An efficient two-step iterative method for solving a class of complex symmetric linear systems

In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems.     

Efficient preconditioned NHSS iteration methods for solving complex symmetric linear systems

Based on the new HSS (NHSS) iteration method introduced by Pour and Goughery (2015), we propose a preconditioned variant of NHSS (P*NHSS) and an efficient parameterized P*NHSS (PPNHSS) iteration methods for solving a class of complex symmetric linear systems. The convergence properties of the P*NHSS and the PPNHSS iteration methods show that the iterative sequences are convergent to the unique solution of the linear system for any initial guess when the parameters are properly chosen. Moreover, we discuss the quasi-optimal parameters which minimize the upper bounds for the spectral radius of the iteration matrices. Numerical results show that the PPNHSS iteration method is superior to several iteration methods whether the experimental optimal parameters are used or not.     

The improvements of the generalized shift-splitting preconditioners for non-singular and singular saddle point problems

ABSTRACT

To solve the saddle point problems with symmetric positive definite (1,1) parts, the improved generalized shift-splitting (IGSS) preconditioner is established in this paper, which yields the IGSS iteration method. Theoretical analysis shows that the IGSS iteration method is convergent and semi-convergent unconditionally. The choices of the iteration parameters are discussed. Moreover, some spectral properties, including the eigenvalue and eigenvector distributions of the preconditioned matrix are also investigated. Finally, numerical results are presented to verify the robustness and the efficiency of the proposed iteration method and the corresponding preconditioner for solving the non-singular and singular saddle point problems.     

Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations

In this paper, to solve a broad class of complex symmetric linear systems, we recast the complex system in a real formulation and apply the generalized successive overrelaxation (GSOR) iterative method to the equivalent real system. We then investigate its convergence properties and determine its optimal iteration parameter as well as its corresponding optimal convergence factor. In addition, the resulting GSOR preconditioner is used to precondition Krylov subspace methods such as the generalized minimal residual method for solving the real equivalent formulation of the system. Finally, we give some numerical experiments to validate the theoretical results and compare the performance of the GSOR method with the modified Hermitian and skew-Hermitian splitting iteration.     

Parameterized generalized shift-splitting preconditioners for nonsymmetric saddle point problems

To solve nonsymmetric saddle point problems, the parameterized generalized shift-splitting (PGSS) preconditioner is presented and analyzed. The corresponding PGSS iteration method can be applied not only to the nonsingular saddle point problems but also to the singular ones. The convergence and semi-convergence of the PGSS iteration method are discussed carefully. Meanwhile, the spectral properties of the preconditioned matrix and the strategy of the choices of the parameters are given. Numerical experiments further demonstrate that the PGSS iteration method and the PGSS preconditioner are efficient and have better performance than some existing iteration methods and newly proposed preconditioners, respectively, for solving both the nonsingular and singular nonsymmetric saddle point problems.     

A new preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

In this paper, based on the preconditioners presented by Cao [A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices, Journal of Computational and Applied Mathematics 238(15) (2013), pp. 109–115], we introduce and study a new augmentation block preconditioners for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and its minimal polynomial. Finally, numerical examples show that the eigenvalue distribution with presented preconditioner has the same spectral clustering with preconditioners in the literature when choosing the optimal parameters and the preconditioner in this paper and in the literature improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solve the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.     

On parameterized matrix splitting preconditioner for the saddle point problems

In this paper, we present a parameterized matrix splitting (PMS) preconditioner for the large sparse saddle point problems. The preconditioner is based on a parameterized splitting of the saddle point matrix, resulting in a fixed-point iteration. The convergence theorem of the new iteration method for solving large sparse saddle point problems is proposed by giving the restrictions imposed on the parameter. Based on the idea of the parameterized splitting, we further propose a modified PMS preconditioner. Some useful properties of the preconditioned matrix are established. Numerical implementations show that the resulting preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as generalized minimal residual method.     

Comparison results on the preconditioned GAOR method for generalized least squares problems

Recently, Zhou et al. [Preconditioned GAOR methods for solving weighted linear least squares problems, J. Comput. Appl. Math. 224 (2009), pp. 242–249] have proposed the preconditioned generalized accelerated over relaxation (GAOR) methods for solving generalized least squares problems and studied their convergence rates. In this paper, we propose a new type of preconditioners and study the convergence rates of the new preconditioned GAOR methods for solving generalized least squares problems. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods presented by Zhou et al. whenever these methods are convergent. Lastly, numerical experiments are provided in order to confirm the theoretical results studied in this paper.     

Multi space reduced basis preconditioners for parametrized Stokes equations

We introduce a two-level preconditioner for the efficient solution of large scale saddle-point linear systems arising from the finite element (FE) discretization of parametrized Stokes equations. This preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning method proposed in Dal Santo et al. (2018); it combines an approximated block (fine grid) preconditioner with a reduced basis (RB) solver which plays the role of coarse component. A sequence of RB spaces, constructed either with an enriched velocity formulation or a Petrov–Galerkin projection, is built. Each RB coarse component is defined to perform a single iteration of the iterative method at hand. The flexible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time. Numerical test cases for Stokes flows in three dimensional parameter-dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings.     

A modified Hestenes–Stiefel conjugate gradient method with an optimal property

In this paper, based on the numerical efficiency of Hestenes–Stiefel (HS) method, a new modified HS algorithm is proposed for unconstrained optimization. The new direction independent of the line search satisfies in the sufficient descent condition. Motivated by theoretical and numerical features of three-term conjugate gradient (CG) methods proposed by Narushima et al., similar to Dai and Kou approach, the new direction is computed by minimizing the distance between the CG direction and the direction of the three-term CG methods proposed by Narushima et al. Under some mild conditions, we establish global convergence of the new method for general functions when the standard Wolfe line search is used. Numerical experiments on some test problems from the CUTEst collection are given to show the efficiency of the proposed method.     

Convergence analysis of modified PGSS methods for singular saddle-point problems

Recently, variants of shift-splitting iteration method have been proposed for solving singular saddle-point problems. However, these methods can only be proved to converge to one of the solutions of the consistent singular linear system, not knowing any further information about this solution. In this work, we consider a modified preconditioned generalized shift-splitting (MPGSS) iteration method for solving both consistent and inconsistent singular saddle-point problems. This method is proved to converge to the best least squares solution. Moreover, based on the iteration form, a preconditioner is obtained to accelerate Krylov subspace methods. Theoretical analysis shows that the preconditioned GMRES method also converges to the best least squares solution of the consistent singular saddle-point problem. In addition, numerical results are presented to show the effectiveness and robustness of the proposed iteration method and preconditioner.     

求解对称特征值问题的块Chebyshev—Davidson方法

块Davidson方法是求解大型对称矩阵特征值问题块Lancz08方法的预处理变形．为了加速块Davidson方法的收敛性,我们组合块Chebyshev迭代法和块Davidson方法,提出了求解大型对称矩阵若干极端特征值的块Chebyshev—Davidson方法,并将收缩技术应用到该方法中．数值结果表明,块Chebyshev—Davidson方法优于块Davidson方法和Chebyshev—Davidson方法．     

The Uzawa-PPS iteration methods for nonsingular and singular non-Hermitian saddle point problems

In this paper, based on the positive-definite and positive-semidefinite splitting (PPS) iteration scheme, we establish a class of Uzawa-PPS iteration methods for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite. Theoretical analyses show that the convergence and semi-convergence properties of the proposed methods can be guaranteed under suitable conditions. Furthermore, we consider acceleration of the Uzawa-PPS methods by Krylov subspace (like GMRES) methods and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners.     

Gradient-based iterative algorithms for generalized coupled Sylvester-conjugate matrix equations

By applying the hierarchical identification principle, the gradient-based iterative algorithm is suggested to solve a class of complex matrix equations. With the real representation of a complex matrix as a tool, the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices. Also, we solve the problem which is proposed by Wu et al. (2010). Finally, some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper.     

On the scaled newton method for the symmetric eigenvalue problem

For solving symmetric eigenvalue problems the scaled Newton method is proposed. The new algorithm is equivalent to the Rayleigh quotient iteration and its rate of convergence is cubic. Numerical experiments indicate that the scaled Newton method is very competitive with the projected Newton method, and produces, in the main, insignificantly smaller residua than the Rayleigh quotient iteration.     

A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems

Recently, Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891] proposed a numerical technique based on cubic B-spline for solving a Bratu-type problem. This method provides a second-order convergent approximation to the solution of the problem. In this paper, we develop a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane–Emden problems. The error analysis of the quartic B-spline interpolation is carried out. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and a numerical method based on cubic B-spline approach. Comparison reveals that our method produces more accurate results than the method proposed by Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891].     

An iterative algorithm for approximate orthogonalisation of symmetric matrices

In a previous article, one of the authors presented an extension of an iterative approximate orthogonalisation algorithm, due to Z. Kovarik, for arbitrary rectangular matrices. In the present article, we propose a modified version of this extension for the class of arbitrary symmetric matrices. For this new algorithm, the computational effort per iteration is much smaller than for the initial one. We prove its convergence and also derive an error reduction factor per iteration. In the second part of the article, we show that we can eliminate the matrix inversion required by the previous algorithm in each iteration, by replacing it with a polynomial matrix expression. Some numerical experiments are also presented for a collocation discretisation of a first kind integral equation.     

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.     

An Algorithmic Framework of Generalized Primal–Dual Hybrid Gradient Methods for Saddle Point Problems

The primal–dual hybrid gradient method (PDHG) originates from the Arrow–Hurwicz method, and it has been widely used to solve saddle point problems, particularly in image processing areas. With the introduction of a combination parameter, Chambolle and Pock proposed a generalized PDHG scheme with both theoretical and numerical advantages. It has been analyzed that except for the special case where the combination parameter is 1, the PDHG cannot be casted to the proximal point algorithm framework due to the lack of symmetry in the matrix associated with the proximal regularization terms. The PDHG scheme is nonsymmetric also in the sense that one variable is updated twice while the other is only updated once at each iteration. These nonsymmetry features also explain why more theoretical issues remain challenging for generalized PDHG schemes; for example, the worst-case convergence rate of PDHG measured by the iteration complexity in a nonergodic sense is still missing. In this paper, we further consider how to generalize the PDHG and propose an algorithmic framework of generalized PDHG schemes for saddle point problems. This algorithmic framework allows the output of the PDHG subroutine to be further updated by correction steps with constant step sizes. We investigate the restriction onto these step sizes and conduct the convergence analysis for the algorithmic framework. The algorithmic framework turns out to include some existing PDHG schemes as special cases, and it immediately yields a class of new generalized PDHG schemes by choosing different step sizes for the correction steps. In particular, a completely symmetric PDHG scheme with the golden-ratio step sizes is included. Theoretically, an advantage of the algorithmic framework is that the worst-case convergence rate measured by the iteration complexity in both the ergodic and nonergodic senses can be established.     

A new matrix splitting preconditioner for generalized saddle point problems

For generalized saddle point problems, we establish a new matrix splitting preconditioner and give the implementing process in detail. The new preconditioner is much easier to be implemented than the modified dimensional split (MDS) preconditioner. The convergence properties of the new splitting iteration method are analyzed. The eigenvalue distribution of the new preconditioned matrix is discussed and an upper bound for the degree of its minimal polynomial is derived. Finally, some numerical examples are carried out to verify the effectiveness and robustness of our preconditioner on generalized saddle point problems discretizing the incompressible Navier–Stokes equations.