Hiding Global Communication Latency in the GMRES Algorithm on Massively Parallel Machines

In the Generalized Minimal Residual Method (GMRES), the global all-to-all communication required in each iteration for orthogonalization and normalization of the Krylov base vectors is becoming a performance bottleneck on massively parallel machines. Long latencies, system noise and load imbalance cause these global reductions to become very costly global synchronizations. In this work, we propose the use of non-blocking or asynchronous global reductions to hide these global communication latencies by overlapping them with other communications and calculations. A pipelined variation of GMRES is presented in which the result of a global reduction is only used one or more iterations after the communication phase has started. This way, global synchronization is relaxed and scalability is much improved at the expense of some extra computations. The numerical instabilities that inevitably arise due to the typical monomial basis by powering the matrix are reduced and often annihilated by using Newton or Chebyshev bases instead. We model the performance on massively parallel machines with an analytical model.