Locality-aware parallel block-sparse matrix-matrix multiplication using the Chunks and Tasks programming model

TL;DR

This paper introduces a scalable parallel method for block-sparse matrix multiplication on distributed systems, leveraging a quadtree representation and the Chunks and Tasks model to optimize data locality and reduce communication.

Contribution

The authors develop a locality-aware parallel multiplication approach using quadtree matrices and the Chunks and Tasks model, enabling efficient scaling and dynamic sparsity detection.

Findings

Achieves favorable weak and strong scaling of communication costs.

Reduces communication significantly compared to non-locality-aware methods.

Effectively utilizes CPUs and GPUs for leaf-level computations.

Abstract

We present a method for parallel block-sparse matrix-matrix multiplication on distributed memory clusters. By using a quadtree matrix representation, data locality is exploited without prior information about the matrix sparsity pattern. A distributed quadtree matrix representation is straightforward to implement due to our recent development of the Chunks and Tasks programming model [Parallel Comput. 40, 328 (2014)]. The quadtree representation combined with the Chunks and Tasks model leads to favorable weak and strong scaling of the communication cost with the number of processes, as shown both theoretically and in numerical experiments. Matrices are represented by sparse quadtrees of chunk objects. The leaves in the hierarchy are block-sparse submatrices. Sparsity is dynamically detected by the matrix library and may occur at any level in the hierarchy and/or within the submatrix leaves. In case graphics processing units (GPUs) are available, both CPUs and GPUs are used for leaf-level multiplication work, thus making use of the full computing capacity of each node. The performance is evaluated for matrices with different sparsity structures, including examples from electronic structure calculations. Compared to methods that do not exploit data locality, our locality-aware approach reduces communication significantly, achieving essentially constant communication per node in weak scaling tests.