An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems
An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems
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DOI：
10.1137/120900800
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发表时间：
2012-11
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影响因子：
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通讯作者：
Zhongxiao Jia;Qian Zhang
中科院分区：
文献类型：
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作者：
Zhongxiao Jia;Qian Zhang
研究了自适应稀疏近似逆(SPAI)和自适应幂稀疏近似逆(PSAI)的两个重要特征：(I)对于具有不规则稀疏的$A$的大型线性系统$Ax=b$，即$A$具有相对密集的列，SPAI的实现可能非常昂贵，并且所得到的稀疏近似逆对于预条件可能是无效的。PSAI对于预适应是有效的，但可能需要过多的存储，并且耗时过长，令人无法接受。(Ii)当$A$是正则稀疏的，即它的所有列都是稀疏的时，情况会有很大的改善。在这种情况下，SPAI和PSAI都是有效的。此外，SPAI，尤其是PSAI更有可能构建有效的预条件。受这些特征的启发，我们提出了一种方法，使SPAI和PSAI在$Ax=b$和$A$不规则稀疏的情况下更实用。我们首先将$A$分解成一个正则稀疏的$\tide A$和一个低阶数的矩阵$S$。然后剥削..。
We investigate the adaptive sparse approximate inverse (SPAI) and adaptive power sparse approximate inverse (PSAI) preconditioning procedures and shed light on two important features of them: (i) For the large linear system $Ax=b$ with $A$ irregular sparse, i.e., with $A$ having $s$ relatively dense columns, SPAI may be very costly to implement, and the resulting sparse approximate inverses may be ineffective for preconditioning. PSAI can be effective for preconditioning but may require excessive storage and be unacceptably time consuming. (ii) The situation is improved drastically when $A$ is regular sparse, that is, all its columns are sparse. In this case, both SPAI and PSAI are efficient. Moreover, SPAI and, especially, PSAI are more likely to construct effective preconditioners. Motivated by these features, we propose an approach to making SPAI and PSAI more practical for $Ax=b$ with $A$ irregular sparse. We first split $A$ into a regular sparse $\tilde A$ and a matrix of low rank $s$. Then exploitin...