GSoFa: Scalable Sparse LU Symbolic Factorization on GPUs
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Decomposing a matrix A into a lower matrix L and an upper matrix U, which is also known as LU decomposition, is an important operation in numerical linear algebra. For a sparse matrix, LU decomposition often introduces more nonzero entries in the L and U factors than the original matrix. Symbolic factorization step is needed to identify the nonzero structures of L and U matrices. Attracted by the enormous potentials of Graphics Processing Units (GPUs), an array of efforts has surged to deploy various steps of LU factorization on GPUs except, to the best of our knowledge, symbolic factorization.This paper introduces GSoFa, a 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 propose multi-source concurrent symbolic factorization to improve the utilization of GPUs with focus on balancing the workload. Third, we introduce a three-pronged optimization to reduce the excessive space requirement faced by multi-source concurrent symbolic factorization. Taken together, this work scales LU symbolic factorization towards 1,000 GPUs with superior performance over the state-of-the-art CPU algorithm.
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