Preconditioning of GMRES by the skew-Hermitian iterations

L. A. Krukier ; T. S. Martynova

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 19 (2016) no. 3, pp. 267-279.

Résumé

A class of preconditioners for solving non-Hermitian positive definite systems of linear algebraic equations is proposed and investigated. It is based on the Hermitian and skew-Hermitian splitting of the initial matrix. The generalization for saddle point systems which have semidefinite or singular $(1,1)$ blocks is given. Our approach is based on an augmented Lagrangian formulation. It is shown that such preconditioners are effective for the iterative solution of systems of linear algebraic equations by the GMRES.

Bibliographie

[1] Benzi M., Golub G. H., Liesen J., “Numerical solution of saddle point problems”, Acta Numerica, 14 (2005), 1–137 | DOI | MR | Zbl

[2] Bychenkov Yu. V., Chizhonkov E. V., Iteratsionnye metody resheniya sedlovykh zadach, BINOM, M., 2010

[3] Krukier L. A., “Neyavnye raznostnye skhemy i iteratsionnyi metod ikh resheniya dlya odnogo klassa sistem kvazilineinykh uravnenii”, Izv. VUZov. Matem., 1979, no. 7, 41–52 | MR | Zbl

[4] Krukier L. A., “Kososimmetrichnye iteratsionnye metody resheniya statsionarnogo uravneniya konvektsii-diffuzii s malym parametrom pri starshei proizvodnoi”, Izv. VUZov. Matem., 1997, no. 4, 77–85 | MR | Zbl

[5] Krukier L. A., “Convergence acceleration of triangular iterative methods based on the skew-symmetric part of the matrix”, Appl. Numer. Mathem., 30 (1999), 281–290 | DOI | MR | Zbl

[6] Bochev M. A., Krukier L. A., “Ob iteratsionnom reshenii silno nesimmetrichnykh sistem lineinykh algebraicheskikh uravnenii”, Zhurn. vychisl. matem. i mat. fiziki, 31:11 (1997), 1283–1293 | MR | Zbl

[7] Chikina L. G., Krukier B. L., “Solution of linear equation systems with a dominant skew-symmetric part using the product triangular iterative method”, Comput. Methods in Appl. Mathem., 3:4 (2003), 647–650 | MR | Zbl

[8] Bai Z. Z., Krukier L. A., Martynova T. S., “Dvukhshagovye iteratsionnye metody resheniya statsionarnogo uravneniya konvektsii-diffuzii s malym parametrom pri starshei proizvodnoi na ravnomernoi setke”, Zhurn. vychisl. matem. i mat. fiziki, 46:2 (2006), 295–306 | MR | Zbl

[9] Krukier L. A., Martinova T. S., Bai Z.-Z., “Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems”, J. of Comput. and Appl. Mathem., 232:1 (2009), 3–16 | DOI | MR | Zbl

[10] Krukier L. A., Krukier B. L., Ren Z.-R., “Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems”, Numer. Linear Algebra Appl., 21 (2014), 152–170 | DOI | MR | Zbl

[11] Saad Y., Iterative Methods for Sparse Linear Systems, PWS Publishing Company, 1995

[12] Bai Z.-Z., Golub G. H., Pan J.-Y., “Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems”, Numerische Mathematik, 98 (2004), 1–32 | DOI | MR | Zbl

[13] Benzi M., Golub G. H., “A preconditioner for generalized saddle point problems”, SIAM J. Matrix Anal. Appl., 26 (2004), 20–41 | DOI | MR | Zbl

[14] Golub G. H., Greif C., “On solving block-structured indefinite linear systems”, SIAM J. on Scientific Computing, 24 (2003), 2076–2092 | DOI | MR | Zbl

[15] Bai Z.-Z., Sun J., Wang D., “A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations”, Comp. Math. Appl., 32:12 (1996), 51–76 | DOI | MR | Zbl

[16] Samarskii A. A., Vabischevich P. N., Chislennye metody resheniya zadach konvektsii-diffuzii, Editorial URSS, M., 1999

[17] Bai Z.-Z., Parlett B. N., Wang Z.-Q., “On generalized successive overrelaxation methods for augmented linear systems”, Numerische Mathematik, 102 (2005), 1–38 | DOI | MR | Zbl