MN-PGSOR Method for Solving Nonlinear Systems with Block Two-by-Two Complex Symmetric Jacobian Matrices

For solving the large sparse linear systems with 2 × 2 block structure, the generalized successive overrelaxation (GSOR) iteration method is an efficient iteration method. Based on the GSOR method, the PGSOR method introduces a preconditioned matrix with a new parameter for the coefficient matrix which can enhance the efficiency. To solve the nonlinear systems in which the Jacobian matrices are complex and symmetric with the block two-by-two form, we try to use the PGSOR method as an inner iteration, with the help of the modified Newton method as an efficient outer iteration method. This new method is called the modified Newton-PGSOR (MN-PGSOR) method. Local convergence properties of the MN-PGSOR are analyzed under the Hölder condition. Finally, we give the comparison of our new method with some previous methods in the numerical results. The MN-PGSOR method is superior in both iteration steps and computing time.

New Preconditioned Iteration Method Solving the Special Linear System from the PDE-Constrained Optimal Control Problem

In many fields of science and engineering, partial differential equation (PDE) constrained optimal control problems are widely used. We mainly solve the optimization problem constrained by the time-periodic eddy current equation in this paper. We propose the three-block splitting (TBS) iterative method and proved that it is unconditionally convergent. At the same time, the corresponding TBS preconditioner is derived from the TBS iteration method, and we studied the spectral properties of the preconditioned matrix. Finally, numerical examples in two-dimensions is applied to demonstrate the advantages of the TBS iterative method and TBS preconditioner with the Krylov subspace method.

Superior properties of the PRESB preconditioner for operators on two-by-two block form with square blocks

Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear.

On Euler preconditioned SHSS iterative method for a class of complex symmetric linear systems

In this paper, we propose an Euler preconditioned single-step HSS (EP-SHSS) iterative method for solving a broad class of complex symmetric linear systems. The proposed method can be applied not only to the non-singular complex symmetric linear systems but also to the singular ones. The convergence (semi-convergence) properties of the proposed method are carefully discussed under suitable restrictions. Furthermore, we consider the acceleration of the EP-SHSS method by preconditioned Krylov subspace method and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments verify the effectiveness of the EP-SHSS method either as a solver or as a preconditioner for solving both non-singular and singular complex symmetric linear systems.

A relaxed block splitting preconditioner for complex symmetric indefinite linear systems

In this paper, we propose a relaxed block splitting preconditioner for a class of complex symmetric indefinite linear systems to accelerate the convergence rate of the Krylov subspace iteration method and the relaxed preconditioner is much closer to the original block two-by-two coefficient matrix. We study the spectral properties and the eigenvector distributions of the corresponding preconditioned matrix. In addition, the degree of the minimal polynomial of the preconditioned matrix is also derived. Finally, some numerical experiments are presented to illustrate the effectiveness of the relaxed splitting preconditioner.

Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems

We discuss an efficient preconditioner and iterative numerical method to solve large complex linear algebraic systems of the form (W + iT)u = c, where W and T are symmetric matrices, and at least one of them is nonsingular. When the real part W is dominantly stronger or weaker than the imaginary part T, we propose a block multiplicative (BM) preconditioner or its variant (VBM), respectively. The BM and VBM preconditioned iteration methods are shown to be parameter-free, in terms of eigenvalue distributions of the preconditioned matrix. Furthermore, when the relationship between W and T is obscure, we propose a new preconditioned BM method (PBM) to overcome this difficulty. Both convergent properties of these new iteration methods and spectral properties of the corresponding preconditioned matrices are discussed. The optimal value of iteration parameter for the PBM method is determined. Numerical experiments involving the Helmholtz equation and some other applications show the effectiveness and robustness of the proposed preconditioners and corresponding iterative methods.

A Modified Relaxed Positive-Semidefinite and Skew-Hermitian Splitting Preconditioner for Generalized Saddle Point Problems

Based on the relaxed factorization techniques studied recently and the idea of the simple-like preconditioner, a modified relaxed positive-semidefinite and skew-Hermitian splitting (MRPSS) preconditioner is proposed for generalized saddle point problems. Some properties, including the eigenvalue distribution, the eigenvector distribution and the minimal polynomial of the preconditioned matrix are studied. Numerical examples arising from the mixed finite element discretization of the Oseen equation are illustrated to show the efficiency of the new preconditioner.

On Preconditioned MHSS Real-Valued Iteration Methods for a Class of Complex Symmetric Indefinite Linear Systems

A generalized preconditioned modified Hermitian and skew-Hermitian splitting (GPMHSS) real-valued iteration method is proposed for a class of complex symmetric indefinite linear systems. Convergence theory is established and the spectral properties of an associated preconditioned matrix are analyzed. We also give several variants of the GPMHSS preconditioner and consider the spectral properties of the preconditioned matrices. Numerical examples illustrate the effectiveness of our proposed method.

A relaxed splitting preconditioner for saddle point problems

In this paper, we introduce a relaxed splitting preconditioner for saddle point problems. Spectral properties of the preconditioned matrix are analyzed and compared with the closely related preconditioner in recent paper [New preconditioners for saddle point problems, Appl. Math. Comput. 172 (2006), 762-771] by Pan et al. Numerical experiments are given to illustrate the efficiency of the proposed precoditioner.