SHF:Small: Solving the Problem of Scalable Multi-Precision Matrix Arithmetic on GPUs

SHF:Small：解决 GPU 上可扩展多精度矩阵算术问题

• 批准号: 1217590
• 负责人: Charles Weems
• 金额: $ 45万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217590&HistoricalAwards=false
• 关键词: SHF; Small; Solving; Problem; Scalable

Computers directly support arithmetic that is typically limited to 64 bits (about 19 decimal digits) of precision. Applications that need more precision must implement arithmetic through computationally expensive software. Beyond about 256 bits of precision, such calculations become quite costly. The RSA encryption algorithm, for example, can require arithmetic with up to 4096 bits of precision. Applications in areas such as experimental mathematics and number theory can require millions of bits of precision. One multiplication with 10 million bits of precision can take a tenth of a second to compute on a modern processor, which means that matrix arithmetic using such large values can take days to weeks to execute. In previous work the investigators have shown that it is possible to obtain a factor of 20 improvement in performance by utilizing the parallel processing capabilities of a commodity graphics processing unit (GPU) in place of the traditional CPU. However, programming a GPU to achieve this level of performance is quite difficult, and the resulting code requires considerable hand-tuning to move it to new generations of GPU and gain the advantage of their performance, which is scaling up at a rate that exceeds CPU performance scaling.This project is working to develop a framework that automatically generates and tunes multi-precision arithmetic libraries to execute on successive generations of GPUs. The libraries include both scalar and basic matrix arithmetic routines. They support scaling in precision as well as matrix size. The problem is challenging because different parallel algorithms must be automatically selected for different levels of precision, which must be balanced with the exploitation of the alternate dimension of parallelism inherent in matrix arithmetic. In addition, the work seeks to employ distributed parallelism across a cluster of computers enhanced with GPUs, so that the libraries can be used on a new generation of GPU-based supercomputers that is beginning to be deployed at national laboratories. The work is significant because it enables easier exploitation of low-cost commodity graphics processors to achieve more than an order of magnitude increase in performance for multi-precision scalar and matrix arithmetic. One important application is enhancing performance of RSA encryption to support longer, more secure keys, at greater data rates, so that it becomes feasible to encrypt greater volumes of internet traffic. Another important use is experimental mathematics, where computationally expensive functions (e.g., integrals, infinite series) are computed at high precision and compared to other functions and high precision constants to help identify more efficient closed-form solutions. Results from experimental mathematics have found applications in particle physics, chaos theory, and calculation of fundamental constants. The resulting software framework offers a significant performance enhancement for multi-precision arithmetic to systems that range from individual researcher workstations to large supercomputers.

计算机直接支持通常限于64位（约19位十进制数）精度的算术。需要更高精度的应用程序必须通过计算昂贵的软件来实现算术。超过256位的精度，这种计算变得相当昂贵。例如，RSA加密算法可能需要高达4096位精度的算术。在实验数学和数论等领域的应用可能需要数百万位的精度。在现代处理器上，一次1000万位精度的乘法运算可能需要十分之一秒的时间，这意味着使用如此大的值的矩阵运算可能需要几天到几周的时间才能执行。在以前的工作中，调查人员已经表明，它是可能的，以获得一个因素的20提高性能，通过利用并行处理能力的商品图形处理单元（GPU）在传统的CPU。然而，对GPU进行编程以实现这种级别的性能是相当困难的，并且由此产生的代码需要大量的手动调整才能将其移动到新一代GPU并获得其性能的优势，该项目正在以超过CPU性能扩展的速度扩展。该项目正在开发一个框架，可以自动生成和调整多个精确的算术库在连续几代GPU上执行。这些库包括标量和基本矩阵运算例程。它们支持精度和矩阵大小的缩放。这个问题是具有挑战性的，因为不同的并行算法必须自动选择不同级别的精度，这必须与矩阵运算中固有的并行性的交替维度的开发相平衡。此外，这项工作还试图在一个由GPU增强的计算机集群中采用分布式并行，以便这些库可以在新一代基于GPU的超级计算机上使用，这些超级计算机正开始在国家实验室部署。这项工作是重要的，因为它可以更容易地利用低成本的商品图形处理器，以实现多精度标量和矩阵运算的性能提高一个数量级以上。一个重要的应用是增强RSA加密的性能，以更高的数据速率支持更长、更安全的密钥，从而可以加密更大量的互联网流量。另一个重要的用途是实验数学，其中计算昂贵的函数（例如，积分、无穷级数）以高精度计算，并与其他函数和高精度常数进行比较，以帮助识别更有效的闭合形式解。实验数学的结果在粒子物理学、混沌理论和基本常数的计算中得到了应用。由此产生的软件框架提供了一个显着的性能增强多精度算术系统，从个人研究工作站到大型超级计算机。