Preconditioned Modified Hermitian And skew-Hermitian Splitting Research Articles

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.

Preconditioned modified Hermitian and skew-Hermitian splitting iteration methods for fractional nonlinear Schrödinger equations

A variant of preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method is proposed for a class of Toeplitz-like complex linear equations arising from the space fractional coupled nonlinear Schrödinger equations. Theoretical analysis show that the new method is convergent unconditionally. Moreover, the choice of the theoretical optimal parameter is further studied in detail, which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the Toeplitz matrix and the extreme absolute values of the diagonal matrix. In addition, the splitting can be used as a preconditioner for Krylov subspaces method. Numerical experiments show that the proposed method is efficient, and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem.

How to compute the optimal parameters of the parameterized PMHSS preconditioner?

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method and its alternating-directional variant, a parameterized PMHSS (P2MHSS) preconditioner is proposed for a class of block two-by-two systems of linear equations. To optimize the preconditioning effect of the P2MHSS preconditioner, practical formulas for computing the optimal parameters of the P2MHSS preconditioner are derived by employing the scaled norm minimization method. Finally, we test the feasibility and the effectiveness of the P2MHSS preconditioner.

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.

MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices

Preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) method is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equation. Motivated by the PMHSS method, we develop a new method of solving a class of linear equations with block two-by-two complex coefficient matrix by introducing two coefficients, noted as DPMHSS. By making use of the DPMHH iteration as the inner solver to approximately solve the Newton equations, we establish modified Newton-DPMHSS (MN-DPMHSS) method for solving large systems of nonlinear equations. We analyze the local convergence properties under the Holder continuous conditions, which is weaker than Lipschitz assumptions. Numerical results are given to confirm the effectiveness of our method.

Two variants of the PMHSS iteration method for a class of complex symmetric indefinite linear systems

For a class of complex symmetric indefinite linear systems, we present two variants of the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method. One is a generalization of the PMHSS iteration method and the other is a generalization of the lopsided PMHSS iteration method. We study the convergence properties of these two variants by analyzing the upper bounds of spectral radiuses of the corresponding iteration matrices and compare these upper bounds. Numerical examples are illustrated to show the effectiveness of the proposed iteration methods.

On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric jacobian matrices

Preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) method is an unconditionally convergent iterative method for solving large sparse complex symmetric systems of linear equations. By making use of the PMHSS iteration as the inner solver to approximately solve the Newton equations, we establish a modified Newton-PMHSS method for solving large systems of nonlinear equations. Motivated by the idea in Chen et al. (2014), we analyze the local convergence properties under the Holder continuous condition, which is weaker than the assumptions used in modified Newton-HSS method proposed by Wu and Chen (2013). Numerical results are given to confirm the effectiveness of our method.

Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

A GENERALIZED PRECONDITIONED MHSS METHOD FOR A CLASS OF COMPLEX SYMMETRIC LINEAR SYSTEMS

Based on the MHSS (Modified Hermitian and skew-Hermitian splitting) and preconditioned MHSS methods, we will present a generalized preconditioned MHSS method for solving a class of complex symmetric linear systems. The new method (GPMHSS) is essentially a two-parameter iteration method where the iterative sequence is unconditionally convergent to the unique solution of the linear system. A parameter region of the convergence for our method is provided. An efficient preconditioner is presented for the actual implementation of the new method. Some numerical results are given to show its effectiveness.

Lopsided PMHSS iteration method for a class of complex symmetric linear systems

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, we introduce a lopsided PMHSS (LPMHSS) iteration method for solving a broad class of complex symmetric linear systems. The convergence properties of the LPMHSS method are analyzed, which show that, under a loose restriction on parameter α, the iterative sequence produced by LPMHSS method is convergent to the unique solution of the linear system for any initial guess. Furthermore, we derive an upper bound for the spectral radius of the LPMHSS iteration matrix, and the quasi-optimal parameter α ⋆ which minimizes the above upper bound is also obtained. Both theoretical and numerical results indicate that the LPMHSS method outperforms the PMHSS method when the real part of the coefficient matrix is dominant.

Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems

We construct a preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration scheme for solving and preconditioning a class of block two-by-two linear systems arising from the Galerkin finite element discretizations of a class of distributed control problems. The convergence theory of this class of PMHSS iteration methods is established and the spectral properties of the PMHSS-preconditioned matrix are analysed. Numerical experiments show that the PMHSS preconditioners can be quite competitive when used to precondition Krylov subspace iteration methods such as GMRES.