Communication-hiding pipelined BiCGSafe methods for solving large linear systems

Recently, a new variant of the BiCGStab method, known as the pipeline BiCGStab, has been proposed. This method can achieve a higher degree of scalability and speed-up rates through a mechanism in which the communication phase for the computation of the inner product can be overlapped with the computation of the matrix-vector product. On the other hand, there exist several generalized iteration methods with better convergence behavior than BiCGStab such as ssBiCGSafe, BiCGSafe, GPBi-CG. Of these methods, ssBiCGSafe, which requires a single phase of computing inner products per one iteration, is best suited for high-performance computing systems. In this paper, inspired by the success of the pipelined BiCGStab method, we propose variations of the ssBiCGSafe method, in which only one phase of inner product computation per iteration is required and this inner product computation phase can be overlapped with the matrix-vector computation. Through numerical experiments, we show that the proposed methods lead to improvements in convergence behavior and execution time compared to the pipelined BiCGStab and ssBiCGSafe methods.