An efficient sparse approximate inverse preconditioning algorithm on GPU

Abstract

Summary The sparse approximate inverse (SPAI) preconditioner has proven to be effective in accelerating the convergence of iterative methods. Recently, accelerating it on the graphics processing unit (GPU) has attracted considerable attention due to the fact that the cost of constructing it is high. This motivates us to investigate how to accelerate the construction of SPAI preconditioners on GPU in this paper. We propose an efficient sparse approximate inverse algorithm on GPU, called SPAI‐Adaptive. For our proposed SPAI‐Adaptive, there are the following novelties: (1) an adaptive thread allocation strategy for SPAI‐Adaptive is proposed to assign the optimal thread number for each column of the preconditioner, and (2) Each component of the preconditioner, which includes finding indices I and J, constructing local submatrix, decomposing the local matrix into QR, and solving the upper triangular linear system, is computed in parallel inside a thread group of GPU. Experimental results show that the proposed SPAI‐Adaptive is effective, and has good performance and high parallelism.

A GPU-accelerated adaptive FSAI preconditioner for massively parallel simulations

“…Benzi and Tûma (1999) and Benzi (2002) for a thorough review. Approximate inverse preconditioners are becoming popular again due to their potential in concurrent computations especially on GPU hardware (Anzt et al, 2018; Bertaccini and Filippone, 2014; He et al, 2019; Moutafis et al, 2020a, 2020b; Xu et al, 2011).…”

A new diagonal storage for efficient implementation of sparse matrix–vector multiplication on graphics processing unit

“…Processing big data by using GPUs has drawn much attention over the recent years. Following the introduction of the compute unified device architecture (CUDA), a programming model that supports the joint CPU/GPU execution of applications, by NVIDIA in 2007, 9 GPUs have become strong competitors as general‐purpose parallel programming systems, and have been increasingly used as tools for high‐performance computation in many fields 10–17 …”

GPU Preconditioning for Block Linear Systems Using Block Incomplete Sparse Approximate Inverses

“…Although the inexact preconditioning step results in more number of solver iterations, it shows an advantage over the exact preconditioning in terms of the total compute time of the linear solver. Some methods are based on sparse approximate inverses (SAI) [15][16][17][18][19] where the matrix inverse of the triangular factors is estimated through solving least squares problems on a preset pattern. And the resulting inverses can transform the preconditioning step into sparse matrix-vector multiplications with much more possible of concurrency computing on parallel computers.…”

“…Incomplete Sparse Approximate Inverses. For a given sparse matrix A n×n with n rows and n columns, the SAI algorithm [1,[15][16][17][18][19] gives an approximation of the inverse of A by minimizing the Frobenius norm of (AW − I) as…”