J. Chen, L. C. Mcinnes, H. Zhang, "Analysis and Practical Use of Flexible Bicgstab," Preprint ANL/MCS-P3039-0912, September 2012. [pdf]
A flexible version of the BiCGStab algorithm (FBiCGStab) for solving a linear system of equations is analyzed. We show that under flexible preconditioning, the perturbation to the outer residual norm is of the same order as that to the preconditioner. In addition, we formulate a mathematically equivalent variant, FBiCGStab-R, in order to reduce the global synchronization cost for performing inner product calculations. FBiCGStab-R is numerically stable and sometimes far outperforms FBiCGStab if high accuracy of the solution is required. Both analysis and numerical experiments show that a strong preconditioner is often favored for the practical use of flexible BiCGStab. We demonstrate in a large-scale reacting flow application (PFLOTRAN) that the use of flexible BiCGStab leads to significantly accelerated simulation time on extreme-scale computers using O(10^4) -(10^5) processor cores.