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Numerical Linear Algebra with Applications, 1994
AbstractThe generalized minimal residual (GMRES) method is widely used for solving very large, nonsymmetric linear systems, particularly those that arise through discretization of continuous mathematical models in science and engineering. By shifting the Arnoldi process to begin with Ar0 instead of r0, we obtain simpler Gram–Schmidt and Householder ...
Walker, Homer F., Zhou, Lu
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AbstractThe generalized minimal residual (GMRES) method is widely used for solving very large, nonsymmetric linear systems, particularly those that arise through discretization of continuous mathematical models in science and engineering. By shifting the Arnoldi process to begin with Ar0 instead of r0, we obtain simpler Gram–Schmidt and Householder ...
Walker, Homer F., Zhou, Lu
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SIAM Journal on Matrix Analysis and Applications, 1997
The GMRES algorithm for solving non-Hermitian linear systems \(Ax=b\) \((A\in\mathbb{C}^{N\times N}\), \(b\in \mathbb{C}^{N}\) is studied. The ideal GMRES problem is obtained if one consideres minimization of \(|p(A) |\) instead of \(|p(A)b|\) as in the GMRES algorithm.
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The GMRES algorithm for solving non-Hermitian linear systems \(Ax=b\) \((A\in\mathbb{C}^{N\times N}\), \(b\in \mathbb{C}^{N}\) is studied. The ideal GMRES problem is obtained if one consideres minimization of \(|p(A) |\) instead of \(|p(A)b|\) as in the GMRES algorithm.
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GMRES with Deflated Restarting
SIAM Journal on Scientific Computing, 2002A new version of the generalized minimal residuum (GMRES) algorithm for solving large systems of linear equations is described. It uses a ``deflated restarting'' and at each cycle a recurrence similar to the Arnoldi's one is generated. The new algorithm has about the same storage and expense requirements as GMRES with implicitly restarted Arnoldi ...
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Flexible GMRES with Deflated Restarting
SIAM Journal on Scientific Computing, 2010In many situations, it has been observed that significant convergence improvements can be achieved in preconditioned Krylov subspace methods by enriching them with some spectral information. On the other hand, effective preconditioning strategies are often designed where the preconditioner varies from one step to the next so that a flexible Krylov ...
Giraud, Luc +3 more
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On GMRES-Equivalent Bounded Operators
SIAM Journal on Matrix Analysis and Applications, 2000The author studies the generalized minimal residual (GMRES) method applied to some operator equation \(Ax= r\) in a Hilbert space \(H\), where the operator \(A\in L(H)\) is supposed to be linear and bounded. At the \(k\)th step, the GMRES produces an approximate solution which minimizes the residual norm \(\|Ax-r\|\) over the Krylov subspace \(K^k(A,r):
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BIT Numerical Mathematics, 1995
The generalized minimal residual (GMRES) method is one of the most popular methods for solving systems of linear equations with nonsymmetric coefficient matrices. The authors study the numerical stability of GMRES when the computation of approximations is based on constructing an orthonormal basis of Krylov subspaces (Arnoldi basis) and after that the ...
Drkošová, J. +3 more
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The generalized minimal residual (GMRES) method is one of the most popular methods for solving systems of linear equations with nonsymmetric coefficient matrices. The authors study the numerical stability of GMRES when the computation of approximations is based on constructing an orthonormal basis of Krylov subspaces (Arnoldi basis) and after that the ...
Drkošová, J. +3 more
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2020
FOM (Full Orthogonalization Method) and GMRES (Generalized Minimal RESidual) is a pair of Q-OR/Q-MR methods using an orthonormal basis for the Krylov subspaces. In fact, as we have seen in Chapter 3, the FOM residual vectors are proportional to the basis vectors. Thus, FOM is a Q-OR method for which the residual vectors are orthogonal to each other. It
Gérard Meurant, Jurjen Duintjer Tebbens
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FOM (Full Orthogonalization Method) and GMRES (Generalized Minimal RESidual) is a pair of Q-OR/Q-MR methods using an orthonormal basis for the Krylov subspaces. In fact, as we have seen in Chapter 3, the FOM residual vectors are proportional to the basis vectors. Thus, FOM is a Q-OR method for which the residual vectors are orthogonal to each other. It
Gérard Meurant, Jurjen Duintjer Tebbens
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Making global simpler GMRES more stable
Numerical Linear Algebra with Applications, 2018SummaryThe global GMRES method is well known for the solution of nonsymmetric linear systems with multiple right hands. In this paper, the condition number for evaluating the stability of the global simpler GMRES method is defined. With this condition number, it is shown that Zong et al.'s global simpler method with a simple basis of the Krylov matrix ...
Qiaohua Liu, Dongmei Shen, Ziwei Jia
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Parallel Adaptive Deflated GMRES
2013Many scientific libraries are currently based on the GMRES method as a Krylov subspace iterative method for solving large linear systems. The restarted formulation known as GMRES(m) has been extensively studied and several approaches have been proposed to reduce the negative effects due to the restarting procedure. A common effect in GMRES(m) is a slow
Nuentsa Wakam, Désiré +2 more
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Restarted GMRES for Shifted Linear Systems
SIAM Journal on Scientific Computing, 1998For a nonsingular and non-Hermitian matrix \(A\in\mathbb{C}^{n\times n}\) and the shifted matrix \(\widehat A=A+\alpha I,\;\alpha\in\mathbb{C}\), the following two systems are considered: \[ Ax=b\text{ as seed system, and }\widehat A\widehat x=b\text{ as add system.} \] In the GMRES method, a Krylov subspace method in which the basis for the Krylov ...
Frommer, Andreas, Glässner, Uwe
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