Low synchronization GMRES algorithms.

Communication-avoiding and pipelined variants of Krylov solvers are critical for the scalability of linear system solvers on future exascale architectures. We present low synchronization variants of iterated classical (CGS) and modified Gram-Schmidt (MGS) algorithms that require one and two global reduction communication steps. Derivations of low synchronization iterated CGS algorithms are based on previous work by Ruhe. Our main contribution is to introduce a backward normalization lag into the compact $WY$ form of MGS resulting in a ${\cal O}(\eps)\kappa(A)$ stable GMRES algorithm that requires only one global synchronization per iteration. The reduction operations are overlapped with computations and pipelined to optimize performance. Further improvements in performance are achieved by accelerating GMRES BLAS-2 operations on GPUs.