Splitting Preconditioner Research Articles

A new relaxed splitting preconditioner for the generalized saddle point problems from the incompressible Navier–Stokes equations

Based on the modified relaxed splitting (MRS) preconditioner proposed by Fan and Zhu (Appl Math Lett 55:18–26, 2016), a new relaxed splitting preconditioner is proposed for the generalized saddle point problems arising from the incompressible Navier–Stokes equations. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are analyzed. Numerical results show that the proposed preconditioner is superior to the MRS preconditioner when they are used as preconditioners to accelerate the convergence rate of the Krylov subspace methods, such as GMRES.

A note on “A dimensional split preconditioner for Stokes and linearized Navier–Stokes equations”

In this note, a technical error is pointed out in the proof of Theorem 3 in the paper M. Benzi and X.-P. Guo [1]. A correct proof of this theorem is given.

Two new variants of the HSS preconditioner for regularized saddle point problems

Two new preconditioners, which can be viewed as improved variants of the Hermitian and skew-Hermitian splitting (HSS) preconditioner, are presented for regularized saddle point problems. The unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed. Moreover, eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained. Compared with the HSS preconditioner, they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions. Numerical experiments from the Stokes problem are presented, which illustrates the advantages of the two preconditioners over the HSS preconditioner and some other preconditioners to accelerate the convergence rate of GMRES.

A splitting preconditioner for implicit Runge-Kutta discretizations of a partial differential-algebraic equation

In this paper we study efficient iterative methods for solving the system of linear equations arising from the fully implicit Runge-Kutta discretizations of a class of partial differential-algebraic equations. In each step of the time integration, a block two-by-two linear system is obtained and needed to be solved numerically. A preconditioning strategy based on an alternating Kronecker product splitting of the coefficient matrix is proposed to solve such linear systems. Some spectral properties of the preconditioned matrix are established and numerical examples are presented to demonstrate the effectiveness of this approach.

A new iterative method for solving a class of complex symmetric system of linear equations

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761---776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523---2538, 2013), when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.

A class of generalized relaxed PSS preconditioners for generalized saddle point problems

In this paper we introduce a class of generalized relaxed positive semi-definite and skew-Hermitian splitting (GRPSS) preconditioners for generalized saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related positive semi-definite and skew-Hermitian (PSS) preconditioner introduced by Shen (2014). Numerical results of a model Navier–Stokes equation demonstrate the effectiveness of the proposed preconditioners.

A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems

For nonsymmetric saddle point problems, Pan et al. (Appl Math Comput 172:762–771, 2006) proposed a deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner. In this paper, a variant of the DPSS preconditioner is proposed to accelerate the convergence of the associated Krylov subspace methods. The new preconditioner is much closer to the coefficient matrix than the DPSS preconditioner. The spectral properties of the new preconditioned matrix are analyzed. Theorem which provides the dimension of the Krylov space for the preconditioned matrix is obtained. Numerical experiments of a model Navier–Stokes problem are presented to illustrate the effectiveness of the new preconditioner.

Augmented block splitting preconditioner for singular saddle point problems

In this paper, the augmented block splitting preconditioner is considered to solve singular saddle point problems. This preconditioner will result in a proper splitting for the saddle point matrix and the corresponding iteration method is proved to converge to the least squares solution of the singular saddle point problem. Besides, GMRES with this preconditioner will also converge to the least squares solution. Numerical experiments are presented to illustrate its effectiveness for solving singular saddle point problems.

Spectral properties of a class of matrix splitting preconditioners for saddle point problems

Based on the accelerated Hermitian and skew-Hermitian splitting iteration scheme (Bai and Golub, 2007), we propose a new two-parameter matrix splitting preconditioner in this paper. Spectral properties of the preconditioned matrix are analyzed in detail. Furthermore, based on this preconditioner, an improved version of matrix splitting preconditioner is presented and analyzed. Finally, performance of the preconditioners is compared by using GMRES(m) as an iterative solver on linear systems arising from the discretization of Stokes and Navier–Stokes equations.

A modified relaxed splitting preconditioner for generalized saddle point problems from the incompressible Navier–Stokes equations

Based on a new matrix splitting of the original coefficient matrix, a modified relaxed splitting (MRS) preconditioner for generalized saddle point problems from the incompressible Navier–Stokes equations is considered. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied. The proposed preconditioner is closer to the original matrix than the generalized relaxed splitting (GRS) preconditioner in the sense of certain norm, which straightforwardly results in a MRS iteration method. Finally, numerical results are given to demonstrate the theoretical analysis. The results show that this novel preconditioner is competitive with and more effective than some of the best existing preconditioners.

A simplified HSS preconditioner for generalized saddle point problems

For generalized saddle point problems, we propose a simplified Hermitian and skew-Hermitian splitting (SHSS) preconditioner which is much closer to the generalized saddle point matrix than the HSS preconditioner. It is proved that all eigenvalues of the SHSS preconditioned matrix are real and nonunit eigenvalues are located in a positive interval. We also study the eigenvector distribution and the degree of the minimal polynomial of the preconditioned matrix. Numerical examples of a model Stokes problem show the effectiveness of the SHSS preconditioner.

A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for non-Hermitian generalized saddle point problems

For non-Hermitian saddle point linear systems, Pan, Ng and Bai presented a positive semi-definite and skew-Hermitian splitting (PSS) preconditioner (Pan et al. Appl. Math. Comput. 172, 762---771 2006), to accelerate the convergence rate of the Krylov subspace iteration methods like the GMRES method. In this paper, a relaxed positive semi-definite and skew-Hermitian (RPSS) splitting preconditioner based on the PSS preconditioner for the non-Hermitian generalized saddle point problems is considered. The distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed. Moreover, an upper bound on the degree of the minimal polynomial is also studied. Finally, numerical experiments of a model Navier-Stokes equation are presented to illustrate the efficiency of the RPSS preconditioner compared to the PSS preconditioner, the block diagonal preconditioner (BD), and the block triangular preconditioner (BT) in terms of the number of iteration and computational time.

A modified positive-definite and skew-Hermitian splitting preconditioner for generalized saddle point problems from the Navier-Stokes equation

In this paper, we extend the relaxed positive-definite and skew-Hermitian splitting preconditioner (RPSS) for generalized saddle-point problems in [J.-L. Zhang, C.-Q. Gu and K. Zhang, Appl. Math. C...

An inexact relaxed DPSS preconditioner for saddle point problem

Based on the relaxed deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner, in this paper, we proposed a class of relaxed deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioner for solving the saddle point problem. The proposed RDPSS preconditioner is a technical modification of the deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner [36]. The PSS preconditioner is a straightforward application of the positive-definite and skew-Hermitian splitting (PSS) iteration method for solving non-Hermitian positive definite linear systems initially established by Bai et al. [37]. Numerical results have shown that the proposed RDPSS preconditioner is advantageous over the existing DPSS preconditioner.

A new splitting preconditioner for the iterative solution of complex symmetric indefinite linear systems

In this paper, we consider preconditioned simplified Hermitian normal splitting (PSHNS) iteration method for solving complex symmetric indefinite linear systems, analyze the convergence of the PSHNS iteration method and discuss the spectral properties of the PSHNS preconditioned matrix. Using discrete Sine transform (DST), we apply a fast algorithm to solve the subsystem during the preconditioning process. Numerical experiments arising from the Helmholtz equation show the effectiveness and robustness of the PSHNS preconditioner. In addition, the GMRES method with the PSHNS preconditioner demonstrates meshsize-independent and wavenumber-insensitive convergence behavior.

An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

A generalized relaxed positive-definite and skew-Hermitian splitting preconditioner for non-Hermitian saddle point problems

For non-Hermitian saddle point problems with the non-Hermitian positive definite (1,1)-block, Zhang et al. (2014) presented a relaxed positive-definite and skew-Hermitian splitting (RPSS) preconditioner to accelerate the convergence rates of the Krylov subspace iteration methods such as GMRES. In this paper, the convergence property of the GRPSS iteration method is proved and a generalized RPSS (GRPSS) preconditioner is proposed. The GRPSS preconditioner is much closer to the coefficient matrix than the RPSS preconditioner in certain norm, which straightforwardly results in an GRPSS iteration method. We employing the GRPSS preconditioner to accelerate some Krylov subspace methods (like GMRES). The spectral distribution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained. Finally, numerical experiments of a model Navier–Stokes equation are presented to illustrate the efficiency of the GRPSS preconditioner.

A relaxed splitting preconditioner for saddle point problems

AbstractIn this paper, we introduce a relaxed splitting preconditioner for saddle point problems. Spectral properties of the preconditioned matrix are analyzed and compared with the closely related preconditioner in recent paper [New preconditioners for saddle point problems, Appl. Math. Comput. 172 (2006), 762-771] by Pan et al. Numerical experiments are given to illustrate the efficiency of the proposed precoditioner.

A relaxed positive-definite and skew-Hermitian splitting preconditioner for saddle point problems

For saddle point linear system whose coefficient matrix is non-Hermitian of positive definite Hermitian part, Pan et al. (2006) proposed a positive-definite and skew-Hermitian splitting (PSS) preconditioner. In this paper, based on the PSS preconditioner, we present a relaxed positive-definite and skew-Hermitian splitting (RPSS) preconditioner. We analyze the spectral properties of the new preconditioned matrix and conclude that the RPSS preconditioner has much better properties than PSS. Numerical experiments of a model Navier–Stokes problem are presented to illustrate the effectiveness of the new preconditioner.

Some new estimates on the complex eigenvalues of the HSS preconditioned matrix

In this paper, we derive some new bounds on the complex eigenvalues of the Hermitian and skew-Hermitian splitting (HSS) preconditioner for saddle point problems and improve those stated in previously published works.