Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization

Abstract

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. K E Y W O R D Seffectiveness, robustness, scaling and compression strategy, SIF preconditioning, spectral analysis

Second Order Accurate Hierarchical Approximate Factorization of Sparse SPD Matrices

“…Contrast with related work. Our work is closely related to that of Xia [35,38,33], Li [22,21], and Xin [39] (which concentrate however on the general, not sparse, case). In particular, our approach includes the double-sided scaling proposed in those works, as well as the implicit Schur compensation.…”

“…Most recently, Xia [34] used similar ideas, to improve the SIF algorithm for general (typically dense) SPD matrices [39]. Below, we use our methods to improve spaND [4], which is an algorithm for sparse matrices, related to, but different from SIF.…”

Efficient construction of an HSS preconditioner for symmetric positive definite $\mathcal{H}^2$ matrices

“…Recently, a scaling-and-compression technique has been developed for SPD matrices to compress matrix blocks into low-rank form as part of the construction of certain rank-structured preconditioners [8,28,27,30,32]. The resulting preconditioners can be more effective than if this technique is not used.…”

Robust and effective eSIF preconditioning for general SPD matrices

“…One type is in [11,12,13,18] based on low-rank strategies for approximating A −1 . Another type is in [1,6,8,9,14,19,21,22] where approximate Cholesky factorizations are computed using low-rank approximations of relevant off-diagonal blocks. Both types of methods have been shown useful for many applications.…”

“…This makes a significant difference as compared with standard rank-structured preconditioners that are based on direct off-diagonal compression. Accordingly, the SIF preconditioner has some attractive features, such as the convenient analysis of the performance, the convenient control of the approximation accuracy, and the nice effectiveness for preconditioning [21,22]. In fact, if only r largest singular values of C are kept in its low-rank approximation, then the resulting preconditioner (called a one-level or prototype preconditioner) approximates A with a relative accuracy bound σ r+1 .…”