Prioritized Multiplicative Schwarz Procedures for Solving Linear Systems

Abstract

We describe a new algorithm designed to quickly and robustly solve general linear problems of the form Ax = b. We describe both serial and parallel versions of the algorithm, which can be considered a prioritized version of an Alternating Multiplicative Schwarz procedure. We also adopt a general view of alternating Multiplicative Schwarz procedures which motivates their use on arbitrary problems (even which may not have arisen from problems that are naturally decomposable) by demonstrating that, even in a serial context, algorithms should use many, many partitions to accelerate convergence; having such an over-partitioned system also allows easy parallelization of the algorithm, and scales extremely well. We present extensive empirical evidence which demonstrates that our algorithm, with a companion subsolver, can often improve performance by several orders of magnitude over the subsolver by itself and over other algorithms.

Fast iterative graph computation with block updates

“…However, block-level scheduling results in less scheduling overhead and better cache utilization, and researchers have successfully used block scheduling with block priorities defined by aggregation in several applications [8,13,41]. Our framework is general enough to support all the aggregation methods used in these papers.…”