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Multipreconditioned Gmres for Shifted Systems [PDF]
SIAM Journal of Scientific Computing, 2017 An implementation of GMRES with multiple preconditioners (MPGMRES) is proposed for solving shifted linear systems with shift-and-invert preconditioners. With this type of preconditioner, the Krylov subspace can be built without requiring the matrix-vector product with the shifted matrix.
Daniel Szyld +2 more
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Proxy-GMRES: Preconditioning via GMRES in Polynomial Space
SIAM Journal on Matrix Analysis and Applications, 2021zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Xin Ye, Yuanzhe Xi, Yousef Saad
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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 ...
Homer F Walker
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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 ...
Homer F Walker
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Theoretical and numerical comparisons of GMRES and WZ-GMRES
Computers and Mathematics With Applications, 2004 The authors study the numerical stability of the WZ-GMRES method proposed by \textit{H. F. Walker} and \textit{L. Zhou} [Numer. Linear Algebra Appl. 1, No. 6, 571--581 (1994; Zbl 0838.65030)] and compare the stability of the WZ-GMRES method with that of the GMRES method for solving systems of linear equations \(Ax = b\) with a non-symmetric matrix \(A\)
Chen, G. Z., Jia, Z. X.
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Polynomial Preconditioned GMRES and GMRES-DR
SIAM Journal of Scientific Computing, 2015Summary: We look at solving large nonsymmetric systems of linear equations using polynomial preconditioned Krylov methods. We give a simple way to find the polynomial. It is shown that polynomial preconditioning can significantly improve restarted GMRES for difficult problems, and the reasons for this are examined.
Ronald Morgan, Walter Wilcox
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GMRES On (Nearly) Singular Systems [PDF]
SIAM Journal on Matrix Analysis and Applications, 1997 The authors' purpose is to examine the behavior of the GMRES method when the matrix \(A\) is singular or nearly so, i.e., ill-conditioned, and to formulate practically effective ways of recognizing the singularity or the ill-conditioning when it might significantly affect the performance of the method.
Peter N Brown, Homer F Walker
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The Tortoise and the Hare Restart GMRES [PDF]
SIAM Review, 2003 Summary: When solving large nonsymmetric systems of linear equations with the restarted generalized minimal residual (GMRES) algorithm, one is inclined to select a relatively large restart parameter in the hope of mimicking the full GMRES process.
Mark Embree
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International Journal of Computer Mathematics, 1996
GMRES method [4] is an effective conjugate gradient-like iterative method for solving linear systems of equations. In each loop of the GMRES we have to solve a special least squares problem (1). A classical way of solving such least squares problems is to factor H into QR using Givens plane rotations.
Changjun Li, David J. Evans 0001
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GMRES method [4] is an effective conjugate gradient-like iterative method for solving linear systems of equations. In each loop of the GMRES we have to solve a special least squares problem (1). A classical way of solving such least squares problems is to factor H into QR using Givens plane rotations.
Changjun Li, David J. Evans 0001
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Adaptively Preconditioned GMRES Algorithms
SIAM Journal on Scientific Computing, 1998Summary: The restarted GMRES algorithm proposed by \textit{Y. Saad} and \textit{M. H. Schultz} [SIAM J. Sci. Statist. Comput. 7, 856-869 (1986; Zbl 0599.65018)] is one of the most popular iterative methods for the solution of large linear systems of equations \(Ax=b\) with a nonsymmetric and sparse matrix. This algorithm is particularly attractive when
James Baglama +3 more
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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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