We present an efficient preconditioner for two-by-two block system of linear equations with the coefficient matrix $ \begin{pmatrix} F & -G^H G & F \end{pmatrix}$ where $F\in\mathbb{C}^{n\times n}$ is Hermitian positive definite and $G\in\mathbb{C}^{n\times n}$ is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration of a Krylov subspace method, like GMRES, for solving the preconditioned system in conjunction with proposed preconditioner two subsystems with Hermitian positive definite coefficient matrix should be solved which can be accomplished exactly using the Cholesky factorization or inexactly using the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner.

翻译：我们提出了一类针对系数矩阵为 $ \begin{pmatrix} F & -G^H \\ G & F \end{pmatrix} $ 的二乘二块线性方程组的高效预处理器，其中 $F\in\mathbb{C}^{n\times n}$ 为厄米正定矩阵，$G\in\mathbb{C}^{n\times n}$ 为半正定矩阵。分析了预处理后矩阵的谱特征。在使用Krylov子空间方法（如GMRES）求解与所提预处理器结合的预处理系统时，每次迭代需求解两个具有厄米正定系数矩阵的子系统，这些子系统可通过Cholesky分解精确求解，或通过共轭梯度法非精确求解。本文将该预处理器应用于由PDE约束优化问题的有限元离散生成的系统中，并通过数值算例验证了该预处理器的有效性。