Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization
Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization
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DOI：
10.1002/nla.2294
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发表时间：
2020-03
影响因子：
4.3
通讯作者：
Zi-Xing Xing-Zi-Xing-Xing-2250347;J. Xia;S. Cauley;V. Balakrishnan
中科院分区：
文献类型：
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作者：
Zi-Xing Xing-Zi-Xing-Xing-2250347;J. Xia;S. Cauley;V. Balakrishnan
在这项工作中，我们提供了新的分析对称正定矩阵的预处理技术称为结构不完全因子分解（SIF）。在这种技术中，缩放和压缩策略被应用于构造SIF预处理器，其中原始矩阵的非对角块首先被缩放，然后被低秩形式近似。应用预处理后的一些频谱行为。通过一种二维和三维离散化模型问题验证了该方法的有效性。我们进一步表明，以前的研究的鲁棒性过于保守。实际上，预处理器的实际多级版本具有鲁棒性增强效果，并且对于模型问题是无条件鲁棒的（或无故障的），而不管缩放非对角块的压缩精度如何。这些研究为SIF预处理技术提供了新的见解，并证实了它是设计结构化预处理器的有效和可靠的方法。这些研究也为分析其他结构化预条件子提供了有用的工具。各种谱分析结果可用于表征其他结构化算法和研究更一般的问题。
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.