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

Abstract

a b s t r a c tOptimization 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 timeperiodic 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.

A new iterative method for solving a class of two-by-two block complex linear systems

“…then B = P − Q and T α = P −1 Q. Hence, the matrix P can be used as preconditioner for the system (6). Since G −1 = −G, in the implementation of the preconditioner P in a Krylov subspace method like GMRES we only need solving two systems with the matrices αI + K and αI + M along with two matrix multiplications with the matrices G and I + G. These systems can be solved exactly using the Cholesky factorization of the matrices αI + K and αI + M , or inexactly using the conjugate gradient (CG) method.…”

“…Numerical results presented in [6] show that the BAS iteration method outperform the GM-RES method [4]. They also showed that the parameter α = 1 + νω 2 often gives quite suitable results.…”

“…In [2], Krendl proposed the real block diagonal and the alternative indefinite preconditioners for the system (2). Zheng et al in [6] proposed the block alternating splitting (BAS) iteration method which can be summarized as…”

On the solution of the distributed optimal control problem with time‐periodic parabolic equations

“…Krendl et al [3] proposed the real block diagonal and the alternative indefinite preconditioners for the system (1). The block alternating splitting (BAS) iteration method was presented by Zheng et al [12] for solving the system (1) which can be written as…”

A robust structured preconditioner for time-harmonic parabolic optimal control problems

“…The alternative method for solving the linear systems (1.8) and (1.9), which is also our choice in this paper, tackles the complex linear system directly. Recently, some special preconditioning and iterative techniques have appeared in the literature to solve the linear system (1.8), see [21,33,34]. In this paper, based on the works in [3,9], we construct a new structured two-by-two block preconditioner, which is called P str-II , for the linear system (1.9).…”