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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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Analysis of an Implicitly Restarted Simpler GMRES Variant of Augmented GMRES
2010We analyze a Simpler GMRES variant of augmented GMRES with implicit restarting for solving nonsymmetric linear systems with small eigenvalues. The use of a shifted Arnoldi process in the Simpler GMRES variant for computing Arnoldi basis vectors has the advantage of not requiring an upper Hessenberg factorization and this often leads to cheaper ...
Ravindra Boojhawon +3 more
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GPU-Accelerated Preconditioned GMRES Solver
2016 IEEE 2nd International Conference on Big Data Security on Cloud (BigDataSecurity), IEEE International Conference on High Performance and Smart Computing (HPSC), and IEEE International Conference on Intelligent Data and Security (IDS), 2016Linear solvers and parallel computing are crucial for developing a new generation reservoir simulator. To acquire satisfied acceleration performance, we study the parallel algorithms of linear solvers with preconditioners on GPUs (Graphics Processing Units).
Bo Yang +3 more
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Applied Mathematics and Computation, 1997
The GMRES method for a nonsymmetric linear system arizing from discretization of a general second-order elliptic boundary value problem is preconditioned by a multigrid iteration on the whole system. It is proved that the preconditioned system is nonsymmetric positive definite for sufficiently small coarsest mesh.
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The GMRES method for a nonsymmetric linear system arizing from discretization of a general second-order elliptic boundary value problem is preconditioned by a multigrid iteration on the whole system. It is proved that the preconditioned system is nonsymmetric positive definite for sufficiently small coarsest mesh.
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Shifted GMRES for oscillatory integrals
Numerische Mathematik, 2009By applying the generalized minimal residual (GMRES) algorithm to a shifted linear differential operator, the author constructs a new method for approximating the oscillatory integral \(\int_a^b f(x)e^{i\omega g(x)}dx\). Unlike the existing methods, this one satisfies simultaneously the following properties: it has high asymptotic order and stability ...
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