On Convergence Speed of Parallel Variants of GPBiCG Method for Solving Linear Equations

Abstract

Abstract The hybrid Bi-Conjugate Gradient (Bi-CG) methods such as Bi-CG stabilized (Bi-CGSTAB), Generalized Product-type based Bi-CG (GPBiCG), and BiCGstab(ℓ) are well-known for efficiently solving linear equations. GPBiCG and BiCGstab(ℓ) are more effective and robust than Bi-CGSTAB on problems with strongly nonsymmetric matrices. On present petascale high-performance computing hardware, the scalability of Krylov subspace methods has recently become increasingly prominent. The main bottleneck for efficient parallelization is the inner products which require a global reduction. The parallel variants of Bi-CGSTAB reducing the number of global communication phases and hiding the communication latency have been proposed. However, it has been reported that the convergence of the parallel variants of Bi-CGSTAB is affected by rounding errors than that of the standard Bi-CGSTAB, and is not as robust as the standard. In this paper, therefore, following [1], we design parallel variants of GPBiCG, which converges faster and is more robust than Bi-CGSTAB. Then we compare the convergence speed between the standard GPBiCG and the parallel variants by numerical experiments.