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In this paper, we consider the numerical solution of a large complex linear system with a saddle-point form obtained by the discretization of the time-harmonic eddy-current optimal control problem. A new Schur complement is proposed for this algebraic system, extending it to both the block-triangular preconditioner and the structured preconditioner. A theoretical analysis proves that the eigenvalues of block-triangular and structured preconditioned matrices are located in the interval [1/2, 1]. Numerical simulations show that two new preconditioners coupled with a Krylov subspace acceleration have good feasibility and effectiveness and are superior to some existing efficient algorithms. [ABSTRACT FROM AUTHOR]

For the discretized linear systems of the spatial fractional diffusion equations, we construct a class of a modified DTS iteration method and give its asymptotic convergence conditions. Then, we design a fast modified DTS preconditioner by replacing Toeplitz matrix T with the τ matrix to accelerate the convergence rates of GMRES method. Theoretically, we show that the spectrum of fast modified DTS preconditioned matrix is clustered around one. Numerical experiments verify the validity of the constructed fast modified DTS preconditioner for GMRES method. [ABSTRACT FROM AUTHOR]

The spatial fractional diffusion equations are discretized by the finite difference method and linear systems are obtained. Transforming the coefficient matrices of these linear systems to obtain the structure of diagonal-plus-Toeplitz. In order to solve this class of linear systems, we construct a two-parameter DTS iteration method on the basis of the DTS iteration method and give its asymptotic convergence conditions. On this basis, we introduce a two-parameter sine-transform-based splitting preconditioner to improve the convergence rate of the generalized minimal residual (GMRES) iteration method, and theoretically prove that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Finally, numerical experiments demonstrate the efficiency of the new method. [ABSTRACT FROM AUTHOR]

To solve nonsymmetric saddle point problems, a new generalized variant of the DPSS (NGVDPSS) preconditioner is proposed. The invertibility of the NGVDPSS preconditioner, the convergence of the NGVDPSS iteration method and some properties of the corresponding preconditioned matrix are presented. The degree of minimal polynomials of the preconditioned matrix is obtained. Simultaneously, the algorithm implementation of the NGVDPSS preconditioner is discussed. Numerical experiments are performed to illustrate the availability of the new preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, by constructing equivalently the absolute value equation (AVE) as the new block equation of two-by-two, we present a new SOR-like (NSOR) method for solving the AVE. The convergence and optimal parameters of NSOR method for solving the AVE are studied in detail. Numerical experiments are presented to show that the NSOR method is implementable and efficient. [ABSTRACT FROM AUTHOR]

The spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme with the shifted Grünwald formula and the given discretized linear systems have a diagonal-plus-Toeplitz structure. In this paper, based on trigonometric transformation splitting (TTS), we study efficient iterative method called GTTS has two parameters. As a focus, we give the simple and effective optimal forms of these two parameters respectively. We carry out some numerical experiments to illustrate the effectiveness and accuracy of this new algorithm. [ABSTRACT FROM AUTHOR]

Abstract For symmetric indefinite linear systems, we introduce a new triangular preconditioner based on symmetric and triangular (ST) decomposition. A new (1, 1) block is obtained by augmented Lagrangian technique. The new ST preconditioner is introduced by the combination of the new (1, 1) block and symmetric and triangular (ST) decomposition. Then a preconditioned system can be obtained by preconditioning technique, which is superior to the original system in terms of condition number. We study the spectral properties of preconditioned system, such as eigenvalues, the estimation of condition number and then give the quasi-optimal parameter. Numerical examples are given to indicate that the new preconditioner has obvious efficiency advantages. Finally, we conclude that the new ST preconditioner is a better option to deal with large and sparse problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose two different modified Uzawa methods for solving the saddle point problems, and prove the convergence of the proposed methods. We solve a lower-upper triangular system per iteration in the proposed methods, instead of only solving a lower one. Numerical experiments show that our proposed methods with four parameters perform much faster than previous Uzawa methods with three parameters. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter's value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime. [ABSTRACT FROM AUTHOR]

Based on previous research results, we propose a new preprocessing HSS iteration method (PHSS) for the generalized Lyapunov equation. At the same time, the corresponding inexact PHSS algorithm (IPHSS) is given from the angle of application. All the new methods presented in this paper have given the corresponding convergence proof. The numerical experiments are carried out to compare the new method with the existing methods, and the improvement effect is obvious. The feasibility and effectiveness of the proposed method are proved from two aspects of theory and calculation. [ABSTRACT FROM AUTHOR]