sTiles: An Accelerated Computational Framework for Sparse Factorizations of Structured Matrices

Abstract: This paper introduces sTiles, a GPU-accelerated framework for factorizing sparse structured symmetric matrices. By leveraging tile algorithms for fine-grained computations, sTiles uses a structure-aware task execution flow to handle challenging arrowhead sparse matrices with variable bandwidths, common in scientific and engineering fields. It minimizes fill-in during Cholesky factorization using permutation techniques and employs a static scheduler to manage tasks on shared-memory systems with GPU accelerators. sTiles balances tile size and parallelism, where larger tiles enhance algorithmic intensity but increase floating-point operations and memory usage, while parallelism is constrained by the arrowhead structure. To expose more parallelism, a left-looking Cholesky variant breaks sequential dependencies in trailing submatrix updates via tree reductions. Evaluations show sTiles achieves speedups of up to 8.41X, 9.34X, 5.07X, and 11.08X compared to CHOLMOD, SymPACK, MUMPS, and PARDISO, respectively, and a 5X speedup compared to a 32-core AMD EPYC CPU on an NVIDIA A100 GPU. Our generic software framework imports well-established concepts from dense matrix computations but they all require customizations in their deployments on hybrid architectures to best handle factorizations of sparse matrices with arrowhead structures.