In this work, we provide new analysis for a preconditioning technique called structured incomplete factorization (SIF) for symmetric positive definite matrices. In this technique, a scaling and compression strategy is applied to construct SIF preconditioners, where off‐diagonal blocks of the original matrix are first scaled and then approximated by low‐rank forms. Some spectral behaviors after applying the preconditioner are shown. The effectiveness is confirmed with the aid of a type of two‐dimensional and three‐dimensional discretized model problems. We further show that previous studies on the robustness are too conservative. In fact, the practical multilevel version of the preconditioner has a robustness enhancement effect, and is unconditionally robust (or breakdown free) for the model problems regardless of the compression accuracy for the scaled off‐diagonal blocks. The studies give new insights into the SIF preconditioning technique and confirm that it is an effective and reliable way for designing structured preconditioners. The studies also provide useful tools for analyzing other structured preconditioners. Various spectral analysis results can be used to characterize other structured algorithms and study more general problems.

中文翻译：

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基于结构不完全分解的预处理技术的有效性和鲁棒性

在这项工作中，我们为对称正定矩阵的一种称为结构化不完全因子分解（SIF）的预处理技术提供了新的分析。在这项技术中，缩放和压缩策略用于构造SIF预处理器，其中原始矩阵的非对角线块首先被缩放，然后通过低阶形式近似。显示了应用预处理器后的一些光谱行为。借助于二维和三维离散模型问题，可以确认有效性。我们进一步表明，先前关于鲁棒性的研究过于保守。实际上，实用的多级预处理器具有增强鲁棒性的作用，并且对于模型问题是无条件的鲁棒性（或无故障），无论缩放后的对角线块的压缩精度如何。这些研究为SIF预处理技术提供了新的见解，并确认这是设计结构化预处理器的有效且可靠的方法。这些研究还提供了用于分析其他结构化预处理器的有用工具。各种频谱分析结果可用于表征其他结构化算法并研究更一般的问题。