Hermitian And skew-Hermitian Splitting Research Articles

Developing HSS iteration schemes for solving the quadratic matrix equation AX2+BX+C=0$AX^{2}+BX+C=0$

AbstractThe quadratic matrix equation (QME) occurs in the branches such as the quadratic eigenvalue problems and quasi‐birth‐death processes. Also, the numerical solution of QMEs is an essential step in many computational methods for linear‐quadratic and robust control, filtering, controller order reduction, inner‐outer factorization, spectral factorization, and other applications. In this study, schemes are presented to solve the QME based on the Hermitian and skew‐Hermitian splitting (HSS). It is shown that the proposed schemes converge to the solutions of the QME. Finally, some examples are solved to discover the application of the schemes in comparison with Newton's method.

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

A new two-step iteration method for discrete ill-posed problems and image restoration

In this study, for the augmented linear system of discrete ill-posed problems we establish a new two-step (NTS) iteration method containing a parameter and a parameter matrix, which is based on the Hermitian and skew-Hermitian splitting (HSS) and the upper and lower triangular splitting (ULT) of the coefficient matrix. Then, we theoretically study its convergence properties and determine its optimal iteration parameters. It is seen that the NTS method converges faster when the parameters are chosen properly. Experimental examples are carried out to further validate the effectiveness and accuracy of the new method compared to the newly developed methods in terms of the numerical performance and image recovering quality.

Variable-parameter HSS methods for non-Hermitian positive definite linear systems

The Hermitian and skew-Hermitian splitting (HSS) method is a well-known method employed in solving the non-Hermitian positive definite linear systems. However, the performance of the HSS method highly depends on the parameter value. In this paper, we address this issue by introducing two practical selections of this parameter, which are computed based on the Rayleigh quotient of the Hermitian part of the coefficient matrix. On the basis of the new selections, we propose the variable-parameter HSS methods for non-Hermitian positive definite linear systems and provide a convergent analysis under reasonable assumptions. Furthermore, to verify the effectiveness of our selections, we conduct experiments on several large and sparse linear systems. The numerical results show their efficiency over the existing quasi-optimal parameter.

Asynchronous iterations of HSS method for non-Hermitian linear systems

A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix splittings for -matrices. The computational model can be thought of as a two-stage alternating iterative method, which well suits to the well-known Hermitian and skew-Hermitian splitting (HSS) approach, with the particularity here of considering only one inner iteration. Experimental parallel performance comparison is conducted between the generalized minimal residual (GMRES) algorithm, the standard HSS and our asynchronous variant, on both real and complex non-Hermitian linear systems, respectively, arising from convection–diffusion and structural dynamics problems. A significant gain on execution time is observed in both cases.

On the Hermitian and skew-Hermitian splitting-like iteration approach for solving complex continuous-time algebraic Riccati matrix equation

Some matrix equations are important in application and studying in various fields of sciences. One of them is continuous-time algebraic Riccati matrix equation that is an interesting subject for many researchers in engineering and sciences specially applied mathematics. In this work we try to apply Hermitian and skew-Hermitian splitting (HSS) like method to solve this equation. Then we analyze the convergence of the new iterative method in detail. Also we try to determine the optimum parameter in our method which minimizes the upper bounds of absolute error. In the end we test the new algorithm by solving some numerical examples.

Hermitian and skew-Hermitian splitting methods for solving a tensor equation

The present paper deals with the numerical solution of non-Hermitian positive definite tensor equation under the Einstein product. Firstly, we extend the Hermitian and skew-Hermitian splitting (HSS) method to solve the tensor equation. Then we propose a new Hermitian splitting (NHS) method under some certain conditions, which is expected to converge faster than the HSS iteration. We also present the optimal parameters of both the HSS and NHS methods. Moreover, we apply the Smith technique to give two modified methods which can greatly accelerate the convergence rate. The performed numerical examples illustrate that the proposed methods are feasible and efficient.

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.

On the convergence of the minimum residual HSS iteration method

Recently, by applying the minimum residual technique to the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, a minimum residual HSS (MRHSS) iteration method was proposed for solving non-Hermitian positive definite linear systems. Although the MRHSS iteration method is very efficient, it is conditionally convergent. In this work, we further study the convergence of the MRHSS iteration method, and show that it can unconditionally convergent if its parameters are determined by minimizing a new norm of the residual. Numerical results verify that the MRHSS method discussed in this work is also very efficient.

Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems

By applying the minimum residual technique to the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, we introduce a non-stationary iteration method named minimum residual HSS (MRHSS) iteration method to solve non-Hermitian positive definite linear systems. The convergence property of the MRHSS iteration method together with the property of the iteration parameters are carefully studied. Numerical results verify the effectiveness and robustness of the MRHSS iteration method.

The Analysis of Dynamic Buckling of an Impacted Column using Difference Methods

In this paper, the object of our study is two coupled partial differential equation which have a rich physical background. With a reasonable change to the initial value, we use central difference method and compact finite difference method to get the numerical results of the governing equations. In the analysis of the latter, we use the Hermitian and skew-Hermitian splitting (HSS) method to solve a large sparse non-Hermitian positive definite system of linear equations. In this way, the computational efficiency can be improved and the convergence is guaranteed which is proved by Bai [Bai et al. [2003] “Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,” SIAM J. Matrix Anal. Appl. 24, 603–626] and Chen [Chen et al. [2014] “Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition,” J. Comput. Appl. Math. 264, 115–130]. And we also pay attention to the effect of initial conditions, on which we add a small perturbation to observe its influence by comparing numerical results.

Scaled norm minimization method for computing the parameters of the HSS and the two‐parameter HSS preconditioners

SummaryThe performance of the Hermitian and skew‐Hermitian splitting (HSS) preconditioner for the non‐Hermitian positive definite system of linear equations is largely dependent on the choice of its parameter value. In this work, an efficient scaled norm minimization (SNM) method is proposed to compute the parameter value of the HSS preconditioner. In addition, by choosing different parameters for the Hermitian and the skew‐Hermitian matrices in the HSS preconditioner, a two‐parameter HSS preconditioner is proposed. Moreover, an efficient and practical formula for computing the parameter values of this new preconditioner is also derived by using the SNM method. Numerical examples are illustrated to verify the performances of the HSS and the two‐parameter HSS preconditioners when their parameters are computed by the SNM method.

An efficient variant of HSS preconditioner for generalized saddle point problems

SummaryWe present an efficient variant of the Hermitian and skew‐Hermitian splitting (HSS) preconditioner for generalized saddle point problems. By switching the positions of the two splitting matrices in the HSS preconditioner, together with some modifications combined with the relaxation preconditioning technique, we show that the new preconditioner is much closer to the coefficient matrix and easier to be implemented. Theoretical analyses show that the corresponding iteration method converges to the unique solution of the generalized saddle point problem under certain conditions. The spectral properties, including bounds on the eigenvalues and condition numbers of the eigenvectors, for the preconditioned matrix, are also discussed. Finally, numerical experiments are presented to illustrate the effectiveness of the new preconditioner.

On the accelerated modified Newton-HSS method for systems of nonlinear equations

Hermitian and skew-Hermitian splitting (HSS) method converges unconditionally, which is efficient and robust for solving non-Hermitian positive-definite systems of linear equations. For solving systems of nonlinear equations with non-Hermitian positive-definite Jacobian matrices, Bai and Guo proposed the Newton-HSS method and gave numerical comparisons to show that the Newton-HSS method is superior to the Newton-USOR, the Newton-GMRES and the Newton-GCG methods. Recently, Wu and Chen proposed the modified Newton-HSS (MN-HSS) method which outperformed the Newton-HSS method. In this paper, we will establish a new accelerated modified Newton-HSS (AMN-HSS) method and give the local convergence theorem. Moreover, numerical results show that the AMN-HSS method outperforms the MN-HSS method.

Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations

Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants.

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.

Multistep matrix splitting iteration preconditioning for singular linear systems

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.

Modified Newton-NSS method for solving systems of nonlinear equations

By making use of the normal and skew-Hermitian splitting (NSS) method as the inner solver for the modified Newton method, we establish a class of modified Newton-NSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Under proper conditions, the local convergence theorem is proved. Furthermore, the successive-overrelaxation (SOR) technique has been proved quite successfully in accelerating the convergence rate of the NSS or the Hermitian and skew-Hermitian splitting (HSS) iteration method, so we employ the SOR method in the NSS iteration, and we get a new method, which is called modified Newton SNSS method. Numerical results are given to examine its feasibility and effectiveness.

On Preconditioners Based on HSS for the Space Fractional CNLS Equations

AbstractThe space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1,0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.