A Robust GMRES-Based Adaptive Polynomial Preconditioning Algorithm for Nonsymmetric Linear Systems

Joubert, Wayne

doi:10.1137/0915029

Cited by 44 publications

(29 citation statements)

References 15 publications

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“…As shown independently in [4] and [8], for each normal matrix A and each step i, there exists an initial residual r (i) 0 so that equality holds in (3.2). Clearly, the ith GMRES residual corresponding to r (i) 0 is an ith worst-case GMRES residual for A in the sense of Definition 3.1.…”

Section: Worst-case Residual Normmentioning

confidence: 99%

“…The result of [4,8], which will play an important role in our further development, can in this notation be phrased as follows: For each normal matrix A ∈ C n×n (with n distinct eigenvalues) and each i = 1, . .…”

Section: Worst-case Residual Normmentioning

confidence: 99%

“…Ideal GMRES approximation. The proofs of (3.4) in [4,8] are based on intricate constructions. For the special case i = n − 1 we now give a simple proof of (3.4), i.e.…”

Section: Worst Case In Stepmentioning

confidence: 99%

“…Being independent of the initial residual, the standard bound is in fact a bound on the "worst-case" GMRES residual norms for the given system matrix. For normal matrices the standard bound has been shown to be sharp in the sense that for each GMRES iteration step there exists an initial residual (depending on the matrix and the iteration step) for which the bound is attained [4,8]. In addition, for normal matrices the worst-case GMRES and the "average" GMRES behavior agree to within a factor of n 1/2 (n = matrix size).…”

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confidence: 99%

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The Worst-Case GMRES for Normal Matrices

Liesen

Tichý

2004

BIT Numerical Mathematics

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Abstract.We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GMRES-related quantities for which this bound is attained (worst-case GMRES). In this paper we completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow us to study the worst-case GMRES residual norm as a function of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a factor of 4/π of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an "average" residual norm, our results are of relevance beyond the worst case.AMS subject classification (2000): 15A06, 15A09, 15A18, 65F10, 65F15, 65F20, 41A10.

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Section: Worst-case Residual Normmentioning

confidence: 99%

Section: Worst-case Residual Normmentioning

confidence: 99%

“…Ideal GMRES approximation. The proofs of (3.4) in [4,8] are based on intricate constructions. For the special case i = n − 1 we now give a simple proof of (3.4), i.e.…”

Section: Worst Case In Stepmentioning

confidence: 99%

mentioning

confidence: 99%

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The Worst-Case GMRES for Normal Matrices

Liesen

Tichý

2004

BIT Numerical Mathematics

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“…Uma outra possibilidade é a de se utilizar informações das iterações anteriores para melhorar a qualidade dos precondicionadores, ver [7], [47], [72]. Vamos analisar, nesta seção, métodos de Krylov cujos os precondicionadores são eles próprios métodos de Krylov, podendo até ser o mesmo.…”

Section: Precondicionadores Flexíveisunclassified

Avanços em Métodos de Krylov para Solução de Sistemas Lineares de Grande Porte

2009

Notas Em Matemática Aplicada

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On IOM(q): The Incomplete Orthogonalization Method for Large Unsymmetric Linear Systems

Jia

1996

Numer. Linear Algebra Appl.

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The incomplete orthogonalization method (IOM(q)), a truncated version of the full orthogonalization method (FOM) proposed by Saad, has been used for solving large unsymmetric linear systems. However, no convergence analysis has been given. In this paper, IOM(q) is analysed in detail from a theoretical point of view. A number of important results are derived showing how the departure of the matrix A from symmetric affects the basis vectors generated by IOM(q), and some relationships between the residuals for IOM(q) and FOM are established. The results show that IOM(q) behaves much like FOM once the basis vectors generated by it are well conditioned. However, it is proved that IOM(q) may generate an ill‐conditioned basis for a general unsymmetric matrix such that IOM(q) may fail to converge or at least cannot behave like FOM. Owing to the mathematical equivalence between IOM(q) and the truncated ORTHORES(q) developed by Young and Jea, insights are given into the convergence of the latter. A possible strategy is proposed for choosing the parameter q involved in IOM(q). Numerical experiments are reported to show convergence behaviour of IOM(q) and of its restarted version.

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A Robust GMRES-Based Adaptive Polynomial Preconditioning Algorithm for Nonsymmetric Linear Systems

Cited by 44 publications

References 15 publications

The Worst-Case GMRES for Normal Matrices

The Worst-Case GMRES for Normal Matrices

Avanços em Métodos de Krylov para Solução de Sistemas Lineares de Grande Porte

On IOM(q): The Incomplete Orthogonalization Method for Large Unsymmetric Linear Systems

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