Aggregation of clans to speed-up solving linear systems on parallel architectures

Abstract

The paper further refines the clan composition technique that is considered a way of matrix partitioning into a union of block-diagonal and block-column matrices. This enables solving the individual systems for each horizontal block on a separate computing node, followed by solving the composition system. The size of minimal clans, obtained as a result of matrix decomposition, varies considerably. For load balancing, early versions of ParAd software were using dynamic scheduling of jobs. The present paper studies a task of static balancing the clan size. Rather good results are obtained using a fast bin packing algorithm with the first fit on a sorted array which are considerably improved applying a multi-objective graph partitioning with software package METIS. Aggregation of clans allows us to obtain up to three times extra speed-up, including systems over fields of real numbers, on matrices from Model Checking Contest and Matrix Market.