Volume 2, Issue 2
A Variable Preconditioning Using the SOR Method for GCR-Like Methods

Int. J. Numer. Anal. Mod., 2 (2005), pp. 147-162.

Published online: 2005-02

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• Abstract

We propose a variant of variable preconditioning for Generalized Conjugate Residual (CCR)-like methods. The preconditioning is carried out by roughly solving $Az = v$ by an iterative method to a certain degree of accuracy instead of computing $Kz = v$ in a conventional preconditioned algorithm. In our proposal, the number of iterations required for computing $Az = v$ is changed at each iteration by establishing a stopping criterion. This enables the use of a stationary iterative method when applying different preconditioners. The proposed procedure is incorporated into GCR, and the mathematical convergence is proved. In numerical experiments, we employ the Successive Over-Relaxation (SOR) method for computing $Az = v$, and we demonstrate that GCR with the variable preconditioning using SOR is faster and more robust than GCR with an incomplete LU preconditioning, and the FGMRES and GMRESR methods with the variable preconditioning using the Generalized Minimal Residual (GMRES) method. Moreover, we confirm that different preconditioners are applied at each iteration.

• Keywords

linear systems, generalized conjugate residual method, generalized minimal residual method, variable preconditioning, inner-loop and outer-loop.

65F10, 65F50, 65N22

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@Article{IJNAM-2-147, author = {}, title = {A Variable Preconditioning Using the SOR Method for GCR-Like Methods}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2005}, volume = {2}, number = {2}, pages = {147--162}, abstract = {

We propose a variant of variable preconditioning for Generalized Conjugate Residual (CCR)-like methods. The preconditioning is carried out by roughly solving $Az = v$ by an iterative method to a certain degree of accuracy instead of computing $Kz = v$ in a conventional preconditioned algorithm. In our proposal, the number of iterations required for computing $Az = v$ is changed at each iteration by establishing a stopping criterion. This enables the use of a stationary iterative method when applying different preconditioners. The proposed procedure is incorporated into GCR, and the mathematical convergence is proved. In numerical experiments, we employ the Successive Over-Relaxation (SOR) method for computing $Az = v$, and we demonstrate that GCR with the variable preconditioning using SOR is faster and more robust than GCR with an incomplete LU preconditioning, and the FGMRES and GMRESR methods with the variable preconditioning using the Generalized Minimal Residual (GMRES) method. Moreover, we confirm that different preconditioners are applied at each iteration.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/926.html} }
TY - JOUR T1 - A Variable Preconditioning Using the SOR Method for GCR-Like Methods JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 147 EP - 162 PY - 2005 DA - 2005/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/926.html KW - linear systems, generalized conjugate residual method, generalized minimal residual method, variable preconditioning, inner-loop and outer-loop. AB -

We propose a variant of variable preconditioning for Generalized Conjugate Residual (CCR)-like methods. The preconditioning is carried out by roughly solving $Az = v$ by an iterative method to a certain degree of accuracy instead of computing $Kz = v$ in a conventional preconditioned algorithm. In our proposal, the number of iterations required for computing $Az = v$ is changed at each iteration by establishing a stopping criterion. This enables the use of a stationary iterative method when applying different preconditioners. The proposed procedure is incorporated into GCR, and the mathematical convergence is proved. In numerical experiments, we employ the Successive Over-Relaxation (SOR) method for computing $Az = v$, and we demonstrate that GCR with the variable preconditioning using SOR is faster and more robust than GCR with an incomplete LU preconditioning, and the FGMRES and GMRESR methods with the variable preconditioning using the Generalized Minimal Residual (GMRES) method. Moreover, we confirm that different preconditioners are applied at each iteration.

K. Abe & S.-L. Zhang. (1970). A Variable Preconditioning Using the SOR Method for GCR-Like Methods. International Journal of Numerical Analysis and Modeling. 2 (2). 147-162. doi:
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