On Convergence Speed of Parallel Variants of BiCGSTAB for Solving Linear Equations

Abstract

A number of hybrid Bi-Conjugate Gradient (Bi-CG) methods such as the Bi-CG STABilized (BiCGSTAB) method have been developed for solving linear equations. BiCGSTAB has been most often used for efficiently solving the linear equations, but we have sometimes seen the convergence behavior with a long stagnation phase. In such cases, it is important to have Bi-CG coefficients that are as accurate as possible, and the stabilization strategy for improving the accuracy of the Bi-CG coefficients has been proposed. In present petascale high-performance computing hardware, the main bottleneck of Krylov subspace methods for efficient parallelization is the inner products which require a global reduction. The parallel variants of BiCGSTAB such as communication avoiding and pipelined BiCGSTAB reducing the number of global communication phases and hiding the communication latency have been proposed. However, the numerical stability, specifically, the convergence speed of the parallel variants of BiCGSTAB has not previously been clarified on problems with situations where the convergence is slow (strongly affected by rounding errors). In this paper, therefore, we examine the convergence speed between the standard BiCGSTAB and the parallel variants, and the effectiveness of the stabilization strategy by numerical experiments on the problems where the convergence has a long stagnation phase.