Geodesic
A deflated block flexible GMRES-DR method for linear systems with multiple right-hand sides
Electronic transactions on numerical analysis, Tome 41 (2014), pp. 478-496
This study is mainly focused on the iterative solution of multiple linear systems with several right-hand sides. To solve such systems efficiently, we first present a flexible version of block GMRES with deflation of eigenvalues according to [R. B. Morgan, Restarted block-GMRES with deflation of eigenvalues, Appl. Numer. Math., 54 (2005), pp. 222 -- 236] and then apply a modified block Arnoldi vector deflation technique to accelerate the convergence of this new flexible version. Incorporating this deflation technique, the new algorithm can address the possible linear dependence at each iteration during the block Arnoldi procedure and reduce computational expense. Moreover, by analyzing its main mathematical properties, we show that the vector deflation procedure arises from the non-increasing behavior of the singular values of the block residual. In addition, the new approach also inherits the property of deflating small eigenvalues to mitigate convergence slowdown. Finally, the effectiveness of the proposed method is illustrated by some numerical experiments.
Classification : 65F10, 65F50
Keywords: deflated BFGMRES-DR, block Krylov subspace, modified block arnoldi vector deflation, harmonic Ritz vectors, deflated block flexible arnoldi procedure, multiple right-hand sides 
@article{ETNA_2014__41__a1,
     author = {Meng,  Jing and Zhu,  Pei-Yong and Li,  Hou-Biao and Gu,  Xian-Ming},
     title = {A deflated block flexible {GMRES-DR} method for linear systems with multiple right-hand sides},
     journal = {Electronic transactions on numerical analysis},
     pages = {478--496},
     year = {2014},
     volume = {41},
     zbl = {1312.65048},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2014__41__a1/}
}
TY  - JOUR
AU  - Meng,  Jing
AU  - Zhu,  Pei-Yong
AU  - Li,  Hou-Biao
AU  - Gu,  Xian-Ming
TI  - A deflated block flexible GMRES-DR method for linear systems with multiple right-hand sides
JO  - Electronic transactions on numerical analysis
PY  - 2014
SP  - 478
EP  - 496
VL  - 41
UR  - http://geodesic.mathdoc.fr/item/ETNA_2014__41__a1/
LA  - en
ID  - ETNA_2014__41__a1
ER  - 
%0 Journal Article
%A Meng,  Jing
%A Zhu,  Pei-Yong
%A Li,  Hou-Biao
%A Gu,  Xian-Ming
%T A deflated block flexible GMRES-DR method for linear systems with multiple right-hand sides
%J Electronic transactions on numerical analysis
%D 2014
%P 478-496
%V 41
%U http://geodesic.mathdoc.fr/item/ETNA_2014__41__a1/
%G en
%F ETNA_2014__41__a1
Meng,  Jing; Zhu,  Pei-Yong; Li,  Hou-Biao; Gu,  Xian-Ming. A deflated block flexible GMRES-DR method for linear systems with multiple right-hand sides. Electronic transactions on numerical analysis, Tome 41 (2014), pp. 478-496. http://geodesic.mathdoc.fr/item/ETNA_2014__41__a1/