Shift-splitting Preconditioner Research Articles

A class of new extended generalized shift-splitting preconditioners for $$3\times 3$$ block saddle point problems

A class of new extended generalized shift-splitting preconditioners for $$3\times 3$$ block saddle point problems

A ACCELERATED MODIFIED SHIFT-SPLITTING METHOD FOR NONSYMMETRIC SADDLE POINT PROBLEMS

Recently, Huang and Su [A modified shift-splitting method for nonsymmetric saddle, <i>Journal of Computational and Applied Mathematics</i>, 317 (2017) 535-546] introduced a modified shift-splitting (denoted by MSSP) preconditioner. In this paper, based on modified shift-splitting (denoted by MSSP) iteration technique, we establish a accelerated (named after AMSSP) iterative method for nonsymmetric saddle point problems. Furthermore, we theoretically verify the AMSSP iteration method unconditionally converges to the unique solution of the saddle point problems, compute the spectral radius of the AMSSP iteration matrix. Finally, numerical examples show the spectrum of the new preconditioned matrix for the different parameters.

A two-parameter shift-splitting preconditioner for saddle point problems

For saddle point problems, a two-parameter shift-splitting (TPSS) preconditioner is proposed. Estimated lower and upper bounds of the real eigenvalues and the real and imaginary parts of the non-real eigenvalues of the TPSS preconditioned matrix are obtained, respectively. As special cases, bounds obtained are tighter than the ones for the GSS, PGSS, IGSS and NESS preconditioners, respectively. Meanwhile, the corresponding TPSS iteration method is constructed and the convergence analysis is studied. Numerical experiments for two frequently used problems are given to illustrate the great robustness of the TPSS preconditioner.

Lopsided shift-splitting preconditioner for saddle point problems with three-by-three structure

Lopsided shift-splitting preconditioner for saddle point problems with three-by-three structure

A Computational Model for Nonlinear Biomechanics Problems of FGA Biological Soft Tissues

The principal objective of this work was to develop a semi-implicit hybrid boundary element method (HBEM) to describe the nonlinear fractional biomechanical interactions in functionally graded anisotropic (FGA) soft tissues. The local radial basis function collocation method (LRBFCM) and general boundary element method (GBEM) were used to solve the nonlinear fractional dual-phase-lag bioheat governing equation. The boundary element method (BEM) was then used to solve the poroelastic governing equation. To solve equations arising from boundary element discretization, an efficient partitioned semi-implicit coupling algorithm was implemented with the generalized modified shift-splitting (GMSS) preconditioners. The computational findings are presented graphically to display the influences of the graded parameter, fractional parameter, and anisotropic property on the bio-thermal stress. Different bioheat transfer models are presented to show the significant differences between the nonlinear bio-thermal stress distributions in functionally graded anisotropic biological tissues. Numerical findings verified the validity, accuracy, and efficiency of the proposed method.

An efficient relaxed shift-splitting preconditioner for a class of complex symmetric indefinite linear systems

&lt;abstract&gt;&lt;p&gt;In this work, by introducing a scalar matrix $ \alpha I $, we transform the complex symmetric indefinite linear systems $ (W+i T)x = b $ into a block two-by-two complex equations equivalently, and propose an efficient relaxed shift-splitting (ERSS) preconditioner. By adopting the relaxation technique, the ERSS preconditioner is not only a computational advantage but also closer to the original two-by-two of complex coefficient matrix. The eigenvalue distributions of the preconditioned matrix are analysed. An efficient and practical formula for computing the parameter value $ \alpha $ is also derived by computing the Frobenius norm of symmetric indefinite matrix $ T $. Numerical examples on a few model problems are illustrated to verify the performances of the ERSS preconditioner.&lt;/p&gt;&lt;/abstract&gt;

A shift-splitting preconditioner for asymmetric saddle point problems

In this paper, we execute the shift-splitting preconditioner for asymmetric saddle point problems with its (1,2) block's transposition unequal to its (2,1) block under the removed minus of its (2,1) block. The proposed preconditioner is stemmed from the shift splitting (SS) iteration method for solving asymmetric saddle point problems, which is convergent under suitable conditions. The relaxed version of the shift-splitting preconditioner is obtained as well. The spectral distributions of the related preconditioned matrices are given. Numerical experiments from the Stokes problem are offered to show the convergence performance of these two preconditioners.

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

A two-sweep shift-splitting iterative method for complex symmetric linear systems

Recently, Chen and Ma [21] constructed the generalized shift-splitting (GSS) preconditioner, and gave the corresponding theoretical analysis and numerical experiments. In this paper, based on the generalized shift-splitting (GSS) preconditioner, we generalize their algorithms and further study the two-sweep shift-splitting (TSSS) preconditioner for complex symmetric linear systems. Moreover, by similar theoretical analysis, we obtain that the two-sweep shift-splitting iterative method is unconditionally convergent. In finally, one example is provided to confirm the effectiveness.

A class of efficient parameterized shift-splitting preconditioners for block two-by-two linear systems

Based on the effective constructing an equivalent block two-by-two linear systems, we establish a class of efficient parameterized shift-splitting (EPS) preconditioners for solving block two-by-two linear systems. Theoretical analysis shows that the EPS iteration method is unconditionally convergent, and then we study the spectral properties of the corresponding preconditioned matrix. By removing one of the shift term of the EPS preconditioners, two classes of local EPS preconditioners are presented and the eigenvalues distribution of the preconditioned matrices are also studied. Numerical experiments are provided to demonstrate the feasibility and effectiveness of the proposed preconditioners.

New modified shift-splitting preconditioners for non-symmetric saddle point problems

Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shift-splitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks.

Generalized fast shift-splitting preconditioner for nonsymmetric saddle-point problems

For nonsingular and singular nonsymmetric saddle-point problems, Dou, Yin, and Liao proposed a fast shift-splitting preconditioner by combining shift-splitting technique and Hermitian and skew-Hermitian (HSS) splitting technique. In this paper, a generalized fast shift-splitting (GFSS) preconditioner is derived. The convergence and semi-convergence properties of the new shift-splitting iteration method are analyzed for solving the nonsingular and singular nonsymmetric saddle-point problems, respectively. Furthermore, the spectral properties of the GFSS preconditioned matrix are given. Finally, some numerical examples are carried out to demonstrate the robustness and effectiveness of our new preconditioner on nonsingular and singular nonsymmetric saddle-point problems.

Shift-splitting preconditioners for a class of block three-by-three saddle point problems

In this paper, shift-splitting preconditioners are studied for a special class of block three-by-three saddle point problems, which arise from many practical problems and are different from the traditional saddle point problems. It is proved that the block three-by-three saddle point matrix is positive stable and the corresponding shift-splitting stationary iteration method is unconditionally convergent, which leads to a nice clustering property of the eigenvalues of the shift-splitting preconditioned matrix. Numerical results show that the proposed shift-splitting preconditioners outperform much better than some existing block diagonal preconditioners studied recently.

A class of new extended shift-splitting preconditioners for saddle point problems

In this paper, a class of new extended shift-splitting (NESS) preconditioners containing t > 0 are proposed for saddle point problems, which contain the existing shift-splitting-based preconditioners as special cases. The convergence properties of the NESS iteration and the spectral distribution of the NESS preconditioned matrix are investigated. We give an estimated bound for the real eigenvalues of the NESS preconditioned matrix whose left end is positive and show that the non-real eigenvalues of the NESS preconditioned matrix are located in an intersection of two rings, particularly, these non-real eigenvalues are located in an intersection of two rings and one circle if t ≥ 1 2 . Moreover, we show that under appropriate conditions our proposed eigenvalue bounds are tighter than the corresponding bounds given in Ren et al. (2017). Using Fourier analysis, we derive an optimized parameter t ∗ independent of the viscosity ν for the continuous version of the NESS preconditioned GMRES method for the 2D Stokes equation. Moreover, we find that the NESS preconditioned GMRES method with a constant multiple of the optimized parameter t ∗ is effective and robust to solve 2D Stokes problems in practice. Numerical experiments are used to verify our theoretical results.

A parameterized shift-splitting preconditioner for saddle point problems.

Recently, Chen and Ma [A generalized shift-splitting preconditioner for saddle point problems, Applied Mathematics Letters, 43 (2015) 49-55] introduced a generalized shift-splitting preconditioner for saddle point problems with symmetric positive definite (1,1)-block. In this paper, I establish a parameterized shift-splitting preconditioner for solving the large sparse augmented systems of linear equations. Furthermore, the preconditioner is based on the parameterized shift-splitting of the saddle point matrix, resulting in an unconditional convergent fixed-point iteration, which has the intersection with the generalized shift-splitting preconditioner. In final, one example is provided to confirm the effectiveness.

Generalized Shift-Splitting Preconditioner for Saddle Point Problems with Block Three-by-Three Structure

We propose a generalized shift-splitting iteration method for saddle point problems with block three-by-three structure. As a new iteration method, the method converges to the unique solution of the saddle point problem unconditionally. When exploited as a preconditioner, the spectral distribu-tion of the preconditioned matrix is investigated. Numerical experiments show that the new variant is efficient in speeding up GMRES for solving the block three-by-three saddle point problem.

A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models

Based on a general splitting of the (1,1) leading block matrix, we first construct a general class of shift-splitting (GCSS) preconditioners for non-Hermitian saddle point problems. Convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the GCSS preconditioned saddle point matrices are analyzed. Then the GCSS preconditioner is specifically applied to the non-Hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time-harmonic eddy current models. With suitable choices of the splittings, the new GCSS preconditioners are easier to implement and have faster convergence rates than the existing shift-splitting preconditioner and its modified variant. Two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners.

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

ABSTRACTTo 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.

Computational Bases for $$\mathcal {S}_{r}\Lambda ^{1}(\mathbb {R}^{2})$$ S r Λ 1 ( R 2 ) and Their Application in Mixed Finite Element Method

In this article, computational bases for finite element spaces $$\mathcal {S}_{2}\Lambda ^{0}(\mathcal {T}_{h})$$ and $$\mathcal {S}_{r}\Lambda ^{1}(\mathcal {T}_{h})$$ in each step of the h-adaptive method are derived. We also implement a mixed method for the Hodge Laplacian equation. In discretization of the mixed method, the pair which consists of the serendipity elements and the rectangular Brezzi–Douglas–Marini (BDM) elements are used. The corresponding saddle point matrix, in each step of the h-adaptive method, is $$\begin{aligned} \mathcal {A}=\begin{pmatrix} A&{}B^{T}\\ -B&{} C \end{pmatrix}, \end{aligned}$$ where $$ C\ne 0 $$ and B is rank deficient. The modified generalized shift-splitting (MGSS) preconditioner for solving this saddle point matrix is considered. The major advantage of our approach is that the MGSS preconditioner can easily be implemented. Numerical results show the effectiveness of the proposed iteration method and the good behavior of corresponding splitting preconditioner.

A generalized shift-splitting preconditioner for complex symmetric linear systems

Abstract In this paper, the generalized shift-splitting (GSS) preconditioner is implemented for solving a class of generalized saddle point problems which stem from the solution of complex symmetric linear systems. The GSS preconditioner is induced by the generalized shift-splitting iterative method. Theoretical analysis shows that the generalized shift-splitting iterative method is unconditionally convergent. Some numerical experiments are provided to show the effectiveness of the proposed preconditioner.