SIAM Journal on Scientific Computing, Ahead of Print.
We propose an unconditionally robust and highly effective preconditioner for general dense symmetric positive definite matrices based on structured incomplete factorization (SIF), called the enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing to the matrix and then compressing appropriate scaled off-diagonal blocks. Here, we use an enhanced scaling-and-compression strategy to design the new eSIF preconditioner. Some subtle modifications are made, such as the use of two-sided block triangular preprocessing. A practical multilevel eSIF scheme is then designed. We give rigorous analysis for both the enhanced scaling-and-compression strategy and the multilevel eSIF preconditioner. The new eSIF framework has some significant advantages and overcomes some major limitations of the SIF strategy. (i) With the same tolerance for compressing the off-diagonal blocks, the eSIF preconditioner can approximate the original matrix to a much higher accuracy. (ii) The new preconditioner leads to much more significant reductions of condition numbers due to an accelerated magnification effect for the decay in the singular values of the scaled off-diagonal blocks. (iii) With the new preconditioner, the eigenvalues of the preconditioned matrix are much better clustered around 1. (iv) The multilevel eSIF preconditioner is further unconditionally robust or is guaranteed to be positive definite without the need of extra stabilization, while the multilevel SIF preconditioner has a strict requirement in order to preserve positive definiteness. Comprehensive numerical tests are used to show the advantages of the eSIF preconditioner in accelerating the convergence of iterative solutions.

中文翻译：

---

适用于一般密集 SPD 矩阵的稳健有效的 eSIF 预处理

SIAM 科学计算杂志，提前印刷。
我们为基于结构化不完全分解 (SIF) 的一般稠密对称正定矩阵提出了一种无条件鲁棒且高效的预处理器，称为增强型 SIF (eSIF) 预处理器。最近提出的原始 SIF 策略通过对矩阵应用块对角线预处理然后压缩适当的缩放非对角线块来推导出结构化预处理器。在这里，我们使用增强的缩放和压缩策略来设计新的 eSIF 预处理器。进行了一些细微的修改，例如使用了两侧块三角形预处理。然后设计了一个实用的多级 eSIF 方案。我们对增强的缩放和压缩策略和多级 eSIF 预处理器进行了严格的分析。新的 eSIF 框架具有一些显着的优势，并且克服了 SIF 策略的一些主要限制。(i) 在压缩非对角块的容差相同的情况下，eSIF 预处理器可以将原始矩阵近似到更高的精度。(ii) 由于缩放的非对角线块的奇异值衰减的加速放大效应，新的预处理器导致条件数的显着减少。(iii) 使用新的预处理器，预处理矩阵的特征值更好地聚集在 1 附近。 (iv) 多级 eSIF 预处理器进一步无条件鲁棒或保证是正定的，而无需额外的稳定性，而多级 SIF为了保持正定性，预处理器有严格的要求。