- \n
- 93% parallel efficiency for the non-SIMD code, and<\/li>\n
- 54% for the SIMD-optimized version (which is 2x faster).<\/li>\n<\/ul>\n
The plot in Figure 3<\/strong> shows\u00a0MPI strong scaling from 1 to 2,048 processes, and timing breakdown of the different kernels, tree construction and communications. Test problem: N=108<\/sup> points placed at random in a cube; FMM with order p=3. Calculation time is multiplied by the number of processes, so that equal bar heights would indicate perfect scaling.<\/p>\n
![\"MPI](\"\/exafmm\/files\/2011\/11\/strong_sm.png\" "\\"strong_sm\\"")
<\/a>Figure 3 \u2014MPI strong scaling from 1 to 2,048 processes, and timing breakdown of the different kernels, tree construction and communications. Parallel efficiency is 93% on 2,048 processes. \u00a9 2011 R Yokota, L Barba.<\/figcaption><\/figure>\n <\/p>\n
Weak scaling with 106<\/sup> particles per node achieved 72% efficiency on 32,768 processes of the Kraken supercomputer. The plot in Figure 4 shows\u00a0MPI weak scaling with (SIMD optimizations) from 1 to 32,768 processes, and timing breakdown of the different kernels, tree construction and communications. Test problem: N=106 points per process placed at random in a cube; FMM with order p=3.<\/p>\n
![\"MPI](\"\/exafmm\/files\/2011\/11\/weak_sm.png\" "\\"weak_sm\\"")
<\/a>Figure 4\u2014MPI weak scaling with (SIMD optimizations) from 1 to 32,768 processes. Parallel efficiency is 72% on 32,768 processes. \u00a9 2011 R Yokota, L Barba.<\/figcaption><\/figure>\n <\/p>\n
The results above are detailed in the following publication:<\/p>\n
“A tuned and scalable fast multipole method as a preeminent algorithm for exascale systems”, Rio Yokota and Lorena A Barba,\u00a0Int. J. High-perf. Comput.,<\/em> online 24 Jan. 2012,\u00a0doi:10.1177\/1094342011429952<\/a> \u2014 <\/em>Preprint:\u00a0arXiv:1106.2176<\/a><\/p><\/blockquote>\n
<\/p>\n
Weak scaling on GPU systems<\/h2>\n
The ExaFMM<\/em> code scales excellently to thousands of GPUs. We studied scalability on the Tsubame 2.0 supercomputer<\/a> of Tokyo Institute of Technology (thanks to guest access). A timing breakdown is shown below, on up to 2048 processes, where the communication time is seen to be minor.<\/p>\n
![\"Timing](\"\/exafmm\/files\/2011\/11\/breakdownGPU.png\" "\\"breakdownGPU\\"")
<\/a>Figure 5\u2014Timing breakdown and scalability of ExaFMM on many GPUs.<\/figcaption><\/figure>\n Parallel efficiency of ExaFMM<\/em> on a weak scaling test in Tsubame 2.0 achieved more than 70% on 4096 processes (with GPUs). A similar test of a parallel FFT (without GPUs) showed a dramatic degradation of efficiency at this number of processes.<\/p>\n
![\"Parallel](\"\/exafmm\/files\/2011\/11\/weakGPU.png\" "\\"weakGPU\\"")
<\/a>Figure 6\u2014Parallel efficiency of the FMM on a weak scaling test on Tsubame 2.0 (with GPUs), and of a parallel FFT (without GPUs) on up to 4096 processes.<\/figcaption><\/figure>\n Cite this figure:<\/p>\n