Multi-level parallel multi-layer block reproducible summation algorithm

Reproducibility means getting the bitwise identical floating point results from multiple runs of the same program, which plays an essential role in debugging and correctness checking in many codes (Villa et al., 2009). However, in parallel computing environments, the combination of dynamic scheduling of parallel computing resources. Moreover, floating point nonassociativity leads to non-reproducible results. Demmel and Nguyen proposed a floating-point summation algorithm that is reproducible independent of the order of summation (Demmel and Nguye, 2013; 2015) and accurate by using the 1-Reduction technique. Our work combines their work with the multi-layer block technology proposed by Castaldo et al. (2009), designs the multi-level parallel multi-layer block reproducible summation algorithm (MLP_rsum), including SIMD, OpenMP, and MPI based on each layer of blocks, and then attains reproducible and expected accurate results with high performance. Numerical experiments show that our algorithm is more efficient than the reproducible summation function in ReproBLAS (2018). With SIMD optimization, our algorithm is 2.41, 2.85, and 3.44 times faster than ReproBLAS on the three ARM platforms. With OpenMP optimization, our algorithm obtains linear speedup, showing that our method applies to multi-core processors. Finally, with reproducible MPI reduction, our algorithm’s parallel efficiency is 76% at 32 nodes with 4 threads and 32 processes.