在LU分解中有效地利用了所有问题规模的并行尺度

Md Rakib Hasan, R. C. Whaley
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引用次数: 6

摘要

LU分解是求解线性方程最广泛使用的方法之一,因此它的性能是广泛的科学计算的基础。由于架构趋势已经用并行规模的增加取代了时钟速率的改进,库编写者已经通过使用平铺算法来应对,在平铺算法中,操作数的大小受到限制,以便最大化并行性,正如在著名的PLASMA库中所见。这种方法有两个主要缺点:(1)由于有限的操作数大小而降低了渐近性能;(2)由于并行缓存中不必要的数据移动而降低了中小型问题的性能。在本文中,我们介绍了一种新的方法,其中通过使用ATLAS框架自动生成的特殊低开销内核原语来最大化渐近性能,同时通过使用显式缓存管理来最小化不必要的缓存运动。我们表明,在商用并行英特尔和AMD平台上,这种技术可以在所有问题规模上优于所有已知的库,在12核英特尔至强处理器上的渐近LU性能大约为硬件理论峰值的91%,在32核AMD Opteron上的渐近LU性能为87%。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Effectively Exploiting Parallel Scale for All Problem Sizes in LU Factorization
LU factorization is one of the most widely-used methods for solving linear equations, and thus its performance underlies a broad range of scientific computing. As architectural trends have replaced clock rate improvements with increases in parallel scale, library writers have responded by using tiled algorithms, where operand size is constrained in order to maximize parallelism, as seen in the well-known PLASMA library. This approach has two main drawbacks: (1) asymptotic performance is reduced due to limited operand size, (2) performance of small to medium sized problems is reduced due to unnecessary data motion in the parallel caches. In this paper we introduce a new approach where asymptotic performance is maximized by using special low-overhead kernel primitives that are auto-generated by the ATLAS framework, while unnecessary cache motion is minimized by using explicit cache management. We show that this technique can outperform all known libraries at all problem sizes on commodity parallel Intel and AMD platforms, with asymptotic LU performance of roughly 91% of hardware theoretical peak for a 12-core Intel Xeon, and 87% for a 32-core AMD Opteron.
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