Robustness and Accuracy in Pipelined Bi-Conjugate Gradient Stabilized Method: A Comparative Study

TL;DR

This paper compares pipelined BiCGStab with standard methods, focusing on balancing scalability and accuracy in solving unsymmetric linear systems across various applications.

Contribution

It introduces an ExBLAS-based stabilization technique to improve accuracy in pipelined BiCGStab without sacrificing scalability.

Findings

Pipelined BiCGStab offers significant speedups over standard methods.

Residual replacement stabilizes the accuracy of pipelined BiCGStab.

Validation on SuiteSparse matrices confirms effectiveness.

Abstract

In this article, we propose an accuracy-assuring technique for finding a solution for unsymmetric linear systems. Such problems are related to different areas such as image processing, computer vision, and computational fluid dynamics. Parallel implementation of Krylov subspace methods speeds up finding approximate solutions for linear systems. In this context, the refined approach in pipelined BiCGStab enhances scalability on distributed memory machines, yielding to substantial speed improvements compared to the standard BiCGStab method. However, it's worth noting that the pipelined BiCGStab algorithm sacrifices some accuracy, which is stabilized with the residual replacement technique. This paper aims to address this issue by employing the ExBLAS-based reproducible approach. We validate the idea on a set of matrices from the SuiteSparse Matrix Collection.