Efficient Implementation of 3D Finite Difference Schemes on Recent Processor Architectures

Sammanfattning: Efficient Implementation of 3D Finite Difference Schemes on Recent Processors Abstract In this paper a solver is introduced that solves a problem set modelled by the Burgers equation using the finite difference method: forward in time and central in space. The solver is parallelized and optimized for Intel Xeon Phi 7120P as well as Intel Xeon E5-2699v3 processors to investigate differences in terms of performance between the two architectures. Optimized data access and layout have been implemented to ensure good cache utilization. Loop tiling strategies are used to adjust data access with respect to the L2 cache size. Compiler hints describing aligned memory access are used to support vectorization on both processors. Additionally, prefetching strategies and streaming stores have been evaluated for the Intel Xeon Phi. Parallelization was done using OpenMP and MPI. The parallelisation for native execution on Xeon Phi is based on OpenMP and yielded a raw performance of nearly 100 GFLOP/s, reaching a speedup of almost 50 at a 83\% parallel efficiency. An OpenMP implementation on the E5-2699v3 (Haswell) processors produced up to 292 GFLOP/s, reaching a speedup of almost 31 at a 85\% parallel efficiency. For comparison a mixed implementation using interleaved communications with computations reached 267 GFLOP/s at a speedup of 28 with a 87\% parallel efficiency. Running a pure MPI implementation on the PDC's Beskow supercomputer with 16 nodes yielded a total performance of 1450 GFLOP/s and for a larger problem set it yielded a total of 2325 GFLOP/s, reaching a speedup and parallel efficiency at resp. 170 and 33,3\% and 290 and 56\%. An analysis based on the roofline performance model shows that the computations were memory bound to the L2 cache bandwidth, suggesting good L2 cache utilization for both the Haswell and the Xeon Phi's architectures. Xeon Phi performance can probably be improved by also using MPI. Keeping technological progress for computational cores in the Haswell processor in mind for the comparison, both processors perform well. Improving the stencil computations to a more compiler friendly form might improve performance more, as the compiler can possibly optimize more for the target platform. The experiments on the Cray system Beskow showed an increased efficiency from 33,3\% to 56\% for the larger problem, illustrating good weak scaling. This suggests that problem sizes should increase accordingly for larger number of nodes in order to achieve high efficiency. Frederick Ceder