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
Abstract. We design a distributed-memory randomized structured multifrontal solver for large sparse matrices. Two layers of hierarchical tree parallelism are used. A sequence of innovative parallel methods are developed for randomized structured frontal matrix operations, structured update matrix computation, skinny extend-add operation, selected entry extraction from structured matrices, etc. Several strategies are proposed to reuse computations and reduce communications. Unlike an earlier parallel structured multifrontal method that still involves large dense intermediate matrices, our parallel solver performs the major operations in terms of skinny matrices and fully structured forms. It thus significantly enhances the efficiency and scalability. Systematic communication cost analysis shows that the numbers of words are reduced by factors of about O( √ n/r) in two dimensions and about O(n 2/3 /r) in three dimensions, where n is the matrix size and r is an off-diagonal numerical rank bound of the intermediate frontal matrices. The efficiency and parallel performance are demonstrated with the solution of some large discretized PDEs in two and three dimensions. Nice scalability and significant savings in the cost and memory can be observed from the weak and strong scaling tests, especially for some 3D problems discretized on unstructured meshes.
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