A Structure-Aware Irregular Blocking Method for Sparse LU Factorization

TL;DR

This paper introduces a structure-aware irregular blocking method for sparse LU factorization that adapts block sizes based on local nonzero distribution, significantly improving computational efficiency on GPU architectures.

Contribution

It proposes a novel diagonal block-based feature and an irregular blocking strategy that balances workload by adjusting block sizes according to local matrix structure.

Findings

Achieves 1.50x and 3.32x speedup over PanguLU and SuperLU_DIST on a single GPU.

Achieves 1.40x and 3.84x speedup on 4 GPUs.

Effectively balances workload across blocks in sparse matrices.

Abstract

In sparse LU factorization, nonzero elements after symbolic factorization tend to distribute in diagonal and right-bottom region of sparse matrices. However, regular 2D blocking on this non-uniform distribution structure may lead to workload imbalance across blocks. Besides, existing matrix features fail to guide us effectively in blocking. In this paper, we propose a structure-aware irregular blocking method for numerical factorization. A novel diagonal block-based feature is introduced to effectively characterize the local nonzero distribution of sparse matrices. Based on this, we further propose an irregular blocking method that adjusts block sizes according to the local distribution of nonzeros. The strategy utilizes fine-grained blocks in dense regions and coarse-grained blocks in sparse regions, adequately balancing the nonzeros of blocks both within the same level and across levels in the dependency tree. Experiments demonstrate that, on a single NVIDIA A100 GPU, our proposed irregular blocking method achieves average speedups of 1.50x and 3.32x over PanguLU and the latest SuperLU_DIST, respectively. In addition, it achieves speedups of 1.40x and 3.84x over PanguLU and SuperLU_DIST on 4 NVIDIA A100 GPUs.