Block conjugate gradient methods with error norm estimates for least squares problems
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10509001" target="_blank" >RIV/00216208:11320/25:10509001 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=DzUk1h86CI" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=DzUk1h86CI</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10543-025-01096-3" target="_blank" >10.1007/s10543-025-01096-3</a>
Alternative languages
Result language
angličtina
Original language name
Block conjugate gradient methods with error norm estimates for least squares problems
Original language description
Least squares problems with multiple right-hand sides naturally arise in many practical applications. When the system matrix A is large and sparse, it is often convenient to solve such problems using suitably adapted variants of the block conjugate gradient method (block CGLS) or the block LSQR method. These block methods allow efficient use of modern computational architectures, and the number of iterations needed to achieve the required accuracy is typically much smaller than that required when solving each system separately and successively. However, a known limitation of these block methods is, for some problems, the occurrence of (near) breakdowns caused by (near) rank deficiencies within block vectors. We show how ideas presented in 2001 by A. Dubrulle for block CG can be incorporated into the block CGLS and block LSQR algorithms to avoid numerical instabilities caused by (near) rank deficiencies. For A with a full column rank, this provably prevents breakdowns in block CGLS. For the considered (preconditioned) algorithms, we derive estimates of the ATAdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$A<^>{T}A$$end{document}-norm of the error for each individual system, as well as for the trace of the corresponding bilinear form. These estimates are often well suited for use in stopping criteria. We consider both lower and upper bounds and show how the estimates can be adaptively refined to heuristically achieve a prescribed level of accuracy. Numerical experiments clearly illustrate which block algorithms are the most effective for practical computations and demonstrate that adaptive estimates perform reliably.
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
BIT Numerical Mathematics
ISSN
0006-3835
e-ISSN
1572-9125
Volume of the periodical
66
Issue of the periodical within the volume
December 2025
Country of publishing house
NL - THE KINGDOM OF THE NETHERLANDS
Number of pages
29
Pages from-to
2
UT code for WoS article
001632465500002
EID of the result in the Scopus database
2-s2.0-105024123657