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Volume 3, Issue 3
An Investigation of Restarted GMRES Method by Using Flexible Starting Vectors

Qiang Niu & Lin-Zhang Lu

DOI: 10.4208/nmtma.2010.33.4

Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 338-351.

Published online: 2010-03

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  • Abstract

We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations. The merit of the variant is that previously generated information can be utilized to select a new starting vector, such that the occurrence of stagnation be mitigated or the convergence be accelerated. The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector, which is discussed in detail. Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.

  • Keywords

Linear systems of equations, Arnoldi process, GMRES, harmonic Ritz vector.

  • AMS Subject Headings

65F10, 65F15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-3-338, author = {Qiang Niu and Lin-Zhang Lu}, title = {An Investigation of Restarted GMRES Method by Using Flexible Starting Vectors}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2010}, volume = {3}, number = {3}, pages = {338--351}, abstract = {

We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations. The merit of the variant is that previously generated information can be utilized to select a new starting vector, such that the occurrence of stagnation be mitigated or the convergence be accelerated. The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector, which is discussed in detail. Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2010.33.4}, url = {http://global-sci.org/intro/article_detail/nmtma/6002.html} }
TY - JOUR T1 - An Investigation of Restarted GMRES Method by Using Flexible Starting Vectors AU - Qiang Niu & Lin-Zhang Lu JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 338 EP - 351 PY - 2010 DA - 2010/03 SN - 3 DO - http://doi.org/10.4208/nmtma.2010.33.4 UR - https://global-sci.org/intro/article_detail/nmtma/6002.html KW - Linear systems of equations, Arnoldi process, GMRES, harmonic Ritz vector. AB -

We discuss a variant of restarted GMRES method that allows changes of the restarting vector at each cycle of iterations. The merit of the variant is that previously generated information can be utilized to select a new starting vector, such that the occurrence of stagnation be mitigated or the convergence be accelerated. The more appealing utilization of the new method is in conjunction with a harmonic Ritz vector as the starting vector, which is discussed in detail. Numerical experiments are carried out to demonstrate that the proposed procedure can effectively mitigate the occurrence of stagnation due to the presence of small eigenvalues in modulus.

Qiang Niu and Lin-Zhang Lu. (2010). An Investigation of Restarted GMRES Method by Using Flexible Starting Vectors. Numerical Mathematics: Theory, Methods and Applications. 3 (3). 338-351. doi:10.4208/nmtma.2010.33.4
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