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Article: Galerkin Projection Methods for Solving Multiple Linear Systems

TitleGalerkin Projection Methods for Solving Multiple Linear Systems
Authors
KeywordsMultiple linear systems
Krylov space
Conjugate gradient method
Galerkin projection
Issue Date1999
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISC
Citation
SIAM Journal on Scientific Computing, 1999, v. 21 n. 3, p. 836-850 How to Cite?
AbstractIn this paper, we consider using conjugate gradient (CG) methods for solving multiple linear systems A(i) x(i) = b(i) , for 1 ≤ i ≤ s, where the coefficient matrices A(i) and the right-hand sides b( i) are different in general. In particular, we focus on the seed projection method which generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated until all the systems are solved. Most papers in the literature [T. F. Chan and W. L. Wan, SIAM J. Sci. Comput., 18 (1997), pp. 1698­1721; B. Parlett Linear Algebra Appl., 29 (1980), pp. 323­346; Y. Saad, Math. Comp., 48 (1987), pp. 651­662; V. Simoncini and E. Gallopoulos, SIAM J. Sci. Comput., 16 (1995), pp. 917­933; C. Smith, A. Peterson, and R. Mittra, IEEE Trans. Antennas and Propagation, 37 (1989), pp. 1490­1493] considered only the case where the coefficient matrices A( i) are the same but the right-hand sides are different. We extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right-hand sides. A theoretical error bound is given for the approximation obtained from a projection process onto a Krylov subspace generated from solving a previous linear system. Finally, numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method.
Persistent Identifierhttp://hdl.handle.net/10722/42993
ISSN
2015 Impact Factor: 1.792
2015 SCImago Journal Rankings: 2.166

 

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dc.contributor.authorNg, MKPen_HK
dc.contributor.authorChan, TFen_HK
dc.date.accessioned2007-03-23T04:36:25Z-
dc.date.available2007-03-23T04:36:25Z-
dc.date.issued1999en_HK
dc.identifier.citationSIAM Journal on Scientific Computing, 1999, v. 21 n. 3, p. 836-850en_HK
dc.identifier.issn1064-8275en_HK
dc.identifier.urihttp://hdl.handle.net/10722/42993-
dc.description.abstractIn this paper, we consider using conjugate gradient (CG) methods for solving multiple linear systems A<sup>(i)</sup> x<sup>(i)</sup> = b<sup>(i)</sup> , for 1 ≤ i ≤ s, where the coefficient matrices A<sup>(i)</sup> and the right-hand sides b( i) are different in general. In particular, we focus on the seed projection method which generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated until all the systems are solved. Most papers in the literature [T. F. Chan and W. L. Wan, SIAM J. Sci. Comput., 18 (1997), pp. 1698­1721; B. Parlett Linear Algebra Appl., 29 (1980), pp. 323­346; Y. Saad, Math. Comp., 48 (1987), pp. 651­662; V. Simoncini and E. Gallopoulos, SIAM J. Sci. Comput., 16 (1995), pp. 917­933; C. Smith, A. Peterson, and R. Mittra, IEEE Trans. Antennas and Propagation, 37 (1989), pp. 1490­1493] considered only the case where the coefficient matrices A( i) are the same but the right-hand sides are different. We extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right-hand sides. A theoretical error bound is given for the approximation obtained from a projection process onto a Krylov subspace generated from solving a previous linear system. Finally, numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method.en_HK
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dc.languageengen_HK
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISCen_HK
dc.relation.ispartofSIAM Journal on Scientific Computing-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectMultiple linear systemsen_HK
dc.subjectKrylov spaceen_HK
dc.subjectConjugate gradient methoden_HK
dc.subjectGalerkin projectionen_HK
dc.titleGalerkin Projection Methods for Solving Multiple Linear Systemsen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=1064-8275&volume=21&issue=3&spage=836&epage=850&date=1999&atitle=Galerkin+Projection+Methods+for+Solving+Multiple+Linear+Systemsen_HK
dc.description.naturepublished_or_final_versionen_HK
dc.identifier.scopuseid_2-s2.0-0033293884-
dc.identifier.hkuros52937-

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