The behavior of the Gauss-Radau upper bound of the error norm in CG
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10472974" target="_blank" >RIV/00216208:11320/23:10472974 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=_hhocVg9B1" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=_hhocVg9B1</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11075-023-01522-z" target="_blank" >10.1007/s11075-023-01522-z</a>
Alternative languages
Result language
angličtina
Original language name
The behavior of the Gauss-Radau upper bound of the error norm in CG
Original language description
Consider the problem of solving systems of linear algebraic equations Ax = b with a real symmetric positive definite matrix A using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the A-norm of the error. This quantity cannot be easily computed; however, it can be estimated. In this paper we discuss and analyze the behavior of theGauss-Radau upper bound on the A-norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate mu to the smallest eigenvalue of A. We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of mu. We construct amodel problem that is used to demonstrate and study the behavior of the upper bound in dependence of mu, and developed formulas that are helpful in understanding this behavior. We show that the above-mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of A. It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate mu. We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations.
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
—
OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
<a href="/en/project/GA20-01074S" target="_blank" >GA20-01074S: Adaptive methods for the numerical solution of partial differential equations: analysis, error estimates and iterative solvers</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Numerical Algorithms
ISSN
1017-1398
e-ISSN
1572-9265
Volume of the periodical
94
Issue of the periodical within the volume
2
Country of publishing house
NL - THE KINGDOM OF THE NETHERLANDS
Number of pages
30
Pages from-to
847-876
UT code for WoS article
000973631900002
EID of the result in the Scopus database
2-s2.0-85153079846