Solution of infinite linear systems by automatic adaptive iterations

Favati, Paola; Lotti, Grazia; Menchi, Ornella; Romani, Francesco

doi:10.1016/s0024-3795(00)00177-4

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…Furthermore, there is no accumulation of errors (so the error is independent of the number of iterations). For this reason using directly computed residuals can be necessary in some applications, see e.g., [9] where the authors discuss the approximate solution of infinite dimensional systems. On the other hand, the advantage of the recursively computed residual is that we do not have to estimate the residual reduction in the coming step.…”

Section: The Outer Iteration: Richardson Iterationmentioning

confidence: 99%

Relaxation strategies for nested Krylov methods

Eshof

Sleijpen

Gijzen

2005

Journal of Computational and Applied Mathematics

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There are classes of linear problems for which the matrix-vector product is a time consuming operation because an expensive approximation method is required to compute it to a given accuracy. In recent years different authors have investigated the use of, what is called, relaxation strategies for various Krylov subspace methods. These relaxation strategies aim to minimize the amount of work that is spent in the computation of the matrix-vector product without compromising the accuracy of the method or the convergence speed too much. In order to achieve this goal, the accuracy of the matrix-vector product is decreased when the iterative process comes closer to the solution. In this paper we show that a further significant reduction in computing time can be obtained by combining a relaxation strategy with the nesting of inexact Krylov methods. Flexible Krylov subspace methods allow variable preconditioning and therefore can be used in the outer most loop of our overall method. We analyze for several flexible Krylov methods strategies for controlling the accuracy of both the inexact matrix-vector products and of the inner iterations. The results of our analysis will be illustrated with an example that models global ocean circulation.

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Section: The Outer Iteration: Richardson Iterationmentioning

confidence: 99%

Relaxation strategies for nested Krylov methods

Eshof

Sleijpen

Gijzen

2005

Journal of Computational and Applied Mathematics

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“…For example, the following hypotheses on A and b (see [5]) guarantee, together with (a) and (b), such a convergence and are always satisfied by stochastic problems:…”

Section: The Enlargement Schemementioning

confidence: 99%

“…In this paper we present an adaptive strategy, similar to the one proposed in [5], which we call enlargement scheme, and whose general framework is given in Section 2. The idea of enlargement for solving a system with an infinite number of equations goes back to Poincaré (1886), a classical book on this subject is [12].…”

Section: Introductionmentioning

confidence: 99%

“…The iterative methods considered here are the Krylov subspace methods. In [5], the Gauss-Seidel method was used to solve the truncated systems and the possibility of substituting it with a more efficient Krylov subspace method was mentioned. This modification is not trivial, since Krylov subspace methods for nonsymmetric systems frequently show an irregular convergence behaviour.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive solution of infinite linear systems by Krylov subspace methods

Favati

Lotti

Menchi

et al. 2007

Journal of Computational and Applied Mathematics

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In this paper we consider the problem of approximating the solution of infinite linear systems, finitely expressed by a sparse coefficient matrix. We analyse an algorithm based on Krylov subspace methods embedded in an adaptive enlargement scheme. The management of the algorithm is not trivial, due to the irregular convergence behaviour frequently displayed by Krylov subspace methods for nonsymmetric systems. Numerical experiments, carried out on several test problems, indicate that the more robust methods, such as GMRES and QMR, embedded in the adaptive enlargement scheme, exhibit good performances.

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