基于GPU的半精度几何多重网格预处理算法
Accelerating Geometric Multigrid Preconditioning with Half-Precision Arithmetic on GPUs
摘要：
借助NVIDIA V100 GPU对半精度运算的硬件支持，高性能计算应用程序可以在适当的位置获得较低的精度，从而加快整体执行时间。在本文中，我们研究了一种混合精度几何多重网格方法来解决大型稀疏方程组源于椭圆偏微分方程离散。虽然最终的解决方案总是计算高精度的准确性，在较低的精度和残差缩放的多重网格预处理的迭代加密方法。我们比较FP 64基线泊松方程纯粹FP 16多重网格预处理和就业的FP 16-FP 32-FP 64组合在一个网格层次。虽然使用较低的精度几乎不会影响迭代计数，但由于减少了内存传输，求解器运行时间大大减少，并且整体求解器的速度提高了2.5倍。我们调查的性能与分层Roofline模型选择内核。
英文摘要：
With the hardware support for half-precision arithmetic on NVIDIA V100 GPUs, high-performance computing applications can benefit from lower precision at appropriate spots to speed up the overall execution time. In this paper, we investigate a mixed-precision geometric multigrid method to solve large sparse systems of equations stemming from discretization of elliptic PDEs. While the final solution is always computed with high-precision accuracy, an iterative refinement approach with multigrid preconditioning in lower precision and residuum scaling is employed. We compare the FP64 baseline for Poisson's equation to purely FP16 multigrid preconditioning and to the employment of FP16-FP32-FP64 combinations within a mesh hierarchy. While the iteration count is almost not affected by using lower accuracy, the solver runtime is considerably decreased due to the reduced memory transfer and a speedup of up to 2.5x is gained for the overall solver. We investigate the performance of selected kernels with the hierarchical Roofline model.
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