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#### Article: Galerkin Projection Methods for Solving Multiple Linear Systems

Title Galerkin Projection Methods for Solving Multiple Linear Systems Ng, MKChan, TF Multiple linear systemsKrylov spaceConjugate gradient methodGalerkin projection 1999 Society for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/sisc.php SIAM Journal on Scientific Computing, 1999, v. 21 n. 3, p. 836-850 How to Cite? In 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. http://hdl.handle.net/10722/42993 1064-82752019 Impact Factor: 1.9762015 SCImago Journal Rankings: 2.166 WOS:000084272500003

DC FieldValueLanguage
dc.contributor.authorNg, MKen_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
dc.format.extent351104 bytes-
dc.format.extent26112 bytes-
dc.format.mimetypeapplication/pdf-
dc.format.mimetypeapplication/msword-
dc.languageengen_HK
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://www.siam.org/journals/sisc.php-
dc.relation.ispartofSIAM Journal on Scientific Computing-
dc.rights© 1999 Society for Industrial and Applied Mathematics. First Published in SIAM Journal on Scientific Computing in volume 21, issue 3, published by the Society for Industrial and Applied Mathematics (SIAM).-
dc.subjectMultiple linear systemsen_HK
dc.subjectKrylov spaceen_HK
dc.subjectGalerkin projectionen_HK
dc.titleGalerkin Projection Methods for Solving Multiple Linear Systemsen_HK
dc.typeArticleen_HK
dc.description.naturepublished_or_final_versionen_HK
dc.identifier.doi10.1137/S1064827598310227-
dc.identifier.scopuseid_2-s2.0-0033293884-
dc.identifier.hkuros52937-
dc.identifier.volume21-
dc.identifier.issue3-
dc.identifier.spage836-
dc.identifier.epage850-
dc.identifier.isiWOS:000084272500003-