Locality Aware Parallel Sparse Matrix Vector and Matrix Transpose Vector Multiplication on Many Core Processors
Sparse matrix vector and matrix transpose vector multiplication (SpMMTV) repeatedly performed as z ATx and y A z (or y A w) for the same sparse matrix A is a kernel operation widely used in various iterative solvers. One important optimization for serial SpMMTV is reusing A matrix nonzeros, which halves the memory bandwidth requirement. However, thread level parallelization of SpMMTV that reuses A matrix nonzeros necessitates concurrent writes to the same output vector entries. These concurrent writes can be handled in two ways: via atomic updates or thread local temporary output vectors that will undergo a reduction operation, both of which are not efficient or scalable on processors with many cores and complicated cache coherency protocols. In this work, we identify five quality criteria for efficient and scalable thread level parallelization of SpMMTV that utilizes one dimensional (1D) matrix partitioning. We also propose two locality aware 1D partitioning methods, which achieve reusing A matrix nonzeros and intermediate z vector entries; exploiting locality in accessing x y and z vector entries; and reducing the number of concurrent writes to the same output vector entries. These two methods utilize rowwise and columnwise singly bordered block diagonal (SB) forms of A. We evaluate the validity of our methods on a wide range of sparse matrices. Experiments on the 60 core cache coherent Intel Xeon Phi processor show the validity of the identified quality criteria and the validity of the proposed methods in practice. The results also show that the performance improvement from reusing A matrix nonzeros compensates for the overhead of concurrent writes through the proposed SB based methods.
IEEE Transactions on Parallel and Distributed Systems