skew-Hermitian Splitting Iteration Research Articles

HSS-progressive interpolation for Loop and Catmull–Clark Subdivision Surfaces

Loop and Catmull–Clark subdivision schemes are the most popular and commonly employed approximation subdivision schemes. However, some features of the given mesh are lost in their refinement process. Thus, the subdivision surfaces of Loop and Catmull–Clark schemes does not interpolate the vertices of the given mesh and shrink in some cases. This paper proposed new progressive iterative approximation (PIA) formats based on Hermitian and skew-Hermitian splitting iteration technique (HSS). The proposed method named H-PIA and its weighted version called WH-PIA force the limit surface of Loop and Catmull–Clark subdivision schemes to interpolate the vertices of the given mesh. The approximate optimal weight of WH-PIA is given, and the convergence of H-PIA and WH-PIA are proved. Various test examples are provided to illustrate the efficiency and effectiveness of the proposed H-PIA and WH-PIA. Experimental results demonstrate that the rate of convergence of H-PIA and WH-PIA are faster than that of the PIA and weighted PIA (W-PIA), respectively.

Novel minimum residual MHSS iteration method for solving complex symmetric linear systems

For solving complex symmetric systems of linear equations, based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme, a minimum residual MHSS (MRMHSS) iteration method was proposed by applying the minimum residual technique to the MHSS iteration process. We have known that the MRMHSS iteration method is very efficient, while the convergence condition is difficult to verify in practice. In this work, we improve the MRMHSS iteration method by determining the involved iteration parameters using a new norm and prove that the so obtained novel variant can be unconditionally convergent. Numerical results are reported to demonstrate the applicability and effectiveness of the proposed method.

Semiconvergence of the extended PSS method for singular generalized saddle point problems

Recently, an extension of the positive definite and skew-Hermitian splitting (EPSS) iteration method for nonsingular generalized saddle point problems has been stated by Masoudi and Salkuyeh. In this article, we study the semiconvergence of the EPSS method for singular generalized saddle problems. Then a special case of the EPSS (SEPSS) preconditioner is applied to accelerate the convergence of the GMRES method for solving singular generalized saddle point problems. Some numerical results are given to demonstrate the robustness of the preconditioner.

A simplified PSS preconditioner for non-Hermitian generalized saddle point problems

The preconditioner for positive-definite and skew-Hermitian splitting (PSS) iteration method has been used to solve saddle point problems. Firstly, we propose a simplified PSS preconditioner for non-Hermitian generalized saddle point problems, which is much closer to original matrix than other PSS preconditioners. Moreover, the spectral properties of preconditioned matrix are also discussed. Finally, we give an example to illustrate the efficiency of the new preconditioner which is used to accelerate the convergence rate of the Krylov subspace, such as GMRES method, it has an obvious advantage than other preconditioners.

Lopsided modified Euler-extrapolated Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems

In this paper, a lopsided modified Euler-extrapolated Hermitian and skew-Hermitian splitting (LME-HS) iteration method is introduced for solving the complex symmetric linear systems. Under a loose restriction on parameter $\theta$, we demonstrate that LME-HS iteration method is convergent. Moreover, we present the optimal parameter ${\theta}^{*}$ of the LME-HS method and discuss the spectral properties of corresponding preconditioned matrix. Finally, the numerical experiments are used to verify the effectiveness of the proposed method.

Convergence of a generalized PMHSS method for a class of singular block two-by-two linear systems

In this work, we introduce a generalization of the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, named as generalized PMHSS (GPMHSS) iteration method, to solve a class of singular block two-by-two system of linear equations. Theoretical analyses show that the GPMHSS iteration method converges unconditionally to the minimum norm least squares solution for any initial guess no matter the system is consistent or inconsistent. Besides, with the preconditioner derived from the GPMHSS iteration method, the preconditioned generalized minimal residual (GMRES) method also determines the minimum norm least squares solution of the consistent singular block two-by-two linear systems at breakdown. Numerical experiments are presented to show the effectiveness and the robustness of the GPMHSS iteration method and the corresponding preconditioner.

On preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting method for complex symmetric linear systems

In this paper, we introduce a preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting (PE-SHSS) iteration method for solving a class of complex symmetric system of linear equations. The convergence properties of the PE-SHSS iteration method are investigated under suitable restrictions. In addition, the spectral properties of the corresponding preconditioned matrix are discussed. Finally, three numerical examples are also used to verify the effectiveness of the PE-SHSS iteration method.

Modified Newton-PHSS method for solving nonlinear systems with positive definite Jacobian matrices

By making use of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration as the inner solver for the modified Newton method, we establish the modified Newton-PHSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Moreover, the local convergence theorem is proved under proper conditions. Numerical results are given to show its feasibility and effectiveness. In addition, the advantages of the modified Newton-PHSS method over some other methods are shown by solving two systems of nonlinear equations.

Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems

For the saddle-point problems, we discuss and analyze the updated preconditioned Hermitian and skew-Hermitian splitting (UPHSS) iteration method in detail. On this basis, we introduce a two-stage iteration method for the UPHSS iteration method. Theoretical analysis shows that the UPHSS and two-stage UPHSS iteration methods are convergent to the unique solution of the saddle-point linear system when the parameter is suitably chosen. Numerical examples show the correctness of the theory and the effectiveness of these methods.

Minimum residual modified HSS iteration method for a class of complex symmetric linear systems

For solving a class of complex symmetric system of linear equations, we apply the minimum residual technique to the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme and propose an iteration method referred to as minimum residual MHSS (MRMHSS) iteration method. Compared with the classical MHSS method, the MRMHSS method has two more iteration parameters, which can be automatically and easily computed. Then, some properties of the MRMHSS iteration method are carefully studied. Finally, we use four examples to test the performance of the MRMHSS iteration method by comparing its numerical results with three other iteration methods.

HSS-iteration-based iterative interpolation of curves and surfaces with NTP bases

Based on the Hermitian and skew-Hermitian splitting iteration technique, a new progressive iteration approximation technique called HPIA for curves and surfaces interpolation with normalized totally positive (NTP) bases and its weighted version, namely WHPIA, are proposed. We take the previous iteration and the current iteration into account simultaneously, and establish a function based on NTP bases as a perturbation term in the iteration process. Convergence analyses and the approximate optimal weight of WHPIA are given. Theoretical and experimental results show that HPIA and WHPIA are effective.

Preconditioned modified Hermitian and skew-Hermitian splitting iteration methods for fractional nonlinear Schrödinger equations

A variant of preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method is proposed for a class of Toeplitz-like complex linear equations arising from the space fractional coupled nonlinear Schrödinger equations. Theoretical analysis show that the new method is convergent unconditionally. Moreover, the choice of the theoretical optimal parameter is further studied in detail, which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the Toeplitz matrix and the extreme absolute values of the diagonal matrix. In addition, the splitting can be used as a preconditioner for Krylov subspaces method. Numerical experiments show that the proposed method is efficient, and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem.

ERHSS iteration method for PDE optimal control problem

An extended regularized Hermitian and skew-Hermitian splitting iteration method, for solving the double saddle point linear system arising from PDE optimal control problem, is proposed. The unconditional convergence of the method is proved. Furthermore, some spectral properties of the preconditioned matrix are investigated. Numerical experiments show that the iteration method is feasible and efficient.

How to compute the minimum norm least squares solution of singular linear system by using the preconditioned HSS method?

For the singular, non-Hermitian, and positive semi-definite system of linear equations Ax = b, we introduce a kind of preconditioners for the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method. Then, the iteration sequence generated by the corresponding PHSS iteration method converges to the minimum norm least squares solution A‡b for any initial guess no matter the singular system of linear equations is consistent or inconsistent. In addition, a kind of PHSS preconditioners are derived from the PHSS iteration method. The PHSS preconditioned generalized minimum residual (PGMRES) methods also determine the minimum norm least squares solution of the consistent singular linear system at breakdown. Numerical experiments are used to verify the effectiveness and the robustness of the PHSS iteration method and the PHSS preconditioner.

Modified special HSS method for discrete ill-posed problems and image restoration

ABSTRACT For solving ill-posed inverse problems, we establish a modified version of special Hermitian and skew-Hermitian splitting (SHSS) iteration method, called the modified SHSS (MSHSS) method in this paper. Based on an augmented system formulation, we apply the modified splitting to solve ill-posed inverse problems, and theoretically study its convergence and determine its optimal iteration parameters. Furthermore, the SOR acceleration scheme for MSHSS iterative method is discussed. Finally, numerical experiments arising from a Fredholm integral equation of the first kind and image restoration are used to further examine the effectiveness of the new method.

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.

Modified Newton–SHSS method for a class of systems of nonlinear equations

It has been proved that Newton–HSS method is efficient and robust for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. In this paper, by utilizing the single-step Hermitian and skew-Hermitian splitting (SHSS) iteration technique, which performs efficiently under certain conditions, as the inner solver of the modified Newton method, we propose a class of modified Newton–SHSS methods. Subsequently, the local and semilocal convergence properties of our method will be discussed under some reasonable assumptions. Furthermore, we introduce the modified Newton–SHSS method with a backtracking strategy and analyze its basic global convergence theorem. Finally, several typical instances are used to illustrate the advantages of our methods when the Hermitian part of the Jacobian matrices are dominant.

An MHSS-like iteration method for two-by-two linear systems with application to FDE optimization problems

In this paper, we exploit the numerical solvers for a fractional differential equations (FDE) optimization problem with constraints given by fractional elliptic state equations. The discretized linear system can be rewritten as a two-by-two linear system. By making use of the strategy of modified Hermitian and skew-Hermitian splitting (MHSS) iteration method proposed by Bai, we propose an MHSS-like iteration method. Comparing with the MHSS iteration method, the MHSS-like method only requires one to solve the two-by-two linear systems by a sparse Cholesky factorization and a fast Fourier transform. Hence, the MHSS-like method has less workload than the MHSS method. The convergence properties and the quasi-optimal parameters are analyzed in detail. Numerical examples are used to testify the efficiency of the new method.

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.

Regularized HSS iteration methods for stabilized saddle-point problems

Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.