skew-Hermitian Splitting Research Articles

Nonlinear Regularized HSS Iteration Method for a Class of Nonlinear Saddle-Point Systems with Applications to Image Deblurring Problems

In this paper, we construct a nonlinear regularized Hermitian and skew-Hermitian splitting (NRHSS) iteration method for a class of nonlinear saddle-point systems. The NRHSS iteration method can be seen as a generalization of the Peaceman-Rachford splitting method with monotone plus skew-symmetric splitting (PRSM/MSS). Besides, if the nonlinear operator reduces to the linear case, the NRHSS iteration method reduces to the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration method. Hence, the NRHSS iteration method can also be seen as a generalization of the RHSS iteration method. To overcome the heavy workload when direct solution methods are used, we also propose an inexact version of the NRHSS iteration method. Detailed theoretical analysis regarding the convergence properties is given. Numerical experiments on image deblurring problems are carried out to illustrate the feasibility and efficiency of the NRHSS iteration method.

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.

A computational macroscopic model of piezomagnetoelectric materials using Generalized Multiscale Finite Element Method

Piezomagnetoelectric materials have many applications in various areas of science and engineering. For example, such materials can be used in sensors, transducers, energy harvesters, etc. The mathematical model of piezomagnetoelectricity consists of a coupled system of partial differential equations for the electric potential, magnetic potential, and mechanical displacements.We develop multiscale algorithms based on the Generalized Multiscale Finite Element Method for solving the piezomagnetoelectricity problem. We consider two ways of constructing multiscale basis functions: split and coupled. In the split approach, we construct the basis functions separately for each field. The coupled approach constructs multiscale basis functions by solving coupled local problems. The split way requires fewer computational resources than the coupled one, but it may give a less accurate solution in the case of a strongly coupled medium. The coupled approach considers the mutual influence of the fields, which is especially important when they are strongly coupled. We use the multiscale partition of unity functions to obtain conforming basis functions for both approaches.We consider two- and three-dimensional model problems to test the effectiveness of the proposed multiscale approaches. On average, the coupled basis functions show better results than the split ones. We also consider the application of the single-step hermitian and skew-hermitian splitting for solving the discrete problem on a coarse grid. The results have demonstrated that one can successfully apply this technique to solving the piezomagnetoelectricity problem on a coarse grid.

A GENERALIZED SIMPLIFIED HERMITIAN AND SKEW-HERMITIAN SPLITTING PRECONDITIONER FOR DOUBLE SADDLE POINT PROBLEMS

In this work, we mainly propose a generalized simplified Hermitian and skew-Hermitian splitting (GSHSS) preconditioner for solving double saddle point problems and the eigenvalue distribution of the GSHSS preconditioner is analyzed in detail. In addition, we also study the eigenvector distribution and the degree of the minimal polynomial of the preconditioned matrix. Finally, numerical experiments show the effectiveness of the proposed preconditioner.

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 General Alternating-Direction Implicit Framework with Gaussian Process Regression Parameter Prediction for Large Sparse Linear Systems

This paper proposes an efficient general alternating-direction implicit (GADI) framework for solving large sparse linear systems. The convergence property of the GADI framework is discussed. Most of the existing ADI methods can be viewed as particular schemes of the developed framework. Meanwhile the GADI framework can derive new ADI methods. Moreover, as the algorithm efficiency is sensitive to the splitting parameters, we offer a data-driven approach, the Gaussian process regression (GPR) method based on the Bayesian inference, to predict the GADI framework's relatively optimal parameters. The GPR method requires a small training data set to learn the regression prediction mapping, which has sufficient accuracy and high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations to obtain relatively optimal splitting parameters. Finally, we use the three-dimensional convection-diffusion equation and continuous Sylvester matrix equation to examine the performance of our proposed methods. Numerical results demonstrate that the proposed framework is faster tens to thousands of times than the existing ADI methods, such as (inexact) Hermitian and skew-Hermitian splitting type methods in which the consumption of obtaining relatively optimal splitting parameters is ignored. Due to the efficiency of the developed methods, we can solve much larger linear systems which these existing ADI methods have been not reached.

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

Accelerated RHSS Iteration Method for Stabilized Saddle-Point Problems

For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretical analysis shows that the ARHSS method converges unconditionally to the unique solution of the saddle point problem. Finally, we use a numerical example to confirm the effectiveness of the method.

Hermitian and skew-Hermitian splitting method on a progressive-iterative approximation for least squares fitting

<abstract><p>To solve the problems of curves and surfaces approximation with normalized totally positive bases, a new progressive and iterative approximation for least square fitting method called HSS-LSPIA is proposed, which is based on the HSS iterative approach for solving linear equations of LSPIA. The HSS-LSPIA format includes two iterations with iterative difference vectors, each of which is distinct from the other. The approximate optimal positive constant, as well as convergence analyses, are provided. Furthermore, the HSS-LSPIA method can be faster than the ELSPIA, LSPIA, and WHPIA methods in terms of convergence speed. Numerical results verify this phenomenon.</p></abstract>

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.

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.

The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation

In this paper, we propose the generalized modified Hermitian and skew-Hermitian splitting (GMHSS) approach for computing the generalized Lyapunov equation. The GMHSS iteration is convergent to the unique solution of the generalized Lyapunov equation. Moreover, we discuss the convergence analysis of the GMHSS algorithm. Further, the inexact version of the GMHSS (IGMHSS) method is formulated to improve the GMHSS method. Finally, some numerical experiments are carried out to demonstrate the effectiveness and competitiveness of the derived methods

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.