Parallelization of Matrix Partitioning in Construction of Hierarchical Matrices using Task Parallel Languages

Abstract

A hierarchical matrix (H-matrix) is an approximated form that represents N × N correlations of N objects. H-matrix construction is achieved by partitioning a matrix into submatrices, followed by calculating the element values of these submatrices. This paper proposes implementations of matrix partitioning using task parallel languages, Cilk Plus and Tascell. Matrix partitioning is divided into two steps: cluster tree construction by dividing objects into clusters hierarchically, and block cluster tree construction by observing all cluster pairs at the same level of the cluster tree that satisfy an admissibility condition. As the two types of trees constructed and traversed in these steps are unpredictably unbalanced, it is expected that we can efficiently parallelize both these steps using task parallel languages. To obtain sufficient parallelism in the cluster tree construction, we not only execute recursive calls in parallel but also parallelize the inside of each recursive step. For the block cluster tree construction, we assigned each worker its own space so that the workers can store the cluster pairs without using locks. As a result, compared to a sequential implementation in C, we achieved up to an 11.5-fold speedup using Cilk Plus and a 12.6-fold speedup by Tascell for the cluster tree construction. For the block cluster tree construction, up to a 37.7-fold speedup by Cilk Plus and a 38.8-fold speedup using Tascell are achieved. In regard to the entire process of matrix partitioning, we achieved up to a 12.2-fold speedup by Cilk Plus and a 14.5-fold speedup using Tascell. All experiments were executed on two 18-core Xeon processors with real datasets to generate coefficient matrices used in the surface charge method.

Parallelization of Matrix Partitioning in Hierarchical Matrix Construction on Distributed Memory Systems

“…We determined the best parameter combination (T N , T S , C) = (10 4 , 15, 10 3 ) using the parameter search algorithm employed in Ref. [11] by 36-worker executions of the linked tree implementation in a single node for the sphere. We extended this parameter combination to all experimental conditions.…”

“…[9], the filling operation was accelerated 214.8-fold by using 16 nodes × 16 cores. We parallelized the partitioning operation in our previous study [11] using the taskparallel language Tascell [12]. However, this approach can only work on SMSs.…”

Analysis of Parallelization Strategies in the context of Hierarchical Matrix Factorizations

“…Chameleon [2,3,38] is originally a dense linear algebra package, and this dissertation enhances it with an extension that allows operating with hierarchical structures (particularly, Tile H-Matrices), thanks to including low-rank arithmetic from the Hmat-oss package. This package utilizes StarPU [17,110] More recent works [21] rely on alternative programming models (particularly, Cilk [45] and Tascell [62]). In case those programming models are employed to parallelize the H-LU, it would be worth it to compare its efficiency with the one achieved by the parallel H-LU from H2Lib or H-Chameleon proposed in this dissertation.…”