Compressed basis GMRES on high-performance graphics processing units
Compressed basis GMRES on high-performance graphics processing units
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高性能图形处理单元上的压缩基础 GMRES
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
A. Tomás
中科院分区：
文献类型：
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作者：
J. Aliaga;H. Anzt;Thomas Grützmacher;E. Quintana;A. Tomás
Krylov方法为许多大型稀疏线性系统的迭代求解提供了一种快速、高度并行的数值工具。在很大程度上，这些方法的实际实现的性能受到当前计算机体系结构中的通信带宽的限制，这促使人们研究复杂的技术，以避免、减少和/或隐藏(在分布式平台中)消息传递成本和(在所有体系结构中)的存储器访问。本文利用Ginkgo的内存访问器，以便将通信减少策略集成到(Krylov)GMRES解算器中，该策略将正交基的存储格式(即，内存中的数据表示)与使用该基进行运算时使用的算术精度分离。假设GMRES解算器的执行时间在很大程度上由存储器访问决定，则数据类型转换的成本可以被大部分隐藏，从而通过减少从存储器检索的位量来加速迭代步骤。再加上正交基的特殊性质(其元素都以1为界)，这为积极定制存储格式铺平了道路，其中包括一些对迭代过程的收敛影响较小的浮点和定点格式。我们使用SuiteSparse矩阵集合中的大量测试问题，开发了银杏稀疏线性代数库中“压缩基GMRES”求解器的高性能实现。与以IEEE双精度存储所有数据的标准GMRES解算器相比，现代NVIDIA V100图形处理器(GPU)具有高达50%的稳健性和性能优势。
Krylov methods provide a fast and highly parallel numerical tool for the iterative solution of many large-scale sparse linear systems. To a large extent, the performance of practical realizations of these methods is constrained by the communication bandwidth in current computer architectures, motivating the investigation of sophisticated techniques to avoid, reduce, and/or hide the message-passing costs (in distributed platforms) and the memory accesses (in all architectures). This article leverages Ginkgo’s memory accessor in order to integrate a communication-reduction strategy into the (Krylov) GMRES solver that decouples the storage format (i.e., the data representation in memory) of the orthogonal basis from the arithmetic precision that is employed during the operations with that basis. Given that the execution time of the GMRES solver is largely determined by the memory accesses, the cost of the datatype transforms can be mostly hidden, resulting in the acceleration of the iterative step via a decrease in the volume of bits being retrieved from memory. Together with the special properties of the orthonormal basis (whose elements are all bounded by 1), this paves the road toward the aggressive customization of the storage format, which includes some floating-point as well as fixed-point formats with mild impact on the convergence of the iterative process. We develop a high-performance implementation of the “compressed basis GMRES” solver in the Ginkgo sparse linear algebra library using a large set of test problems from the SuiteSparse Matrix Collection. We demonstrate robustness and performance advantages on a modern NVIDIA V100 graphics processing unit (GPU) of up to 50% over the standard GMRES solver that stores all data in IEEE double-precision.