Matrices or operators in two-by-two block form with square blocks arise in numerous important applications, such as in optimal control problems for PDEs. The problems are normally of very large scale so iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is of crucial importance. Since some time a very efficient preconditioner, the preconditioned square block, PRESB method has been used by the authors and coauthors in various applications, in particular for optimal control problems for PDEs. It has been shown to have excellent properties, such as a very fast and robust rate of convergence that outperforms other methods. In this paper the fundamental and most important properties of the method are stressed and presented with new and extended proofs. Under certain conditions, the condition number of the preconditioned matrix is bounded by 2 or even smaller. Furthermore, under certain assumptions the rate of convergence is superlinear.

中文翻译：

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PRESB 预处理器的优越特性，适用于带有方形块的 2×2 块形式的运算符

带有方形块的 2×2 块形式的矩阵或运算符出现在许多重要应用中，例如 PDE 的优化控制问题。这些问题通常规模非常大，因此必须使用迭代求解方法。因此，选择高效且稳健的预处理器至关重要。一段时间以来，作者和合著者在各种应用中使用了一种非常有效的预处理器，即预处理方块，PRESB 方法，特别是用于 PDE 的优化控制问题。它已被证明具有出色的特性，例如超越其他方法的非常快速和稳健的收敛速度。在本文中，强调了该方法的基本和最重要的特性，并用新的和扩展的证明进行了介绍。在一定条件下，预处理矩阵的条件数以2为界，甚至更小。此外，在某些假设下，收敛速度是超线性的。