Guo-Feng Zhang

A block alternating splitting iteration method for a class of block two-by-two complex linear systems☆

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering. In this paper, we focus on solving a class of block two-by-two complex linear systems arising from the distributed optimal control with time-periodic parabolic equations. A new block alternating splitting (BAS) iteration method is presented for solving the class of complex linear systems. The convergence theory and the spectral properties of the BAS iteration method are discussed. Numerical experiments are presented to illustrate the efficiency of the BAS iteration as a solver as well as a preconditioner for Krylov subspace methods.

On generalized parameterized inexact Uzawa method for a block two-by-two linear system

Recently, Chen and Jiang F.?Chen, Y.-L.?Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765-771] presented a parameterized inexact Uzawa (PIU) algorithm for solving symmetric saddle point problems, where the (1, 2)- and the (2, 1)-blocks are the transpose of each other. In this paper, we extend the PIU method to the block two-by-two linear system by allowing the (1, 2)-block to be not equal to the transpose of the (2, 1)-block and the (2, 2)-block may not be zero. We prove that the iteration method is convergent under certain conditions. With different choices of the parameter matrices, we obtain several new algorithms for solving the block two-by-two linear system. Numerical experiments confirm our theoretical results and show that our method is feasible and effective.

SIMPLE-like preconditioners for saddle point problems from the steady Navier-Stokes equations

The semi implicit method for pressure linked equations (SIMPLE) preconditioner is further generalized to a class of SIMPLE-like (SL) preconditioners for solving saddle point problems from the steady Navier-Stokes equations. The SL preconditioners can be also viewed as a generalization of the relaxed deteriorated PSS (RDPSS) preconditioner proposed by Cao et al. (2015). Convergence analysis of the corresponding SL iteration is presented and the optimal iteration parameter is obtained by minimizing the spectral radius of the SL iteration matrix. Moreover, Krylov subspace acceleration of the SL preconditioning is studied. The SL preconditioned saddle point matrix is analyzed. Results about eigenvalue and eigenvector distributions and the minimal polynomial are derived. Numerical experiments from leaky two dimensional lid-driven cavity problems are given, which show the advantages of the SL preconditioned GMRES over the DPSS and RDPSS preconditioned ones for solving saddle point problems.

A note on preconditioners for complex linear systems arising from PDE-constrained optimization problems

Abstract In this note, the block-diagonal preconditioner proposed and the block triangular proposed in Krendl etxa0al. (2013) and Pearson and Wathen (2012), respectively, are further studied and optimized. Two improved preconditioners are proposed for solving a class of complex linear systems arising from optimal control with time-periodic parabolic equation. Theoretical analyses show that the eigenvalues of the improved block-diagonal and block triangular preconditioned matrices are located in [ − 1 , − 2 / 2 ) ∪ ( 2 / 2 , 1 ] and ( 1 / 2 , 1 ] , respectively. Numerical experiments illustrate the feasibility and effectiveness of these preconditioners for Krylov subspace methods.

On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations

Abstract For Toeplitz system of weakly nonlinear equations, by using the separability and strong dominance between the linear and the nonlinear terms and using the circulant and skew-circulant splitting (CSCS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-CSCS and nonlinear CSCS-like iteration methods, respectively. The advantage of these methods is that they do not require accurate computation and storage of Jacobian matrix, and only need to solve linear sub-systems of constant coefficient matrices. Therefore, computational workloads and computer storage may be saved in actual implementations. Theoretical analysis shows that these new iteration methods are local convergent under suitable conditions. Numerical results show that both Picard-CSCS and nonlinear CSCS-like iteration methods are feasible and effective for some cases.

General constraint preconditioning iteration method for singular saddle-point problems

For the singular saddle-point problems with nonsymmetric positive definite ( 1 , 1 ) block, we present a general constraint preconditioning (GCP) iteration method based on a singular constraint preconditioner. Using the properties of the Moore-Penrose inverse, the convergence properties of the GCP iteration method are studied. In particular, for each of the two different choices of the ( 1 , 1 ) block of the singular constraint preconditioner, a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the GCP iteration method. Moreover, the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual (GMRES) and quasi-minimal residual (QMR) methods are also tested.

Convergence behavior of generalized parameterized Uzawa method for singular saddle-point problems

In this paper, we will seek the least squares solution for singular saddle-point problems. The parameterized Uzawa (PU) method is further studied and a generalized PU (GPU) proper splitting is proposed. The convergence behavior of the corresponding GPU iteration is studied. It is proved that the GPU iteration method can converge to the best least squares solutions of the singular saddle-point problems. In addition, we prove that the GPU preconditioned GMRES for singular saddle-point problems will also determine the least squares solution at breakdown. The eigenvalue distributions of the GPU preconditioned matrix are derived. Numerical experiments are presented, which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the GPU iteration has better convergence behavior than the PU iteration, both as a solver and a preconditioner of GMRES.

Optimizing and improving of the C-to-R method for solving complex symmetric linear systems

Abstract For complex symmetric linear systems, Axelsson et al. (2014) proposed the C-to-R method. In this paper, by further studying the C-to-R method with W and T being symmetric positive semidefinite, the optimal iteration parameter for the C-to-R method α o p t = 2 2 2 is obtained and the C-to-R method is optimized. Furthermore, based on the optimized C-to-R method, we further propose an optimized preconditioner. Eigenvalue properties of the optimized preconditioned matrix are analyzed, which show that all the eigenvalues of the preconditioned matrix are located in tighter interval. Numerical results are presented, not only confirm the validity of the theoretical analysis, but also demonstrate the feasibility and effectiveness of the proposed optimized C-to-R method.

A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations

For Toeplitz systems of weakly nonlinear equations, combining the separability and strong dominance between the linear and the nonlinear terms with the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-cSSS and nonlinear cSSS-like iteration methods, which are based on a special case of the HSS, where the symmetric part H = 1 2 ( A + A T ) is a centrosymmetric matrix and the skew-symmetric part H = 1 2 ( A - A T ) is a skew-centrosymmetric matrix. The advantages of these methods are that they can transfer the linear sub-systems involved in inner iteration to two linear systems of half an order, besides, fast methods are available for computing the two half-steps involved in the inner iteration. Numerical results are provided, to further show that both Picard-cSSS and nonlinear cSSS-like iteration methods are feasible and effective.

Parameterized approximate block LU preconditioners for generalized saddle point problems

Abstract In this paper, we are concerned with the iteration solution of generalized saddle point problems. Based on the exact block LU factorization of the coefficient matrix, we construct a class of parameterized approximate block LU factorization preconditioners, which rely on suitable approximations of the Schur complement of the (1,1) block of the coefficient matrix. Convergence of the corresponding iteration methods is analyzed and the optimal iteration parameters minimizing the spectral radii are deduced. Algebraic characteristics of the related preconditioned matrices are discussed, including eigenvalue and eigenvector distributions and upper bounds for degree of the minimal polynomial. The established results extend those of the approximate factorization and variants of the Hermitian and skew-Hermitian splitting and positive and skew-Hermitian splitting preconditioners for saddle point problems. Numerical experiments are demonstrated to illustrate the efficiency of the new preconditioners.