ON THE BACKWARD STABILITY OF S-STEP GMRES
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10509865" target="_blank" >RIV/00216208:11320/25:10509865 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=rO5XDsC93s" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=rO5XDsC93s</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/24M1690485" target="_blank" >10.1137/24M1690485</a>
Alternative languages
Result language
angličtina
Original language name
ON THE BACKWARD STABILITY OF S-STEP GMRES
Original language description
Communication, i.e., data movement, is a critical bottleneck for the performance of classical Krylov subspace method solvers on modern computer architectures. Variants of these methods which avoid communication have been introduced, which, while equivalent in exact arithmetic, can be unstable in finite precision. In this work, we address the backward stability of s-step GMRES, also known as communication-avoiding GMRES. Building upon the ''modular framework"" proposed in [A. Buttari et al., preprint, hal-04525918v2, 2024.], we present an improved framework for simplifying the analysis of s-step GMRES, which includes standard GMRES (s = 1) as a special case, by isolating the effects of rounding errors in the QR factorization and the solution of the least squares problem. The key advantage of this new framework is that it is evident how the orthogonalization method affects the backward error, and it is not necessary to reevaluate anything other than the orthogonalization itself when modifying the orthogonalization used in GMRES. Using this framework, we analyze s-step GMRES with popular block orthogonalization methods: block modified Gram-Schmidt and reorthogonalized block classical Gram--Schmidt algorithms. An example illustrates the resulting instability of s-step GMRES when paired with the classical s-step Arnoldi process and shows the limitations of popular strategies for resolving this instability. To address this issue, we propose a modified s-step Arnoldi process that allows for much larger block size s while maintaining satisfactory accuracy, as confirmed by our numerical experiments. An example illustrates the resulting instability of s-step GMRES when paired with the classical s-step Arnoldi process and shows the limitations of popular strategies for resolving this instability. To address this issue, we propose a modified Arnoldi process that allows for much larger block size s while maintaining satisfactory accuracy, as confirmed by our numerical experiments.
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
10102 - Applied mathematics
Result continuities
Project
—
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2025
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 Matrix Analysis and Applications
ISSN
0895-4798
e-ISSN
1095-7162
Volume of the periodical
46
Issue of the periodical within the volume
3
Country of publishing house
US - UNITED STATES
Number of pages
33
Pages from-to
2008-2040
UT code for WoS article
001580457700004
EID of the result in the Scopus database
2-s2.0-105014621532