skew-Hermitian Splitting Preconditioner Research Articles

A generalized variant of modified relaxed positive-semidefinite and skew-Hermitian splitting preconditioner for generalized saddle point problems

A generalized variant of modified relaxed positive-semidefinite and skew-Hermitian splitting preconditioner for generalized saddle point problems

A modified variant of HSS preconditioner for generalized saddle point problems

Recently, Zhang [Numerical Linear Algebra with Applications, 2018: e2166] constructed an efficient variant of Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems, and gave the corresponding theoretical analysis and numerical experiment. In this paper, based on the efficient variant of the HSS (EVHSS) preconditioner, we relax the strong condition on the iterative parameter [Formula: see text], generalize the algorithms and further present the modified variant of Hermitian and skew-Hermitian splitting (HSS) (MVHSS) preconditioner for solving the generalized saddle point problem. Moreover, by similar theoretical analysis, we obtain that the MVHSS iterative method is convergent under weaker conditions. Finally, an example is given to verify the effectiveness of the proposed method.MSC: 65F10; 65F15; 65F50

On the Preconditioning Properties of RHSS Preconditioner for Saddle-Point Linear Systems

In this paper, for the regularized Hermitian and skew-Hermitian splitting (RHSS) preconditioner introduced by Bai and Benzi (BIT Numer Math 57: 287–311, 2017) for the solution of saddle-point linear systems, we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters. We accurately describe the numbers of eigenvalues clustered at (0, 0) and (2, 0), if the iteration parameter is close to 0. An estimate about the condition number of the corresponding eigenvector matrix, which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method, is also studied in this work.

On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems

In this work, with respect to the regularization matrix of the new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for non-Hermitian saddle-point problems, we provide a class of Hermitian positive semidefinite matrices, which depend on certain parameters, for practical computations. A precise description about the eigenvalue distribution of the corresponding preconditioned matrix is given. The condition number of the eigenvector matrix, which partly determines the convergence rate of the related preconditioned Krylov subspace method, is also discussed in this work. Finally, some numerical experiments are carried out to identify the effectiveness of the presented special choices for the regularization matrix to solve the non-Hermitian saddle-point problems.

Regularized DPSS preconditioners for generalized saddle point linear systems

By introducing a regularization matrix and an additional iteration parameter, a new class of regularized deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioners are proposed for generalized saddle point linear systems. Compared with the well-known Hermitian and skew-Hermitian splitting (HSS) preconditioner and the regularized HSS preconditioner (Bai, 2019) studied recently, the new RDPSS preconditioners have much better computing efficiency especially when the (1,1) block matrix is non-Hermitian. It is proved that the corresponding RDPSS stationary iteration method is unconditionally convergent. In addition, clustering property of the eigenvalues of the RDPSS preconditioned matrix is studied in detail. Two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the Navier–Stokes equation show the effectiveness of the new proposed preconditioners.

Regularized DPSS Preconditioners for Singular Saddle Point Problems

Recently, Cao proposed a regularized deteriorated positive and skewHermitian splitting (RDPSS) preconditioner for the non-Hermitian nonsingular saddle point problem. In this paper, we consider applying RDPSS preconditioner to solve the singular saddle point problem. Moreover, we propose a two-parameter accelerated variant of the RDPSS (ARDPSS) preconditioner to further improve its efficiency. Theoretical analysis proves that the RDPSS and ARDPSS methods are semi-convergent unconditionally. Some spectral properties of the corresponding preconditioned matrices are analyzed. Numerical experiments indicate that better performance can be achieved when applying the ARDPSS preconditioner to accelerate the GMRES method for solving the singular saddle point problem.

A parameterized deteriorated PSS preconditioner and its optimization for nonsymmetric saddle point problems

Based on the deteriorated positive definite and skew-Hermitian splitting (DPSS) preconditioner, we develop a parameterized DPSS (PDPSS) preconditioner for the nonsymmetric saddle point linear systems. The spectral properties of the PDPSS preconditioned coefficient matrix are analyzed. Moreover, an upper bound of the degree of the minimal polynomial of the PDPSS preconditioned matrix is also obtained. Inasmuch as the efficiency of the PDPSS preconditioner depends on the values of its parameters, we further derive a fast and effective method to compute the optimal parameter values of the PDPSS preconditioner. Finally, numerical examples are employed to illustrate the feasibility and the efficiency of the PDPSS preconditioner.

A generalized variant of simplified HSS preconditioner for generalized saddle point problems

Based on the simplified Hermitian and skew-Hermitian splitting (SHSS) preconditioner developed by Cao, Ren and Shi (BIT Numer. Math., 2016), a generalized variant of the SHSS (GSHSS) preconditioner is presented for solving generalized saddle point problems. The convergence of the GSHSS iteration method and the spectral properties of the preconditioned matrix are studied in details. Eigenvalue bounds and the degree of minimal polynomials of the preconditioned matrix are obtained. In particular, a practical and feasible choice strategy of iteration parameter is given, which also provides a feasible way to select the accelerated parameters for some other preconditioners. Numerical experiments are presented, which show the effectiveness of the proposed preconditioner and the feasibility of the practical choice strategy of accelerated parameter.

Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems

In this article, to better implement the modified positive-definite and skew-Hermitian splitting preconditioners studied recently (Numer. Algor., 72 (2016) 243–258) for generalized saddle point problems, a class of inexact modified positive-definite and skew-Hermitian splitting preconditioners is proposed with improved computing efficiency. Some spectral properties, including the eigenvalue distribution, the eigenvector distribution, and an upper bound of the degree of the minimal polynomial of the inexact modified positive-definite and skew-Hermitian splitting preconditioned matrices are studied. In addition, a theoretical optimal inexact modified positive-definite and skew-Hermitian splitting preconditioner is obtained. Numerical experiments arising from a model steady incompressible Navier–Stokes problem are used to validate the theoretical results and illustrate the effectiveness of this new class of proposed preconditioners.

On spectral clustering of HSS preconditioner for generalized saddle-point matrices

For the nonsingular generalized saddle-point matrix of a Hermitian positive definite or semidefinite leading block, we rigorously analyze clustering property for the eigenvalues of the corresponding preconditioned matrix with respect to the Hermitian and skew-Hermitian splitting preconditioner. The result shows that these eigenvalues are clustered around 0+, 2−, and a few points located on the unit circle centered at 1, as the iteration parameter is close to 0.

Variants of the deteriorated PSS preconditioner for saddle point problems

Two new preconditioners, which can be viewed as variants of the deteriorated positive definite and skew-Hermitian splitting preconditioner, are proposed for solving saddle point problems. The corresponding iteration methods are proved to be convergent unconditionally for cases with positive definite leading blocks. The choice strategies of optimal parameters for the two iteration methods are discussed based on two recent optimization results for extrapolated Cayley transform, which result in faster convergence rate and more clustered spectrum. Compared with some preconditioners of similar structures, the new preconditioners have better convergence properties and spectrum distributions. In addition, more practical preconditioning variants of the new preconditioners are considered. Numerical experiments are presented to illustrate the advantages of the new preconditioners over some similar preconditioners to accelerate GMRES.

Parameterized approximate block LU preconditioners for generalized saddle point problems

In this paper, we are concerned with the iteration solution of generalized saddle point problems. Based on the exact block LU factorization of the coefficient matrix, we construct a class of parameterized approximate block LU factorization preconditioners, which rely on suitable approximations of the Schur complement of the (1,1) block of the coefficient matrix. Convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced. Algebraic characteristics of the related preconditioned matrices are discussed, including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial. The established results extend those of the approximate factorization and variants of the Hermitian and skew-Hermitian splitting and positive and skew-Hermitian splitting preconditioners for saddle point problems. Numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners.

A variant of the HSS preconditioner for complex symmetric indefinite linear systems

Using the equivalent block two-by-two real linear systems, we establish a new variant of the Hermitian and skew-Hermitian splitting (HSS) preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is not only a better approximation to the block two-by-two real coefficient matrix than the well-known HSS preconditioner, but also resulting in an unconditional convergent fixed-point iteration. The quasi-optimal parameter, which minimizes an upper bound of the spectral radius of the iteration matrix, is analyzed. Eigen-properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed. Finally, two numerical examples are provided to show the efficiency of the new preconditioner.

Spectral properties of the matrix splitting preconditioners for generalized saddle point problems

In this paper, we study the spectral properties of two different kinds of preconditioners for generalized saddle point problems. One is based on the two-parameter matrix splitting preconditioner for saddle point problems by Wang et al., we generalize this preconditioner to generalized saddle point problems and analyze the spectral properties of the corresponding preconditioned matrix. The other is based on the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems by Huang et al., we study the spectral properties of the HSS preconditioner with two different parameters α and β (the generalized HSS preconditioner) for generalized saddle point problems. In addition, some numerical tests are given to verify the validity of the presented theoretical results.

New variant of the HSS iteration method for weighted Toeplitz regularized least-squares problems from image restoration

For fast solving weighted Toeplitz least-squares problems from image restoration, Ng and Pan (2014) studied a new Hermitian and skew-Hermitian splitting (NHSS) preconditioner. In this paper, a generalization of the NHSS preconditioner and the corresponding iterative method are presented. Convergence for the new iteration method is studied and optimal choice of the parameters is discussed. Bounds on the eigenvalues and the corresponding eigenvector distributions are proposed. The degree of the minimal polynomial of the preconditioned matrix is obtained. Numerical experiments arising from image restoration are provided, which show the proposed iteration method is effective and confirm our theoretical results are correct.

A Modified Relaxed Positive-Semidefinite and Skew-Hermitian Splitting Preconditioner for Generalized Saddle Point Problems

AbstractBased on the relaxed factorization techniques studied recently and the idea of the simple-like preconditioner, a modified relaxed positive-semidefinite and skew-Hermitian splitting (MRPSS) preconditioner is proposed for generalized saddle point problems. Some properties, including the eigenvalue distribution, the eigenvector distribution and the minimal polynomial of the preconditioned matrix are studied. Numerical examples arising from the mixed finite element discretization of the Oseen equation are illustrated to show the efficiency of the new preconditioner.

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

Based on the variant of the deteriorated positive-definite and skew-Hermitian splitting (VDPSS) preconditioner developed by Zhang and Gu (BIT Numer. Math. 56:587---604, 2016), a generalized VDPSS (GVDPSS) preconditioner is established in this paper by replacing the parameter ź in (2,2)-block of the VDPSS preconditioner by another parameter β. This preconditioner can also be viewed as a generalized form of the VDPSS preconditioner and the new relaxed HSS (NRHSS) preconditioner which has been exhibited by Salkuyeh and Masoudi (Numer. Algorithms, 2016). The convergence properties of the GVDPSS iteration method are derived. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. We also study the upper bounds on the degree of the minimum polynomial of the preconditioned matrix. Numerical experiments are implemented to illustrate the effectiveness of the GVDPSS preconditioner and verify that the GVDPSS preconditioned generalized minimal residual method is superior to the DPSS, relaxed DPSS, SIMPLE-like, NRHSS, and VDPSS preconditioned ones for solving saddle point problems in terms of the iterations and computational times.

Two improvements of the deteriorated PSS preconditioner for generalized saddle point problems

For generalized saddle point problems, we present two improved variants of the deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner to accelerate the convergence rate of the Krylov subspace iteration method. The new preconditioners are not only better approximations to the generalized saddle point matrix than the DPSS preconditioner but also easier to implement than the DPSS preconditioner. Theoretical analyses show that the corresponding splitting iteration methods are also convergent unconditionally. The quasi-optimal choices and practical estimations of the iteration parameters are discussed. Moreover, eigenproperties of the preconditioned matrices are described and upper bounds of the degree of the minimal polynomial of the preconditioned matrices are obtained. Finally, numerical experiments arising from the discretization of a model Navier-Stokes equation are presented to show the efficiency of the proposed preconditioners.

A new block preconditioner for complex symmetric indefinite linear systems

Using the equivalent block two-by-two real linear systems and relaxing technique, we establish a new block preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is much closer to the original block two-by-two coefficient matrix than the Hermitian and skew-Hermitian splitting (HSS) preconditioner. We analyze the spectral properties of the new preconditioned matrix, discuss the eigenvalue distribution and derive an upper bound for the degree of its minimal polynomial. Finally, some numerical examples are provided to show the effectiveness and robustness of our proposed preconditioner.

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.