We consider the low-rank alternating directions implicit (ADI) iteration for approximately solving large-scale algebraic Sylvester equations. Inside every iteration step of this iterative process a pair of linear systems of equations has to be solved. We investigate the situation when those inner linear systems are solved inexactly by an iterative methods such as, for example, preconditioned Krylov subspace methods. The main contribution of this work are thresholds for the required accuracies regarding the inner linear systems which dictate when the employed inner Krylov subspace methods can be safely terminated. The goal is to save computational effort by solving the inner linear system as inaccurate as possible without endangering the functionality of the low-rank Sylvester-ADI method. Ideally, the inexact ADI method mimics the convergence behaviour of the more expensive exact ADI method, where the linear systems are solved directly. Alongside the theoretical results, also strategies for an actual practical implementation of the stopping criteria are developed. Numerical experiments confirm the effectiveness of the proposed strategies.

翻译：考虑低秩交替方向隐式（ADI）迭代法，用于近似求解大规模代数Sylvester方程。在该迭代过程的每一步中，需求解一对线性方程组。本文研究当这些内部线性系统采用迭代方法（例如预处理Krylov子空间方法）非精确求解时的情况。主要贡献在于提出了内部线性系统所需精度的阈值，该阈值决定了何时可安全终止所使用的内部Krylov子空间方法。目标是在不危及低秩Sylvester-ADI方法功能的前提下，以尽可能低的精度求解内部线性系统，从而节省计算开销。理想情况下，非精确ADI方法应能模拟计算代价更高的精确ADI方法（其中线性系统直接求解）的收敛行为。除理论结果外，本文还开发了面向实际实现的停止准则实施策略。数值实验验证了所提策略的有效性。