Robust Dropping Criteria for F-norm Minimization Based Sparse Approximate Inverse Preconditioning

Dropping tolerance criteria play a central role in Sparse Approximate Inverse preconditioning. Such criteria have received, however, little attention and have been treated heuristically in the following manner: If the size of an entry is below some empirically small positive quantity, then it is set to zero. The meaning of "small" is vague and has not been considered rigorously. It has not been clear how dropping tolerances affect the quality and effectiveness of a preconditioner $M$. In this paper, we focus on the adaptive Power Sparse Approximate Inverse algorithm and establish a mathematical theory on robust selection criteria for dropping tolerances. Using the theory, we derive an adaptive dropping criterion that is used to drop entries of small magnitude dynamically during the setup process of $M$. The proposed criterion enables us to make $M$ both as sparse as possible as well as to be of comparable quality to the potentially denser matrix which is obtained without dropping. As a byproduct, the theory applies to static F-norm minimization based preconditioning procedures, and a similar dropping criterion is given that can be used to sparsify a matrix after it has been computed by a static sparse approximate inverse procedure. In contrast to the adaptive procedure, dropping in the static procedure does not reduce the setup time of the matrix but makes the application of the sparser $M$ for Krylov iterations cheaper. Numerical experiments reported confirm the theory and illustrate the robustness and effectiveness of the dropping criteria.Comment: 27 pages, 2 figure

Sparse approximate inverse preconditioners on high performance GPU platforms

Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

A Sparse-Sparse Iteration for Computing a Sparse Incomplete Factorization of the Inverse of an SPD Matrix

In this paper, a method via sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix is proposed. The resulting factorized sparse approximate inverse is used as a preconditioner for solving symmetric positive definite linear systems of equations by using the preconditioned conjugate gradient algorithm. Some numerical experiments on test matrices from the Harwell-Boeing collection for comparing the numerical performance of the presented method with one available well-known algorithm are also given.Comment: 15 pages, 1 figur

Communication-aware sparse patterns for the factorized approximate inverse preconditioner

The Conjugate Gradient (CG) method is an iterative solver targeting linear systems of equations Ax=b where A is a symmetric and positive definite matrix. CG convergence properties improve when preconditioning is applied to reduce the condition number of matrix A. While many different options can be found in the literature, the Factorized Sparse Approximate Inverse (FSAI) preconditioner constitutes a highly parallel option based on approximating A-1. This paper proposes the Communication-aware Factorized Sparse Approximate Inverse preconditioner (FSAIE-Comm), a method to generate extensions of the FSAI sparse pattern that are not only cache friendly, but also avoid increasing communication costs in distributed memory systems. We also propose a filtering strategy to reduce inter-process imbalance. We evaluate FSAIE-Comm on a heterogeneous set of 39 matrices achieving an average solution time decrease of 17.98%, 26.44% and 16.74% on three different architectures, respectively, Intel Skylake, Fujitsu A64FX and AMD Zen 2 with respect to FSAI. In addition, we consider a set of 8 large matrices running on up to 32,768 CPU cores, and we achieve an average solution time decrease of 12.59%.Marc Casas is supported by Grant RYC-2017-23269 funded by MCIN/AEI/ 10.13039/501100011033 and by “ESF Investing in your future”. This work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 955606. This work has been supported by the Computación de Altas Prestaciones VIII (BSC-HPC8) project. It has also been partially supported by the EXCELLERAT project funded by the European Commission’s ICT activity of the H2020 Programme under grant agreement number: 823691 and by the Spanish Ministry of Science and Innovation (Nucleate, Project PID2020-117001GB-I00).Peer ReviewedPostprint (author's final draft

A Novel Factorized Sparse Approximate Inverse Preconditioner with Supernodes

AbstractKrylov methods preconditioned by Factorized Sparse Approximate Inverses (FSAI) are an efficient approach for the solution of symmetric positive definite linear systems on massively parallel computers. However, FSAI often suffers from a high set-up cost, especially in ill-conditioned problems. In this communication we propose a novel algorithm for the FSAI computation that makes use of the concept of supernode borrowed from sparse LU factorizations and direct methods