BLR-Krylov: A Single-GPU Iterative SpMM Framework with Communication Avoidance and Block Low-Rank Optimization

Sparse General Matrix Multiplication (SpMM) is a fundamental kernel in iterative scientific computing and graph neural network inference. On single-GPU platforms, iterative SpMM suffers from two major performance bottlenecks: excessive memory traffic caused by repeated loading of Krylov basis vectors, and low computational efficiency due to irregular sparsity patterns. To address these challenges, this paper proposes BLR-Krylov , an iterative SpMM optimization framework that integrates Communication-Avoiding Krylov (CA-Krylov) methods with a Block Low-Rank Sparse (BLR-Sparse) matrix format. BLR-Krylov first preprocesses sparse matrices into the BLR-Sparse format, where low-rank blocks are compressed through basis vector decomposition to reduce storage and arithmetic overhead. It then employs GPU-optimized matrix power kernels to batch-compute Krylov subspace basis vectors, effectively minimizing global memory accesses across iterations. Experimental results on multiple GPU architectures demonstrate that BLR-Krylov achieves a 4.0–6.0 × speedup over existing communication-avoiding approaches in iterative workloads, increases global memory bandwidth utilization to \(82\% \) – \(87\% \) , and improves tensor core utilization to \(85\% \) – \(92\% \) , while maintaining numerical accuracy within 1 × 10 − 6 . To facilitate reproducibility, partial source code is publicly available at https://github.com/19547035579zz-tech/BLR-Krylov.