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Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. (English) Zbl 1134.65022

The authors propose a class of accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large saddle-point problems. It has been shown that the new methods converge unconditionally to the unique solution of the saddle-point problem. Moreover, the optimal choices of the iteration parameters involved and the corresponding asymptotic convergence rates of the new methods are computed exactly. Additionally, the authors study the use of AHSS as a preconditioner to Krylov subspace methods such as the generalized minimal residual (GMRES) method and demonstrate the asymptotic convergence rate of the preconditioned GMRES method.

Finally, the authors demonstrate the applicability and effectiveness of the proposed methods for solving sparse saddle-point problem. The AHSS method is superior to GMRES and its restarted variants and can be considered as an attractive iterative method for solving sparse saddle-point problems.

MSC:
65F10Iterative methods for linear systems
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
65F50Sparse matrices (numerical linear algebra)