GSoFa: Scalable Sparse Symbolic LU Factorization on GPUs

TL;DR

This paper presents gSoFa, a GPU-based symbolic LU factorization method that significantly accelerates sparse matrix decomposition, achieving up to 31x speedup and better memory efficiency compared to CPU approaches.

Contribution

gSoFa introduces the first GPU-optimized symbolic LU factorization algorithm with novel parallelization, workload balancing, and space reduction techniques for sparse matrices.

Findings

Up to 31x speedup on Summit supercomputers.

Outperforms state-of-the-art CPU methods by 5x on average.

Achieves 47% of peak memory throughput of V100 GPUs.

Abstract

Decomposing matrix A into a lower matrix L and an upper matrix U, which is also known as LU decomposition, is an essential operation in numerical linear algebra. For a sparse matrix, LU decomposition often introduces more nonzero entries in the L and U factors than in the original matrix. A symbolic factorization step is needed to identify the nonzero structures of L and U matrices. Attracted by the enormous potentials of the Graphics Processing Units (GPUs), an array of efforts have surged to deploy various LU factorization steps except for the symbolic factorization, to the best of our knowledge, on GPUs. This paper introduces gSoFa, the first GPU-based Symbolic factorization design with the following three optimizations to enable scalable LU symbolic factorization for nonsymmetric pattern sparse matrices on GPUs. First, we introduce a novel fine-grained parallel symbolic factorization algorithm that is well suited for the Single Instruction Multiple Thread (SIMT) architecture of GPUs. Second, we tailor supernode detection into a SIMT friendly process and strive to balance the workload, minimize the communication and saturate the GPU computing resources during supernode detection. Third, we introduce a three-pronged optimization to reduce the excessive space consumption problem faced by multi-source concurrent symbolic factorization. Taken together, gSoFa achieves up to 31x speedup from 1 to 44 Summit nodes (6 to 264 GPUs) and outperforms the state-of-the-art CPU project, on average, by 5x. Notably, gSoFa also achieves {up to 47%} of the peak memory throughput of a V100 GPU in Summit.