How to make Simpler GMRES and GCR more Stable
Result description
In this paper we analyze the numerical behavior of several minimum residual methods, which are mathematically equivalent to the GMRES method. Two main approaches are compared: the one that computes the approximate solution in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and the one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of theresulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and they explain the experimentally observed fact that the GCR method delivers very accurate approximate solutions when it converges fast enough without stagnation.
Keywords
large-scale nonsymmetric linear systemsKrylov subspace methodsminimum residual methodsnumerical stabilityrounding errors
The result's identifiers
Result code in IS VaVaI
Result on the web
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DOI - Digital Object Identifier
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Alternative languages
Result language
angličtina
Original language name
How to make Simpler GMRES and GCR more Stable
Original language description
In this paper we analyze the numerical behavior of several minimum residual methods, which are mathematically equivalent to the GMRES method. Two main approaches are compared: the one that computes the approximate solution in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and the one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of theresulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and they explain the experimentally observed fact that the GCR method delivers very accurate approximate solutions when it converges fast enough without stagnation.
Czech name
Jak stabilizovat metody Simpler GMRES and GCR?
Czech description
V této práci analyzujeme numerické chování několika metod minimalizujících normu rezidua, které jsou matematicky ekvivalentní metodě GMRES. Porovnáváme dva základní postupy: postup, který je založen na výpočtu aproximace řešení z horní trojúhelníkové soustavy pro její souřadnice a postup, kde jsou aproximace postupně upravovány jednoduchou rekurzní formulí. Práce ukazuje, že volba báze vypočteného prostoru může podstatným způsoben ovlivnit numerickou stabilitu implementace daných metod. Z důvodu špatnépodmíněnosti zvolené báze jsou implementace Simpler GMRES a ORTHODIR méně stabilní. Na druhé straně se ukazuje, ze pokud metoda konverguje dostatečně rychle, je báze reziduí dobře podmíněná. Tyto výsledky pak vedou k nove implementaci, která je podmíněnězpětně stabilní, a potvrzují experimentálně známý fakt, že metoda GCR v případě dostatečně rychlé konvergence generuje velice přesné aproximace řešení.
Classification
Type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2008
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
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Volume of the periodical
30
Issue of the periodical within the volume
4
Country of publishing house
US - UNITED STATES
Number of pages
17
Pages from-to
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UT code for WoS article
000263103700013
EID of the result in the Scopus database
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Basic information
Result type
Jx - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP
BA - General mathematics
Year of implementation
2008