Error-Minimizing Krylov Subspace Methods

Abstract. Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.

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“…For nonsymmetric problems, Weiss [345], [346], more recently proposed a generalization of this approach by choosing…”

Section: Other Minimization Proceduresmentioning

confidence: 99%

Recent computational developments in Krylov subspace methods for linear systems

Simoncini

Szyld

2006

Numerical Linear Algebra App

312

254

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“…They gave relations between the iterates of these two pairs of methods and extend the procedure to a quasi-minimal residual smoothing (QMRS) which can be applied to any iterative method. Other results relating to the smoothing technique can be found in [36] and [37].…”

Section: Casementioning

confidence: 98%

Hybrid procedures for solving linear systems

Brezinski¹,

Redivo–Zaglia

1994

Numerische Mathematik

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In this paper, we introduce the notion of hybrid procedures for solving a system of linear equations. A hybrid procedure consists in a combination of two arbitrary approximate solutions with coefficients summing up to one. Thus the combination only depends on one parameter whose value is chosen in order to minimize the Euclidean norm of the residual vector obtained by the hybrid procedure. Properties of such procedures are studied in detail. The two approximate solutions which are combined in a hybrid procedure are usually obtained by two iterative methods. Several strategies for combining these two methods together or with the previous iterate of the hybrid procedure itself are discussed and their properties are analyzed. Numerical experiments illustrate the various procedures

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“…Most Krylov subspace methods choose iterates to satisfy certain orthogonality conditions or, equivalently, to minimize certain norms. For example, the 2-norm of the residual vector is minimized in the case of the GMRES method [22] and the 2-norm of the error vector in the case of the generalized minimal error method (GMERR) [28]. In cases where the principal quantity of interest is a linear functional J pr (x) rather than the solution itself, those methods may not give optimal results within the Krylov subspace.…”

Section: Superconvergent Estimates Of Linear Functionalsmentioning

confidence: 99%

A Quasi-Minimal Residual Method for Simultaneous Primal-Dual Solutions and Superconvergent Functional Estimates

Lu¹,

Darmofal²

2003

SIAM J. Sci. Comput.

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The adjoint solution has found many uses in computational simulations where the quantities of interest are the functionals of the solution, including design optimization, error estimation, and control. In those applications where both the solution and the adjoint are desired, the conventional approach is to apply iterative methods to solve the primal and dual problems separately. However, we show that there is an advantage associated with iterating the primal and dual problem simultaneously since this enables the construction of iterative methods where both the primal and the dual iterates may be chosen so that they provide functional estimates that are "superconvergent" in that the error converges at twice the order of the optimal global solution error norm. In particular, we show that the structure of the Lanczos process allows for this superconvergence property and propose a modified QMR method which uses the same Lanczos process to simultaneously solve the primal and dual problems. Thus both the primal and the dual systems are solved at essentially the same computational cost as the conventional QMR method applied to the primal problem alone. Numerical experiments show that our proposed method does indeed exhibit superconvergence behavior.

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Error-Minimizing Krylov Subspace Methods

Cited by 27 publications

References 13 publications

Recent computational developments in Krylov subspace methods for linear systems

Recent computational developments in Krylov subspace methods for linear systems

Hybrid procedures for solving linear systems

A Quasi-Minimal Residual Method for Simultaneous Primal-Dual Solutions and Superconvergent Functional Estimates

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