CapelliniSpTRSV: A Thread-Level Synchronization-Free Sparse Triangular Solve on GPUs

Sparse triangular solves (SpTRSVs) have been extensively used in linear algebra fields, and many GPU-based SpTRSV algorithms have been proposed. Synchronization-free SpTRSVs, due to their short preprocessing time and high performance, are currently the most popular SpTRSV algorithms. However, we observe that the performance of those SpTRSV algorithms on different matrices can vary greatly by 845 times. Our further studies show that when the average number of components per level is high and the average number of nonzero elements per row is low, those SpTRSVs exhibit extremely low performance. The reason is that, they use a warp on the GPU to process a row in sparse matrices, and such warp-level designs have severe underutilization of the GPU. To solve this problem, we propose CapelliniSpTRSV, a thread-level synchronization-free SpTRSV algorithm. Particularly, CapelliniSpTRSV has three novel features. First, unlike the previous studies, CapelliniSpTRSV does not need preprocessing to calculate levels. Second, CapelliniSpTRSV exhibits high performance on matrices that previous SpTRSVs cannot handle efficiently. Third, CapelliniSpTRSV’s optimization does not rely on specific sparse matrix storage format. Instead, it can achieve very good performance on the most popular sparse matrix storage, compressed sparse row (CSR) format, and thus users do not need to conduct format conversion. We evaluate CapelliniSpTRSV with 245 matrices from the Florida Sparse Matrix Collection on three GPU platforms, and experiments show that our SpTRSV exhibits 6.84 GFLOPS/s, which is 4.97x speedup over the state-of-the-art synchronization-free SpTRSV algorithm, and 4.74x speedup over the SpTRSV in cuSPARSE. CapelliniSpTRSV is open-sourced in https://github.com/JiyaSu/CapelliniSpTRSV.