Multilevel approaches for FSAI preconditioning

Abstract

Summary Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree.

Preconditioners for Krylov subspace methods: An overview

“…However, perhaps counterintuitively, newer versions of these algorithms may increase communication and/or reduce parallelism to improve overall efficiency on cutting‐edge computer architectures. For example, domain decomposition methods may also include a multilevel component that induces global coupling [422], while SPAI preconditioners may, for example, increase sequential computation to improve efficiency [423]. Additionally, it may be advantageous to tailor kernels to GPUs (see, eg, [424]).…”

An efficient sparse approximate inverse preconditioning algorithm on GPU

“…Popular preconditioners include the incomplete factorization preconditioners, the sparse approximate inverse (SPAI) preconditioners based on Frobenius norm minimization, the factorized sparse approximate inverse (FSAI) preconditioners, and the preconditioners that consist of an incomplete factorization, followed by an approximate inversion of the incomplete factors . However, the cost of constructing the preconditioners is generally very high for large‐scale problems.…”

“…As compared to the class of the incomplete factorized preconditioners, SPAI preconditioners require only a few sparse matrix‐vector multiplication operations instead of triangular solves. This feature is quite attractive especially from the point of view of a parallel implementation on the GPU architecture because this type of mathematical operation inherently involves a high level of concurrency . Furthermore, FSAI preconditioners can fail due to breakdowns during an incomplete factorization process .…”

“…This feature is quite attractive especially from the point of view of a parallel implementation on the GPU architecture because this type of mathematical operation inherently involves a high level of concurrency. 15 Furthermore, FSAI preconditioners can fail due to breakdowns during an incomplete factorization process. 39 A comparative study of various approximate inverse preconditioners was presented in the work of Benzi and Tuma.…”