On Combining Pipelining and \(\text{s}\)-Step Concepts in Preconditioned Conjugate Gradient Methods

On large-scale parallel computers, global communication becomes a major bottleneck for the Preconditioned Conjugate Gradient (PCG) method, an iterative solver for large sparse linear systems. Scalable PCG variants reduce the number of global synchronization points by a factor of \(\mathcal{O}(s)\) (\(s\)-step methods), or overlap communication with local computations (pipelined methods). The pipelined \(s\)-step PCG method P-sPCGmon combines these two approaches, but introduces significant local computational overhead increasing with step size \(s\). Moreover, it suffers from poor numerical stability.