Load-Balancing-Aware Parallel Algorithms of H-Matrices with Adaptive Cross Approximation for GPUs

Abstract

Hierarchical matrices (H-matrices) are an approximation technique for dense matrices, such as the coefficient matrix of the boundary element method (BEM). An H-matrix is expressed by a set of low-rank approximated and small dense sub-matrices, each of which has various ranks. The use of H-matrices reduces the required memory footprint of dense matrices from O(N^2) to O(NlogN) and is suitable for many-core processors that have relatively small memory capacities compared to traditional CPUs. However, existing parallel adaptive cross approximation (ACA) algorithms, which are low-rank approximation algorithms used to construct H-matrices, are not designed to exploit many-core processors in terms of load balancing. In existing parallel algorithms, the ACA process is independently applied to each sub-matrix. The computational load of the ACA process for each sub-matrix depends on the sub-matrix's rank; however, the rank is defined after the ACA process is applied. This makes it difficult to balance the load. We propose load-balancing-aware parallel ACA algorithms for H-matrices that focus on many-core processors. We implemented the proposed algorithms into HACApK, which is an open-source H-matrix library originally developed for CPU-based clusters. The proposed algorithms were evaluated using BEM problems on an NVIDIA Tesla P100 GPU (P100) and an Intel Xeon Broadwell processor. The evaluation results demonstrate the improved performance of the proposed algorithms in all GPU cases. For example, in a case where it is difficult for existing parallel algorithms to balance the load, the proposed algorithms achieved a 12.9 times performance improvement for the P100.