A Stochastic Performance Model for Pipelined Krylov Methods

TL;DR

This paper develops a stochastic performance model for pipelined Krylov methods, revealing that, contrary to common belief, these algorithms can achieve speedups greater than 2x by accounting for stochastic noise.

Contribution

It introduces an analytical stochastic model for pipelined Krylov methods, challenging the folk theorem and demonstrating potential for higher speedups.

Findings

Speedups greater than 2x are achievable.

Stochastic noise significantly affects latency.

The model accurately predicts performance improvements.

Abstract

Pipelined Krylov methods seek to ameliorate the latency due to inner products necessary for projection by overlapping it with the computation associated with sparse matrix-vector multiplication. We clarify a folk theorem that this can only result in a speedup of $2\times$ over the naive implementation. Examining many repeated runs, we show that stochastic noise also contributes to the latency, and we model this using an analytical probability distribution. Our analysis shows that speedups greater than $2\times$ are possible with these algorithms.