem 9.2 below is an extension of Samokish's result for an HQEP.…”

Section: Convergence Analysismentioning

confidence: 99%

The Hyperbolic Quadratic Eigenvalue Problem

Liang

2015

Forum math. Sigma

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The hyperbolic quadratic eigenvalue problem (HQEP) was shown to admit Courant-Fischer type min-max principles in 1955 by Duffin and Cauchy type interlacing inequalities in 2010 by Veselić. It can be regarded as the closest analog (among all kinds of quadratic eigenvalue problem) to the standard Hermitian eigenvalue problem (among all kinds of standard eigenvalue problem). In this paper, we conduct a systematic study on the HQEP both theoretically and numerically. On the theoretical front, we generalize Wielandt-Lidskii type min-max principles and, as a special case, Fan type trace min/max principles and establish Weyl type and Wielandt-Lidskii-Mirsky type perturbation results when an HQEP is perturbed to another HQEP. On the numerical front, we justify the natural generalization of the Rayleigh-Ritz procedure with existing principles and our new optimization principles, and, as consequences of these principles, we extend various current optimization approaches-steepest descent/ascent and nonlinear conjugate gradient type methods for the Hermitian eigenvalue problem-to calculate a few extreme eigenvalues (of both positive and negative type). A detailed convergence analysis is given for the steepest descent/ascent methods. The analysis reveals the intrinsic quantities that control convergence rates and consequently yields ways of constructing effective preconditioners. Numerical examples are presented to demonstrate the proposed theory and algorithms.2010 Mathematics Subject Classification: 15A42, 65F15 (primary); 15A18, 65F08, 65F50, 65G99 (secondary)

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Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation

Zhou

Bai

Cai

et al. 2023

Numerical Linear Algebra App

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Gradient‐type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD‐id) is one of such methods. The convergence behavior of the PSD‐id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non‐asymptotic estimates indicate a superlinear convergence of the PSD‐id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD‐id using a restricted formulation of the PSD‐id. More importantly, we extend the new convergence analysis of the PSD‐id to a practically preferred block version of the PSD‐id, or BPSD‐id, and show the cluster robustness of the BPSD‐id. Numerical examples are provided to validate the theoretical estimates.

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Sharp Convergence Estimates for the Preconditioned Steepest Descent Method for Hermitian Eigenvalue Problems

Cited by 21 publications

References 11 publications

Jacobi Correction Equation, Line Search, and Conjugate Gradients in Hermitian Eigenvalue Computation I: Computing an Extreme Eigenvalue

Jacobi Correction Equation, Line Search, and Conjugate Gradients in Hermitian Eigenvalue Computation I: Computing an Extreme Eigenvalue

The Hyperbolic Quadratic Eigenvalue Problem

Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation

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