The improved BiCGStab method for large and sparse unsymmetric linear systems on parallel distributed memory architectures

Abstract

In this paper, an improved version of the BiCGStab (IBiCGStab) method for the solutions of large and sparse linear systems of equations with unsymmetric coefficient matrices is proposed. The method combines elements of numerical stability and parallel algorithm design without increasing the computational costs. The algorithm is derived such that all inner products of a single iteration step are independent and communication time required for the inner product can be overlapped efficiently with computation time of vector updates. Therefore, the cost of global communication which represents the bottleneck of the parallel performance can be significantly reduced. The resulting IBiCGStab algorithm maintains the favorable properties of the original method while not increasing computational costs. Data distribution suitable for both irregularly and regularly structured matrices based on the analysis of the nonzero matrix elements is presented. Communication scheme is supported by overlapping execution of computation and communication to reduce waiting times. The efficiency of this method is demonstrated by numerical experimental results carried out on a massively parallel distributed memory system.