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

Authors  
Keywords  Multiple linear systems Krylov space Conjugate gradient method Galerkin projection 
Issue Date  1999 
Publisher  Society for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sambin/dbq/toclist/SISC 
Citation  SIAM Journal on Scientific Computing, 1999, v. 21 n. 3, p. 836850 How to Cite? 
Abstract  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 righthand 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 righthand sides are different. We extend and analyze the method to solve multiple linear systems with varying coefficient matrices and righthand 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 Identifier  http://hdl.handle.net/10722/42993 
ISSN  2015 Impact Factor: 1.792 2015 SCImago Journal Rankings: 2.166 
DC Field  Value  Language 

dc.contributor.author  Ng, MKP  en_HK 
dc.contributor.author  Chan, TF  en_HK 
dc.date.accessioned  20070323T04:36:25Z   
dc.date.available  20070323T04:36:25Z   
dc.date.issued  1999  en_HK 
dc.identifier.citation  SIAM Journal on Scientific Computing, 1999, v. 21 n. 3, p. 836850  en_HK 
dc.identifier.issn  10648275  en_HK 
dc.identifier.uri  http://hdl.handle.net/10722/42993   
dc.description.abstract  In 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 righthand 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 righthand sides are different. We extend and analyze the method to solve multiple linear systems with varying coefficient matrices and righthand 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.extent  351104 bytes   
dc.format.extent  26112 bytes   
dc.format.mimetype  application/pdf   
dc.format.mimetype  application/msword   
dc.language  eng  en_HK 
dc.publisher  Society for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sambin/dbq/toclist/SISC  en_HK 
dc.relation.ispartof  SIAM Journal on Scientific Computing   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.subject  Multiple linear systems  en_HK 
dc.subject  Krylov space  en_HK 
dc.subject  Conjugate gradient method  en_HK 
dc.subject  Galerkin projection  en_HK 
dc.title  Galerkin Projection Methods for Solving Multiple Linear Systems  en_HK 
dc.type  Article  en_HK 
dc.identifier.openurl  http://library.hku.hk:4550/resserv?sid=HKU:IR&issn=10648275&volume=21&issue=3&spage=836&epage=850&date=1999&atitle=Galerkin+Projection+Methods+for+Solving+Multiple+Linear+Systems  en_HK 
dc.description.nature  published_or_final_version  en_HK 
dc.identifier.scopus  eid_2s2.00033293884   
dc.identifier.hkuros  52937   