On preconditioned generalized shift-splitting iteration methods for saddle point problems

New modified shift-splitting preconditioners for non-symmetric saddle point problems

“…Assume that A = P + S is a dominant positive definite and skew-Hermitian splitting. Let a, b, and c be defined as in Theorem 3.3 and α, β > 0 satisfy (12) or (13). Then all eigenvalues of the NMSSpreconditioned matrix P −1 NMSS A are located in a circle centered at (1, 0) with radius strictly less than 1.…”

“…Moreover, semi-convergence of the shift-splitting iteration method and spectral analysis of the shift-splitting preconditioned saddle point matrix have been studied by Cao et al [11] and Ren et al [22], respectively. Cao et al used the generalize shift-splitting matrix as a preconditioner and analyzed eigenvalue distribution of the preconditioned saddle point matrix [12]. Zhou et al [26] and Huang et al [17], respectively, proposed modified shift-splitting (MSS) and generalized modified shift-splitting (GMSS) preconditioners, for solving non-Hermitian saddle point problems.…”

Preconditioned inexact Jacobi–Davidson method for large symmetric eigenvalue problems

“…For saddle point problems, numerous solution techniques have been proposed for solving this type of systems, such as HSS preconditioning, constraint preconditioning, nonlinear Uzawa algorithm, SOR-like method (Benzi et al 2005;Pestana and Wathen 2015). The shift-splitting preconditioner was first studied in Bai et al (2006) for a class of non-Hermitian positive definite linear systems and then extended to saddle point problems (Axelsson 1979;Cao et al 2014Cao et al , 2015Cao et al , 2017. Greif and Schotzau (2006) is the first paper that considered regularization technique for the singular (1,1) block saddle point problem.…”

Generalized fast shift-splitting preconditioner for nonsymmetric saddle-point problems

“…By replacing the parameter in (2,2) block in the SS preconditioner by another parameter, Chen and Ma (2015) established a generalized SS (GSS) preconditioner and studied the unconditional convergence of the corresponding GSS iteration method. Ren et al (2017), discussed the refined eigenvalue distributions of the GSS preconditioned matrix. To improve convergence property of the GSS iteration method, Cao et al (2017) proposed a preconditioned GSS (PGSS) iteration method and a PGSS preconditioner for saddle-point problems (1).…”

“…Ren et al (2017), discussed the refined eigenvalue distributions of the GSS preconditioned matrix. To improve convergence property of the GSS iteration method, Cao et al (2017) proposed a preconditioned GSS (PGSS) iteration method and a PGSS preconditioner for saddle-point problems (1). Using the generalized SS and two-parameters acceleration techniques, Huang and Su (2017) considered the modified SS preconditioner (MSSP) and MGSSP preconditioner, respectively.…”