A modified block preconditioner for complex nonsymmetric indefinite linear systems☆
Introduction
We consider the nonsingular system of linear equations of the formwhere is a symmetric indefinite matrix, is a nonsymmetric positive definite matrix, the vectors x, y, f and g are all in and is the imaginary unit. Linear system (1.1) arises in numerous applications in scientific computing and engineering, including interior point methods, computational fluid dynamics, constrained optimization, weighted least squares problems, incompressible flow problems, structural analysis, and so forth; for more details about the practical backgrounds of this class of problems, we refer to [5], [6], [7], [8] and the references therein.
The problem (1.1) can be equivalently rewritten as the following real block two-by-two linear systemorThe linear system of Eq. (1.2) or (1.3) can also be regarded as a special class of generalized saddle-point problems [2], [3], [4].
In order to solve the system (1.2) or (1.3) efficiently and fast, many methods have been proposed in the past few years [9], [10]. Among all the candidates, iterative methods are more attractive than direct methods in terms of storage requirements and computing time. Recently, many splitting iteration methods based on the Hermitian and skew-Hermitian splitting (HSS) [11], [12] have been proposed to solve the complex symmetric linear system (1.1). When the matrices W and T are symmetric positive semi-definite with at least one of them being positive definite, Bai et al. [13], [15] introduced the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method and the preconditioned MHSS iteration method for solving the system (1.1). It has been proved in [13] that the MHSS iteration method converges to the unique solution of (1.1) unconditionally. Bai et al. also established the convergence theory for the PMHSS iteration method under suitable conditions in [15] and showed the h-independent behavior. There are also some other effective iterative methods, such as the skew-normal splitting (SNS) method [14], the Hermitian normal splitting (HNS) method and its variant simplified HNS (SHNS) method [16], among others.
If the matrices W and T are symmetric positive semi-definite, Bai [18] introduced the rotated block triangular (RBT) preconditioner. And then to accelerate the computation of the RBT preconditioner, Lang and Ren [19] proposed the inexact RBT preconditioner. When, however, the matrix W is symmetric indefinite and T is symmetric positive definite, the MHSS (PMHSS) method and SNS method may be slow or not applicable due to the fact that the coefficient matrices and are indefinite or singular. Under these circumstances, an appropriate preconditioner is needed to circumvent this difficulty, see [8], [17]. Recently, Zhang and Dai in [1] proposed a new block splitting (BS) preconditionerBy observing the residual between the original coefficient matrix in (1.3) and the BS preconditioner, we obtainHowever, one limitation of this preconditioner is the fact that the diagonal blocks tend to zero while the nonzero off-diagonal block becomes unbounded as α approaches 0.
To solve this problem, inspired by the idea of the relaxed preconditioner [1], [3], [20], we construct a modified block (MB) splitting preconditioner for the real block two-by-two system of linear equation (1.3). Theoretical analysis prove that the preconditioned matrix has an eigenvalue 1 with algebraic multiplicity at least n. We also investigate the impact upon the corresponding Krylov subspace method. Numerical experiments are presented to illustrate the effectiveness of our preconditioner.
The remainder of this work is organized as follows. In Section 2, a modified block preconditioner for solving the complex system is provided. In Section 3, we investigate the eigenvalue properties of the preconditioned matrix and the impact upon the convergence of the corresponding Krylov subspace method. In Section 4, some numerical experiments are carried out to validate the effectiveness of the new preconditioner. In Section 5, some conclusions are given.
Section snippets
A modified block preconditioner
In this section, we present a modified block preconditioner for solving the complex system (1.3) with symmetric indefinite matrix and nonsymmetric positive definite matrix . Inspired by the method used in [1], for the linear system (1.3), we can easily obtain the following splitting of the coefficient matrix where and . Analogous to the PSS preconditioner proposed in [2], we setwhere is an identity matrix with appropriate order. Since
Theorectical analysis of the preconditioned matrix
The spectral distribution of the preconditioned matrix relates closely to the convergence rate of Krylov subspace methods. Tightly clustered spectra or positive real spectra of the preconditioned matrix are desirable. In the following, we investigate the spectral properties of the preconditioned matrix . Theorem 3.1 Assume that the coefficient matrix is nonsingular, is symmetric indefinite and is nonsymmetric positive definite. Let α and β be real positive constants. Then for the
Numerical experiments
In this section, we carry out some numerical experiments of complex nonsymmetric linear system (1.1) to validate the effectiveness of our proposed MB preconditioner . For comparison, we also choose the HSS preconditioner and the DPSS preconditioner [1] coupled with GMRES(30). All runs are implemented using double precision float point arithmetic in MATLAB (version R2015a). In practical computation, we set the initial guess and the stopping criterion to be
Conclusion
To solve a class of complex nonsymmetric indefinite linear systems, a modified block splitting preconditioner is proposed. By adopting two iteration parameters and a relaxing technique, the new preconditioner is much closer to the original coefficient matrix. We prove that the preconditioned matrix has an eigenvalue 1 with algebraic multiplicity at least n. Numerical experiments are used to illustrate the competitiveness and effectiveness of the proposed preconditioner compared to other
Acknowledgments
The authors are very much indebted to the anonymous referees and editor-in-chief for their constructive suggestions, which substantially improved the original manuscript of this paper.
References (20)
A note on PSS preconditioners for generalized saddle point problems
Appl. Math. Comput.
(2014)- et al.
A class of generalized relaxed PSS preconditioners for generalized saddle point problems
Appl. Math. Lett.
(2016) - et al.
A generalized relaxed positive-definite and skew-hermitian splitting preconditioner for non-hermitian saddle point problems
Appl. Math. Comput.
(2015) - et al.
A new block preconditioner for complex symmetric indefinite linear systems
Numer. Algorithms
(2017) - et al.
A simplified HSS preconditioner for generalized saddle point problems
BIT
(2016) Optical tomography in medical imaging
Inverse Probl.
(1999)- et al.
A fast algorithm for the electromagnetic scattering from a large cavity
SIAM J. Sci. Comput.
(2005) - et al.
Accurate numerical solutions of the time-dependent Schrödinger equation
Phys. Rev. E.
(2007) - et al.
Block preconditioning of real-valued iterative algorithms for complex linear systems
IMA J. Numer. Anal.
(2008) - et al.
On solving block-structured indefinite linear systems
SIAM J. Sci. Comput.
(2003)
Cited by (0)
- ☆
This work was supported by the National Natural Science Foundation (Nos.11701456, 11801452 and 11571004), Fundamental Research Funds for the Central Universities (Nos. 2452017219, 2452018017), College Students Innovation and Entrepreneurship Training Program (1201810712004), and by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-K201801901).