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

This paper introduces sTiles, a GPU-accelerated software framework that computes factorization of sparse structured symmetric matrices. By leveraging tile algorithms to achieve fine-grained computations, sTiles employs a structure-aware flow of task executions for the Cholesky factorization to tackle challenging arrowhead sparse matrices with variable bandwidths. These matrices, common in various scientific and engineering fields, necessitate an adaptive factorization strategy that operates only on nonzero tiles. Our approach relies first on a collection of permutation techniques to minimize fill-in during factorization. Then, sTiles proceeds with a customized static scheduler to orchestrate computational tasks on shared-memory systems equipped with GPU hardware accelerators. sTiles strikes the required balance between the tile size and the degree of parallelism. The former has a direct impact on the algorithmic intensity, although monitoring the additional floating-point operations and memory footprint becomes crucial for parallel performance. The latter may be limited due to the inherent arrowhead sparse matrix structure. To further expose parallelism, we must adopt a left-looking variant of the Cholesky factorization to break the sequential dependencies during the accumulation operations on the trailing submatrix and operate on them instead using tree reductions. Extensive evaluations demonstrate a boost in performance for the sparse Cholesky factorization on various arrowhead structured matrices against the state-of-the-art sparse libraries, with sTiles achieving up to 8.41X/9.34X/5.07X/11.08X speedups compared to CHOLMOD/SymPACK/MUMPS/PARDISO, respectively. On GPUs, sTiles further exploits the computational throughput by adjusting the two key ingredients aforementioned to attain a 5X speedup compared to a 32-core AMD EPYC CPU when executed on an NVIDIA A100 GPU. Our generic software framework¹¹The software is available at https://github.com/esmail-abdulfattah/sTiles 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.