Shift-splitting Iteration Method Research Articles

Minimum residual shift-splitting iteration method for non-Hermitian positive definite and positive semidefinite linear systems

By applying the minimum residual technique to the shift-splitting (SS) iteration scheme, we introduce a non-stationary iteration method named minimum residual SS (MRSS) iteration method to solve non-Hermitian positive definite and positive semidefinite systems of linear equations. Theoretical analyses show that the MRSS iteration method is unconditionally convergent for both of the two kinds of systems of linear equations. Numerical examples are employed to verify the feasibility and effectiveness of the MRSS iteration method.

Shift-Splitting Iteration Method and Its Variants for Solving Continuous Sylvester Equations

Shift-Splitting Iteration Method and Its Variants for Solving Continuous Sylvester Equations

A preconditioned two-sweep shift splitting method for non-Hermitian positive definite linear systems

In this paper, to find the numerical solution of the linear systems with the non-Hermitian positive definite matrix, a preconditioned two-sweep shift splitting (PTSS) iteration method is introduced and its convergent conditions are discussed. When the PTSS method is used as a preconditioner, the eigenvalue distribution of the corresponding preconditioned matrix is given. Numerical experiments are reported to the efficiency of the PTSS method and the PTSS preconditioner.

A general fast shift-splitting iteration method for nonsymmetric saddle point problems

A general fast shift-splitting iteration method for nonsymmetric saddle point problems

A modified generalized shift-splitting iteration method for complex symmetric linear systems

In this work, we introduce a modified generalized shift-splitting (MGSS) iteration method for a class of complex symmetric linear systems. Meanwhile, we establish the convergence analysis and give the spectral properties of the corresponding preconditioned matrix. Finally, a numerical example illustrates the effectiveness of the MGSS.

Newton-PGSS and Its Improvement Method for Solving Nonlinear Systems with Saddle Point Jacobian Matrices

The preconditioned generalized shift-splitting (PGSS) iteration method is unconditionally convergent for solving saddle point problems with nonsymmetric coefficient matrices. By making use of the PGSS iteration as the inner solver for the Newton method, we establish a class of Newton-PGSS method for solving large sparse nonlinear system with nonsymmetric Jacobian matrices about saddle point problems. For the new presented method, we give the local convergence analysis and semilocal convergence analysis under Hölder condition, which is weaker than Lipschitz condition. In order to further raise the efficiency of the algorithm, we improve the method to obtain the modified Newton-PGSS and prove its local convergence. Furthermore, we compare our new methods with the Newton-RHSS method, which is a considerable method for solving large sparse nonlinear system with saddle point nonsymmetric Jacobian matrix, and the numerical results show the efficiency of our new method.

A Shift Splitting Iteration Method for Generalized Absolute Value Equations

Abstract In this paper, based on the shift splitting technique, a shift splitting (SS) iteration method is presented to solve the generalized absolute value equations. Convergence conditions of the SS method are discussed in detail when the involved matrices are some special matrices. Finally, numerical experiments show the effectiveness of the proposed method.

Shift-splitting iteration methods for a class of large sparse linear matrix equations

<abstract> By utilizing an inner-outer iteration strategy, a shift-splitting (SS) iteration method to solve a class of large sparse linear matrix equation $ AXB = C $ is proposed in this work. Two convergence theorems for differential forms are studied in depth. Moreover, the quasi-optimal parameters which minimize the upper bound for the spectral radius of SS iteration matrix are given. Two numerical examples illustrate the high-efficiency of SS iteration method, especially when coefficient matrices are ill-conditioned. </abstract>

A New BEM Modeling Algorithm for Size-Dependent Thermopiezoelectric Problems in Smart Nanostructures

The main objective of this paper is to introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures. The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to describe the deformation and mechanical behaviors of smart nanostructures subjected to thermal, and piezoelectric loadings. Because of difficulty of experimental research problems associated with the proposed theory. Therefore, we propose a new boundary element method (BEM) formulation and algorithm for the solution of such problems, which involve temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation. The computational performance of the proposed methodology has been demonstrated by using the generalized modified shift-splitting (GMSS) iteration method to solve the linear systems resulting from the BEM discretization. GMSS advantages are investigated and compared with other iterative methods. The numerical results are depicted graphically to show the size-dependent effects of thermopiezoelectricity, thermoelasticity, piezoelectricity, and elasticity theories of nanostructures. The numerical results also show the effects of the size-dependent and piezoelectric on the displacement components. The validity, efficiency and accuracy of the proposed BEM formulation and algorithm have been demonstrated. The findings of the current study contribute to the further development of technological and industrial applications of smart nanostructures.

A new constraint preconditioner based on the PGSS iteration method for non-Hermitian generalized saddle point problems

For the non-Hermitian generalized saddle point problems, we propose a new constraint preconditioner. The new constraint preconditioner is constructed based on the preconditioned generalized shift-splitting (PGSS) iteration method. We also analyze the invertibility condition of the new preconditioner in detail. Moreover, the convergence properties of the new constraint preconditioning iteration method are derived. Finally, the effectiveness of the proposed preconditioner is illustrated by numerical experiments.

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.

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.

Accelerating the shift-splitting iteration algorithm

The dazzling property of shift-splitting iteration method is unconditional convergence for any parameters and Anderson mixing as a simple and classic method can greatly speed up convergence of fix point iterations. Inheriting the merits of them, we propose an accelerated preconditioning shift-splitting algorithm for generalized saddle point problems. Then, we verify its unconditional convergence. Besides, we discuss the spectrum distribution of iteration matrix and then provide the relationship of optimal parameters involved. Finally, numerical experiments underline its superiority both as a solver and preconditioner.

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.

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.

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.

Semi-convergence analysis of the GSS iteration methods for singular saddle point problems

In this paper, we use the generalized shift-splitting (GSS) iteration method to solve nonsymmetric singular saddle point systems with the symmetric positive (semi-)definite (1,1) sub-block and symmetric positive semi-definite (2,2) sub-block, and prove the semi-convergence of the GSS iteration method. Numerical experiments are presented to investigate the correctness of the theoretical analyses and the feasibility of the GSS iteration method.

On preconditioned generalized shift-splitting iteration methods for saddle point problems

We study a preconditioned generalized shift-splitting iteration method for solving saddle point problems. The unconditional convergence theory of the preconditioned generalized shift-splitting iteration method is established. When the splitting matrix is used as a preconditioner, we analyze eigenvalue distribution of the preconditioned saddle point matrix. It is proved that complex eigenvalues having nonzero imaginary parts of the preconditioned matrix are located in an intersection of two circles and the real parts of all eigenvalues of the preconditioned matrix are located in a positive interval. Numerical experiments are used to verify our theoretical results and illustrate effectiveness of the proposed iteration method and the corresponding splitting preconditioner.

A Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems

AbstractBased on the shift-splitting technique and the idea of Hermitian and skew-Hermitian splitting, a fast shift-splitting iteration method is proposed for solving nonsingular and singular nonsymmetric saddle point problems in this paper. Convergence and semi-convergence of the proposed iteration method for nonsingular and singular cases are carefully studied, respectively. Numerical experiments are implemented to demonstrate the feasibility and effectiveness of the proposed method.

Spectral analysis of the generalized shift-splitting preconditioned saddle point problem

A shift-splitting preconditioner was recently proposed for saddle point problems, which is based on a generalized shift-splitting of the saddle point matrix. We provide a new analysis to prove that the corresponding shift-splitting iteration method is unconditional convergent. To further show the efficiency of the shift-splitting preconditioner, the eigenvalue distribution of the shift-splitting preconditioned saddle point matrix is investigated. We show that all eigenvalues having nonzero imaginary parts are located in an intersection of two circles and all real eigenvalues are located in a positive interval. Numerical examples are given to confirm our theoretical results.