PRKP: A Parallel Randomized Iterative Algorithm for Solving Linear Systems

This paper proposes a distributed-memory parallel randomized iterative algorithm for solving linear systems, called the parallel randomized kaczmarz projection (PRKP) algorithm. The algorithm has the property of greedy sampling, alternating projection, and lazy approximation. We derive the alternating projection process from the Randomized Kaczmarz algorithm and develop the greedy sampling and lazy approximation process to improve the convergence rate and reduce the per-process communication volume. Moreover, we develop a sampling residual estimation scheme for our proposed algorithm, which greatly reduces the extra computation cost required to obtain residuals. Our experimental results show that the proposed algorithm significantly outperforms previous parallel iterative algorithms on overdetermined linear systems, and yields up to $\mathbf{159}\times$ and $\mathbf{384}\times$ speedups on average respectively compared to the leading pipelined CG and pipelined GMRES algorithms.