Convergence Analysis of Restarted Krylov Subspace Eigensolvers

Neymeyr, Klaus; Zhou, Ming

doi:10.1137/16m1056481

Cited by 3 publications

(20 citation statements)

References 17 publications

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“…By using an A ‐geometry representation as in section 1.3 in the work of Neymeyr et al, the invert‐Lanczos process and its restarted version can be reformulated in terms of a standard eigenvalue problem. Then, the classical convergence estimates for the Lanczos method by other authors can be applied; see, for example, theorem 2 in the work of Saad .…”

Section: Introductionmentioning

confidence: 99%

“…Instead, an estimate in eq. 1.9 by Knyazev is applicable to and has been generalized in theorem 3.5 in the work of Neymeyr et al The result reads: If ρ ( x ( ℓ ) ) is between two neighboring eigenvalues λ i < λ i +1 of ( A , M ), then it holds that

\frac{ρ ((), x^{false(ℓ + 1 false)}) - λ_{i}}{λ_{i + 1} - ρ ((), x^{false(ℓ + 1 false)})} \leq {([], T_{k - 1} false(1 + 2 γ_{i} false))}^{- 2} \frac{ρ ((), x^{false(ℓ false)}) - λ_{i}}{λ_{i + 1} - ρ ((), x^{false(ℓ false)})}

with the Chebyshev polynomial T k −1 and the gap ratio

γ_{i} = false(λ_{i}^{- 1} - λ_{i + 1}^{- 1} false) false/ false(λ_{i + 1}^{- 1} - λ_{\max}^{- 1} false)

.…”

Section: Introductionmentioning

confidence: 99%

“…In order to present the convergence analysis in a simple and compact form, we transform the generalized eigenvalue problem A x = λ M x into the standard eigenvalue problem H y = λ −1 y with the substitutions H = A −1/2 M A −1/2 and y = A 1/2 x as in section 1.3 in the work of Neymeyr et al Subsequently, the block‐Krylov subspace has the simplified notation

span false{scriptY, H scriptY, ⋯, H^{k - 1} scriptY false}

by using the substitution

scriptY = A^{1 false/ 2} scriptX

, and the iteration corresponds to

Y^{false(ℓ + 1 false)} \leftarrow {RR}_{\max} ((), span false{Y^{false(ℓ false)}, H Y^{false(ℓ false)}, ⋯, H^{k - 1} Y^{false(ℓ false)} false}),

where the Rayleigh–Ritz procedure

{RR}_{\max}

yields a subspace spanned by the Ritz vectors associated with the s largest Ritz values.…”

Section: Introductionmentioning

confidence: 99%

“…Some general settings for our analysis are formulated in the following lemma together with a transformation of the estimate ; compare theorem 3.1 in the work of Neymeyr et al…”

Section: Introductionmentioning

confidence: 99%

“…2.22 in the work by Knyazev cannot be generalized in a certain cluster‐robust form without further assumptions. Instead, a generalization based on the results from the work of Neymeyr et al on block steepest descent iterations and single‐vector restarted Krylov subspace iterations is derived and shown to be sharp. In Section 4, the new estimates derived in the two previous sections are reformulated for invert‐block‐Lanczos methods.…”

Section: Introductionmentioning

confidence: 99%

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Convergence estimates of nonrestarted and restarted block‐Lanczos methods

Zhou

2018

Numerical Linear Algebra App

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Summary The block‐Lanczos method serves to compute a moderate number of eigenvalues and the corresponding invariant subspace of a symmetric matrix. In this paper, the convergence behavior of nonrestarted and restarted versions of the block‐Lanczos method is analyzed. For the nonrestarted version, we improve an estimate by Saad by means of a change of the auxiliary vector so that the new estimate is much more accurate in the case of clustered or multiple eigenvalues. For the restarted version, an estimate by Knyazev is generalized by extending our previous results on block steepest descent iterations and single‐vector restarted Krylov subspace iterations. The new estimates can also be reformulated and applied to invert‐block‐Lanczos methods for solving generalized matrix eigenvalue problems.

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Section: Introductionmentioning

confidence: 99%

\frac{ρ ((), x^{false(ℓ + 1 false)}) - λ_{i}}{λ_{i + 1} - ρ ((), x^{false(ℓ + 1 false)})} \leq {([], T_{k - 1} false(1 + 2 γ_{i} false))}^{- 2} \frac{ρ ((), x^{false(ℓ false)}) - λ_{i}}{λ_{i + 1} - ρ ((), x^{false(ℓ false)})}

with the Chebyshev polynomial T k −1 and the gap ratio

γ_{i} = false(λ_{i}^{- 1} - λ_{i + 1}^{- 1} false) false/ false(λ_{i + 1}^{- 1} - λ_{\max}^{- 1} false)

.…”

Section: Introductionmentioning

confidence: 99%

span false{scriptY, H scriptY, ⋯, H^{k - 1} scriptY false}

by using the substitution

scriptY = A^{1 false/ 2} scriptX

, and the iteration corresponds to

Y^{false(ℓ + 1 false)} \leftarrow {RR}_{\max} ((), span false{Y^{false(ℓ false)}, H Y^{false(ℓ false)}, ⋯, H^{k - 1} Y^{false(ℓ false)} false}),

where the Rayleigh–Ritz procedure

{RR}_{\max}

yields a subspace spanned by the Ritz vectors associated with the s largest Ritz values.…”

Section: Introductionmentioning

confidence: 99%

“…Some general settings for our analysis are formulated in the following lemma together with a transformation of the estimate ; compare theorem 3.1 in the work of Neymeyr et al…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergence estimates of nonrestarted and restarted block‐Lanczos methods

Zhou

2018

Numerical Linear Algebra App

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Cluster robust estimates for block gradient-type eigensolvers

Zhou

Neymeyr

2019

Math. Comp.

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Sharp convergence estimates have been derived in recent years for gradient-type eigensolvers for large and sparse symmetric matrices or matrix pairs. An extension of these estimates to the corresponding block iterative methods can be achieved by applying a similar analysis to an embedded vector iteration. Although the resulting estimates are also sharp in the sense that they are not improvable without further assumptions, they cannot reflect the well-known cluster robustness of block eigensolvers. In the present paper, we analyze the cluster robustness of the preconditioned inverse subspace iteration. The main estimate has a weaker assumption and a simpler form compared to some known cluster robust estimates. In addition, it is applicable to further block gradient-type eigensolvers such as LOBPCG. The analysis is based on an orthogonal splitting for the block power method and a geometric interpretation of preconditioning. As a by-product, a cluster robust Ritz value estimate for the block power method is improved.

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