Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method

Improving strong scaling of the Conjugate Gradient method for solving large linear systems using global reduction pipelining

“…The corresponding dot-products on line 23 are defined as M-inner products in the context of the preconditioned algorithm, see [8] for details.…”

“…Numerical round-off errors may increase the occurrence of these breakdowns in practice, cf. [8,11]. When a breakdown occurs the p(l)-CG iteration is restarted explicitly.…”

“…that are well established for the original Krylov algorithms. Research on analyzing and improving the numerical stability of pipelined Krylov subspace methods has recently been performed in [4,7,8,10]; we point out the references therein for more information on the numerical properties of Krylov subspace methods.…”

“…Strong scaling results obtained from an MPI-based implementation of the l-length pipelined Conjugate Gradient method, p(l)-CG for short, are presented in this work. The p(l)-CG method was recently presented in [8,11] and allows to overlap each global reduction phase with the communication and computational work of l subsequent iterations. This overlap is achieved by exploiting the asynchronous non-blocking global "Iallreduce" MPI operations that have been available from the MPI3 standard onward.…”