GMRES and the minimal polynomial

Campbell, Stephen L.; Ipsen, Ilse C. F.; Kelley, C. T.; Meyer, Carl D.

doi:10.1007/bf01733786

Cited by 95 publications

(85 citation statements)

References 17 publications

(9 reference statements)

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“…The techniques used in [39] to obtain these results are based on particular choices of the polynomialq in (2.4). A similar technique was used in [9] when considering the special case where the spectrum of A has the form σ(A) = σ c ∪ σ o , such that, e.g., σ c ⊂ {z : |z − 1| < ρ} and σ o = {λ 1 , . .…”

Section: Some Analytic Models Of Superlinear Convergencementioning

confidence: 99%

“…The described phenomenon of superlinear convergence of Krylov subspace methods has been widely observed, and some models explaining this behavior have been proposed; see [2], [38], [39], and also [9], [10], [12]. The analysis proposed in most of the cited papers relies on the polynomial representation of the approximate solution in K m ; see section 2.…”

Section: Introductionmentioning

confidence: 99%

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On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

Simoncini¹,

Szyld²

2005

SIAM Rev.

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Abstract. Krylov subspace methods often exhibit superlinear convergence. We present a general analytic model which describes this superlinear convergence, when it occurs. We take an invariant subspace approach, so that our results apply also to inexact methods, and to nondiagonalizable matrices. Thus, we provide a unified treatment of the superlinear convergence of GMRES, conjugate gradients, block versions of these, and inexact subspace methods. Numerical experiments illustrate the bounds obtained.

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Section: Some Analytic Models Of Superlinear Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

Simoncini¹,

Szyld²

2005

SIAM Rev.

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“…a*identity ≈ identity. It is well known that this type of equation is well-adapted to an iterative resolution and that if the spectral behavior of the equation is well restored to the discrete level, then the convergence rate is independent to space and frequency refinement [9,10]. This result has already been proved in [18] for the metallic case i.e.…”

Section: Well-posedness For Smooth Surface and Impedance Operatormentioning

confidence: 65%

A well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition

Pernet¹

2010

ESAIM: M2AN

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Abstract. The construction of a well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition is proposed. A suitable parametrix is obtained by using a new unknown and an approximation of the transparency condition. We prove the well-posedness of the equation for any wavenumber. Finally, some numerical comparisons with well-tried method prove the efficiency of the new formulation.Mathematics Subject Classification. 65R20, 15A12, 65N38, 65F10, 65Z05.

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“…Mesh-independence of the GMRES iteration will follow from the convergence of Ã ¼´ÍAE µ to Ã ¼´Í£ µ in the operator norm [5,6]. This means that then the number of GMRES iterations needed to satisfy (2.5) from an initial iterate of × ¼ is independent of the level AE of discretization.…”

Section: Nested Newton-gmres the Newton-gmres Methods Is An Inexactmentioning

confidence: 99%

A fast solver for the Ornstein–Zernike equations

Kelley

Pettitt

2004

Journal of Computational Physics

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Abstract. In this paper we report on the design and analysis of a multilevel method for the solution of the OrnsteinZernike Equations and related systems of integro-algebraic equations. Our approach is based on an extension of the Atkinson-Brakhage method, with Newton-GMRES used as the coarse mesh solver. We report on several numerical experiments to illustrate the effectiveness of the method.Key words. Multilevel method, Newton-GMRES, Ornstein-Zernike equations, nonlinear equations AMS subject classifications. 65H10, 65N55, 65R20, 65T50, 92E10, 1. Introduction. In this paper we propose a fast multi-level method for the solution of a class of integral equations called the Ornstein-Zernike (OZ) equations, which are useful in calculating probability distributions of matter (atoms) in fluid states [10]. Our approach is faster than a Newton-Krylov approach, such as the one proposed in [2], because the linear solver is only used on a coarse mesh problem.The OZ equations were initially designed to model density fluctuations near the critical point via the equilibrium theory of liquids [8,15]. Since then the range of validity and usefulness has been extended to include the entire fluid range of states. This set of nonlinear coupled integral equations has been derived from the full partition function for atomic systems [20] and while essentially never solved without some approximations has proved a useful tool for understanding liquids at the atomic level for over 50 years. While the OZ equation has two unknowns it is usually closed with another often algebraic relation between the two unknown functions. Two useful approximate closure relations are the Perkus-Yevick equation [16] and the hyper netted chain equation [21]. These equations then provide essentially two equations and two unknowns and when convenient may be substituted into the OZ equation to provide a single nonlinear integral equation for the unknown probability distribution function.The equations, when the physical parameters are reasonably adjusted, have solutions which can be achieved by a variety of techniques [10]. Those methods include Picard iteration with or with out relaxation and basis set (variational) methods. In cases where the physical parameters make the equations stiff, iterative solutions are particularly tedious.The objectives of this paper are to describe an multilevel approach for solving the OZ equations and apply that new approach to two examples. The multilevel method is based on the enhanced version of the Atkinson-Brakhage [1, 3] method from [13]. We show that we can compute

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GMRES and the minimal polynomial

Cited by 95 publications

References 17 publications

On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

A well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition

A fast solver for the Ornstein–Zernike equations

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