We present a modified version of the PRESB preconditioner for two-by-two block system of linear equations with the coefficient matrix $$\textbf{A}=\left(\begin{array}{cc} F & -G^* G & F \end{array}\right),$$ 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 utilizing 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：本文针对系数矩阵为 $$\textbf{A}=\left(\begin{array}{cc} F & -G^* \\ G & F \end{array}\right)$$ 的2×2块线性方程组，提出了一种PRESB预条件子的改进形式，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。分析了预条件后矩阵的谱特征。在使用Krylov子空间方法（如GMRES）结合所提预条件子求解预条件系统的每次迭代中，需求解两个具有厄米正定系数矩阵的子系统，可通过Cholesky分解精确求解，或利用共轭梯度法非精确求解。本文展示了所提预条件子应用于PDE约束优化问题有限元离散系统的实例，并通过数值结果验证了该预条件子的有效性。