Iterated Gauss-Seidel GMRES
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00585518" target="_blank" >RIV/67985840:_____/24:00585518 - isvavai.cz</a>
Alternative codes found
RIV/00216208:11320/24:10493576
Result on the web
<a href="https://doi.org/10.1137/22M1491241" target="_blank" >https://doi.org/10.1137/22M1491241</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/22M1491241" target="_blank" >10.1137/22M1491241</a>
Alternative languages
Result language
angličtina
Original language name
Iterated Gauss-Seidel GMRES
Original language description
The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856--869] is an iterative method for approximately solving linear systems Ax = b, with initial guess x0 and residual r0 = b- Ax0. The algorithm employs the Arnoldi process to generate the Krylov basis vectors (the columns of Vk). It is well known that this process can be viewed as a QR factorization of the matrix Bk = [r0,AVk ] at each iteration. Despite an O(varepsilon)kappa(Bk) loss of orthogonality, for unit roundoff varepsilon and condition number kappa , the modified Gram-Schmidt formulation was shown to be backward stable in the seminal paper by Paige et al. [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 264--284]. We present an iterated Gauss--Seidel formulation of the GMRES algorithm (IGS-GMRES) based on the ideas of Ruhe [Linear Algebra Appl., 52 (1983), pp. 591--601] and S'wirydowicz et al. [Numer. Linear Algebra Appl., 28 (2020), pp. 1--20]. IGS-GMRES maintains orthogonality to the level O(varepsilon)kappa(Bk) or O(varepsilon), depending on the choice of one or two iterations, for two Gauss--Seidel iterations, the computed Krylov basis vectors remain orthogonal to working accuracy and the smallest singular value of Vk remains close to one. The resulting GMRES method is thus backward stable. We show that IGS-GMRES can be implemented with only a single synchronization point per iteration, making it relevant to large-scale parallel computing environments. We also demonstrate that, unlike MGS-GMRES, in IGS-GMRES the relative Arnoldi residual corresponding to the computed approximate solution no longer stagnates above machine precision even for highly nonnormal systems.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA23-06159S" target="_blank" >GA23-06159S: Vortical structures: advanced identification and efficient numerical simulation</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Scientific Computing
ISSN
1064-8275
e-ISSN
1095-7197
Volume of the periodical
46
Issue of the periodical within the volume
2
Country of publishing house
US - UNITED STATES
Number of pages
26
Pages from-to
"S254"-"S279"
UT code for WoS article
001308408900002
EID of the result in the Scopus database
2-s2.0-85192678897