Large Sparse Saddle Point Problems Research Articles

A new generalized shift-splitting method for nonsymmetric saddle point problems

Recently, Huang and Huang [ Journal of Computational and Applied Mathematics, 328 (2018) 381–399] proposed a modified generalized shift-splitting preconditioned (denoted by MGSSP) method for solving large sparse saddle point problems, and gave the corresponding theoretical analysis and numerical experiments. In this paper, based on the modified generalized shift-splitting preconditioned (MGSSP) method, we generalize the MGSSP algorithms and further present the new generalized shift-splitting preconditioned (NGSSP) method for nonsymmetric saddle point problems. Moreover, by similar theoretical analysis, we analyze the convergence conditions of the corresponding matrix splitting iteration methods of the NGSSP preconditioned saddle point matrices. In final, one example is provided to confirm the effectiveness. MSC: 65F10, 65F15, 65F50

On quasi shift-splitting iteration method for a class of saddle point problems

Motivated by a technique of scale-splitting for matrix, we put forward a quasi shift-splitting (named after QSS) iteration method to address a class of large scale sparse saddle point problems, this novel method also naturally leads to the corresponding QSS-preconditioner. In addition, an extrapolated variant of QSS (named after EQSS) iteration method is constructed. Further, some useful convergence properties of the QSS and EQSS iteration methods and spectral properties of QSS-preconditioner are studied. Finally, a saddle point equations stem from model Navier–Stokes problem is supplied to illustrate the effectiveness of our new two methods and QSS-preconditioner.

Modified PHSS iterative methods for solving nonsingular and singular saddle point problems

For large sparse saddle point problems, we establish a new version of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method, called the modified PHSS (MPHSS) method in this paper. Then, we theoretically study its convergence and semi-convergence properties and determine its optimal iteration parameter and corresponding optimal convergence factor. Furthermore, the spectral properties of the MPHSS preconditioned matrix are discussed in detail. Numerical experiments show that the MPHSS iteration method is effective and robust when it is used either as a solver or as a matrix splitting preconditioner for the generalized minimal residual (GMRES) method.

Optimization of a parameterized inexact Uzawa method for saddle point problems

ABSTRACTFor large sparse saddle point problems, Cao et al. studied a modified generalized parameterized inexact Uzawa (MGPIU) method (see [Y. Cao, M.Q. Jiang, L.Q. Yao, New choices of preconditioning matrices for generalized inexact parameterized iterative methods, J. Comput. Appl. Math. 235 (1) (2010) 263–269]). For iterative methods of this type, the choice of the relaxation parameter is crucial for the methods to achieve their best performance. In this paper, for an example of 2D Stokes equations, we derive the optimal relaxation parameter for the continuous version of the MGPIU method, by minimizing the corresponding convergence factor that is obtained using Fourier analysis. In addition, we find that the MGPIU method is mesh parameter independent, however, it depends asymptotically linearly on the viscosity ν, which suggests that the numerical methods for Stokes equations should be investigated with the presence of the viscosity ν, though it can be scaled out from the equations in advance. We use numerical experiments to validate our theoretical findings.

A modified shift-splitting method for nonsymmetric saddle point problems

To solve large sparse saddle point problems, based on modified shift-splitting (denoted by MSSP) iteration technique, a MSSP preconditioner is proposed. We theoretically verify the MSSP iteration method unconditionally converges to the unique solution of the saddle point problems, compute the spectral radius of the MSSP iteration matrix and estimate the sharp bounds of the eigenvalues of the corresponding iteration matrix. Numerical experiments show that the MSSP iteration method is effective and accurate.

On semi-convergence of ULT iterative method for the singular saddle point problems

Zheng and Ma recently proposed an efficient upper and lower triangular (ULT) splitting iterative method for solving the large sparse nonsingular saddle point problems; see Zheng and Ma (2016). In this paper, we further prove the semi-convergence of this method when it is applied to solve the large sparse singular saddle point problems under suitable conditions. The characteristic of eigenvalues of the iteration matrix of the ULT method is analyzed. Also, the pseudo-optimal iteration parameters and the corresponding pseudo-optimal semi-convergence factor for some special cases of the ULT method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the ULT iterative method for solving singular saddle point problems.

Preconditioned AHSS-PU alternating splitting iterative methods for saddle point problems

In order to solve large sparse saddle point problems (SPP) quickly and efficiently, Wang and Zhang recently studied the preconditioned accelerated Hermitian and skew-Hermitian splitting (PAHSS) methods. Through accelerating the PAHSS iteration algorithms by using parameterized Uzawa (PU) method, a preconditioned AHSS-PU alternating splitting iterative method (PAHSS-PU method) for solving saddle point problems is proposed in this paper. The convergence results of this new method are given under some suitable conditions. Moreover, we can obtain that if the parameters are suitable selected, then the PAHSS-PU algorithm will outperform the PAHSS algorithm and some Uzawa-type methods in the same precision condition. Numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the PAHSS-PU method.

On generalized local Hermitian and skew-Hermitian splitting iterative method for block two-by-two linear systems

For large sparse saddle point problems whose (1,2) and (2,1)-blocks are the transpose of each other, Zhu studied a generalized local Hermitian and skew-Hermitian splitting (GLHSS) iterative method (see Zhu (2012)). In this paper, the GLHSS method is extended to the block 2×2 linear system, which allows that the (1,2)-block is not equal to the transpose of the (2,1)-block or the (2,2)-block is non-zero. With different choices of the parameter matrices, the existing methods are included and the new algorithms for solving the block 2×2 linear system are obtained. The conditions for guaranteeing the convergence of the new iterative method are studied and the correctness of the existing corollaries are questioned. Numerical experiments are provided to show that the proposed method is feasible and effective, and it is not necessary to introduce some parameter matrices in some cases.

On HSS-based sequential two-stage method for non-Hermitian saddle point problems

For large sparse saddle point problems with symmetric positive definite (1,1)-block, Li et al. studied an efficient iterative method (see Li et al. (2011)) [25]. By making use of the same preconditioning technique and a new matrix splitting based on the Hermitian and skew-Hermitian splitting (HSS) of the (1,1)-block of the preconditioned non-Hermitian saddle point systems, an efficient sequential two-stage method is proposed for solving the non-Hermitian saddle point problems. Theoretical analysis shows the proposed iterative method is convergent, and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter α approaches infinity. Numerical experiments arising from Naiver–Stokes problem are provided to show that the new iterative method is feasible, effective and robust.

Convergence analysis of the corrected Uzawa algorithm for symmetric saddle point problems

For the large sparse saddle point problems, Pan and Li recently proposed in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31(3): 231–242] a corrected Uzawa algorithm based on a nonlinear Uzawa algorithm with two nonlinear approximate inverses, and gave the detailed convergence analysis. In this paper, we focus on the convergence analysis of this corrected Uzawa algorithm, some inaccuracies in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31(3): 231–242] are pointed out, and a corrected convergence theorem is presented. A special case of this modified Uzawa algorithm is also discussed.

Modified Block Symmetric SOR Preconditioners for Large Sparse Saddle-Point Problems

In this paper, based on the preconditioners presented by Wang [On hybrid preconditioning methods for large sparse saddle-point problems, Linear Algebra Appl., 434 (2011) 2353-2366], we introduce and study the block symmetric SOR (BSSOR) and the modified block symmetric SOR (MBSSOR) preconditioners for the block traingular product approximation to a 2-by-2 block matrix and these preconditioners are the generalization of BSGS and MBSGS preconditioners presented by Wang. Moreover, we analyse the properties of the corresponding preconditioned matrices.

A generalization of the local Hermitian and skew-Hermitian splitting iteration methods for the non-Hermitian saddle point problems

For large sparse saddle point problems, Jiang and Cao studied a class of local Hermitian and skew-Hermitian splitting (LHSS) iteration methods (see M.-Q. Jiang, Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math. 231 (2009) 973–982). In this paper, we generalized these methods and propose a class of generalized local Hermitian and skew-Hermitian splitting (GLHSS) iteration schemes for solving the non-Hermitian saddle point problems. We derive conditions for guaranteeing the convergence of these iterative methods. With different choices of the parameter matrices, the generalized iterative methods lead to a series of existing and new iterative methods. Numerical experiments for a model Stokes problem are provided, further show that the new iteration methods are feasible and effective.

New choices of preconditioning matrices for generalized inexact parameterized iterative methods

For large sparse saddle point problems, Chen and Jiang recently studied a class of generalized inexact parameterized iterative methods (see [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765–771]). In this paper, the methods are modified and some choices of preconditioning matrices are given. These preconditioning matrices have advantages in solving large sparse linear system. Numerical experiments of a model Stokes problem are presented.

Block diagonally preconditioned PIU methods of saddle point problem

For large sparse saddle point problems, we firstly introduce the block diagonally preconditioned Gauss–Seidl method (PBGS) which reduces to the GSOR method [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38] and PIU method [Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932] when the preconditioners equal to different matrices, respectively. Then we generalize the PBGS method to the PPIU method and discuss the sufficient conditions such that the spectral radius of the PPIU method is much less than one. Furthermore, some rules are considered for choices of the preconditioners including the splitting method of the (1, 1) block matrix in the PIU method and numerical examples are given to show the superiority of the new method to the PIU method.

A generalization of the inexact parameterized Uzawa methods for saddle point problems

For large sparse saddle point problems, Bai and Wang recently studied a class of parameterized inexact Uzawa methods (see Z.-Z. Bai, Z.-Q. Wang, On paramaterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932). In this paper, we generalize these methods and propose a class of generalized inexact parameterized iterative schemes for solving the saddle point problems. We derive conditions for guaranteeing the convergence of these iterative methods. With different choices of the parameter matrices, the generalized iterative methods lead to a series of existing and new iterative methods including the classical Uzawa method, the inexact Uzawa method, the GSOR method and the GIAOR method.

On the iterative algorithm for saddle point problems

In this paper, we point out that the generalized iteration method presented in a recent paper by Ling and Hu [X.-F. Ling, X.-Z. Hu, On the iterative algorithm for large sparse saddle point problems, Appl. Math. Comput. 178 (2006) 372–379] belongs to the inexact Uzawa algorithm, and an improper statement in a theorem in that paper is corrected.

On parameterized inexact Uzawa methods for generalized saddle point problems

For the large sparse saddle point problems, Bai et al. recently studied a class of parameterized inexact Uzawa (PIU) methods [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang. On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38]. For a special case that the (1,1)-block is solved exactly, they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method. In this paper, we develop these methods to the large sparse generalized saddle point problems. For the obtained parameterized inexact Uzawa method, we prove its convergence under suitable restrictions on the iteration parameters. In particular, we determine its quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor for the saddle point problems. Furthermore, This PIU method is generalized to obtain a framework of accelerated variants of the parameterized inexact Uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration.

A note on the iterative algorithm for large sparse saddle point problems

In this paper, we discuss some more generalized iterative schemes for solving the large sparse saddle point problems proposed by Ling and Hu [X.F. Ling, X.Z. Hu, On the iterative algorithm for large sparse saddle point problems, Appl. Math. Comput. 178 (2006) 372–379.], and apply the preconditioned GMRES to accelerate the convergence. Numerical experiments are reported that show the effectiveness of the proposed methods.

On the iterative algorithm for large sparse saddle point problems

In this paper, we discuss some iterated method for solving the saddle point problem. We propose some new schemes and prove its convergence. The method has weaker convergence condition than the classic Uzawa method. The analysis is supported by numerical experiments.

A new iterative method for large sparse saddle point problems

In this short note, the convergence of a new iterative method for the Saddle Point Problem is presented.