Analysis of rounding error accumulation in Conjugate Gradients to improve the maximal attainable accuracy of pipelined CG
Résumé
Pipelined Krylov solvers typically offer better scalability in the strong scaling
limit compared to standard Krylov methods. The synchronization bottleneck is mitigated
by overlapping time-consuming global communications with useful computations in the
algorithm. However, to achieve this communication hiding strategy, pipelined methods
feature multiple recurrence relations on additional auxiliary variables to update the guess
for the solution. This paper aims at studying the influence of rounding errors on the
convergence of the pipelined Conjugate Gradient method. It is analyzed why rounding effects have a
significantly larger impact on the maximal attainable accuracy of the pipelined CG algorithm
compared to the traditional CG method. Furthermore, an algebraic model for the accumulation of
rounding errors throughout the (pipelined) CG algorithm is derived.
Based on this rounding error model, we then propose an automated residual
replacement strategy to reduce the effect of rounding errors on the final iterative solution.
The resulting pipelined CG method with automated residual replacement improves the maximal
attainable accuracy of pipelined CG to a precision comparable to that of standard CG,
while maintaining the efficient parallel performance of the pipelined method.
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