Communication-hiding pipelined BiCGSafe methods for solving large linear systems

• The proposed methods developed based on the pipelined BiCGStab and single-synchronized BiCGSafe methods, inherited two outstanding properties from each method: hiding communication latency and using only one phase for global reduction per iteration. • Compared to p-BiCGStab method, p-BiCGSafe method has a greater convergence characteristic; compared to ssBiCGSafe2 method, p-BiCGSafe method is more efficient on HPC systems in terms of computation time. Recently, a new variant of the BiCGStab method, known as the pipelined 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. Meanwhile, several generalized iteration methods with better convergence behavior than BiCGStab exist, such as ssBiCGSafe, BiCGSafe, and GPBi-CG. Among these methods, ssBiCGSafe, which requires a single phase of computing inner products per iteration, is best suited for high-performance computing systems. As described herein, inspired by the success of the pipelined BiCGStab method, we propose pipelined variations of the ssBiCGSafe method in which only one phase of inner product computation per iteration is required and this phase of inner product computation can be overlapped with the matrix-vector computation. Through numerical experimentation, we demonstrate that the proposed methods engender improvements in convergence behavior and execution time compared to the pipelined BiCGStab and ssBiCGSafe methods.