Solving a trillion unknowns per second with HPGMG on Sunway TaihuLight

Benchmarks for supercomputers are important tools, not only for evaluating and ranking modern supercomputers, but also for providing hints for future architecture design. As a new benchmark, HPGMG (high performance geometric multigrid) solves a linear equation set with a full geometric multi-grid algorithm. It involves computation on different scales, data movement with various volumes, global communication and neighbor communication with both large and small messages, etc., and is more correlated to real world applications than traditional benchmarks such as LINPACK. Therefore, it is desirable to examine how well HPGMG can perform on leadership supercomputers such as Sunway Taihulight. Sunway Taihulight, the No. 1 supercomputer in the Top 500 list from June 2016 to June 2018, which uses a specially designed many-core architecture SW26010, is of great interest to the community of high performance computing. With careful analysis and code design, we came up with an efficient implementation of HPGMG on SW26010 processors. We not only employed traditional optimization techniques such as 2.5D partitioning, double buffering, and collective data load, but also introduced a micro-benchmark to help with the choice of optimization direction and parameter tuning. Another contribution is that we proposed a new procedure for the major operations, by granulating and reordering the smooth function and the ghost exchange operation, leading to reduced memory copy and accelerated communication process. Our optimized implementation of HPGMG on Sunway TaihuLight achieved a ground-breaking performance of 1.036× 10^12 Degrees of Freedom per second at the finest level, which is No. 1 on the HPGMG list of Nov 2017.