基于申威众核处理器的1、2级BLAS函数优化研究

BLAS （Basic Linear Algebra Subprograms）是一个以向量和矩阵为操作对象的基础函数库.该库中函数分为3个级别,各个级别分别提供了向量-向量（1级）、向量-矩阵（2级）、矩阵-矩阵（3级）之间的基本运算.本文研究如何在申威众核处理器上BLAS-1、2级函数的并行实现,并充分利用平台特性对它们进行深度的性能调优,归纳总结程序在申威平台上的并行实现与优化技巧.申威26010 CPU采用了异构众核架构,众多计算核心提供的大规模并行处理能力,使单块芯片具有3 TFLOPS的双精度浮点计算性能.实验结果显示BLAS-1、2级函数相对于GotoBLAS参考实现版的平均加速比分别高达11.x和6.x,对于每一优化手段,均有明显的性能加速.     

面向SW26010-Pro的1、2级BLAS函数众核并行优化技术

BLAS (basic linear algebra subprograms)是高性能扩展数学库的一个重要模块,广泛应用于科学与工程计算领域. BLAS 1级提供向量-向量运算, BLAS 2级提供矩阵-向量运算.针对国产SW26010-Pro众核处理器设计并实现了高性能BLAS 1、2级函数.基于RMA通信机制设计了从核归约策略,提升了BLAS 1、2级若干函数的归约效率.针对TRSV、TPSV等存在数据依赖关系的函数,提出了一套高效并行算法,该算法通过点对点同步维持数据依赖关系,设计了适用于三角矩阵的高效任务映射机制,有效减少了从核点对点同步的次数,提高了函数的执行效率.通过自适应优化、向量压缩、数据复用等技术,进一步提升了BLAS 1、2级函数的访存带宽利用率.实验结果显示, BLAS 1级函数的访存带宽利用率最高可达95%,平均可达90%以上, BLAS 2级函数的访存带宽利用率最高可达98%,平均可达80%以上.与广泛使用的开源数学库GotoBLAS相比, BLAS 1、2级函数分别取得了平均18.78倍和25.96倍的加速效果. LU分解、QR分解以及对称特征值问题通过调用...     

国产SW26010-Pro处理器上3级BLAS函数众核并行优化

BLAS (basic linear algebra subprograms)是最基本、最重要的底层数学库之一.在一个标准的BLAS库中,BLAS 3级函数涵盖的矩阵-矩阵运算尤为重要,在许多大规模科学与工程计算应用中被广泛调用.另外, BLAS 3级属于计算密集型函数,对充分发挥处理器的计算性能有至关重要的作用.针对国产SW26010-Pro处理器研究BLAS 3级函数的众核并行优化技术.具体而言,根据SW26010-Pro的存储层次结构,设计多级分块算法,挖掘矩阵运算的并行性.在此基础上,基于远程内存访问(remote memory access, RMA)机制设计数据共享策略,提高从核间的数据传输效率.进一步地,采用三缓冲、参数调优等方法对算法进行全面优化,隐藏直接内存访问(direct memory access, DMA)访存开销和RMA通信开销.此外,利用SW26010-Pro的两条硬件流水线和若干向量化计算/访存指令,还对BLAS 3级函数的矩阵-矩阵乘法、矩阵方程组求解、矩阵转置操作等若干运算进行手工汇编优化,提高了函数的浮点计算效率.实验结果显示,所提出的并行优化技术...     

多核龙芯3A上二级BLAS库的优化

针对龙芯3A体系结构以及二级BLAS库函数的特点,在指令级、存储级和线程级抽取并行方案,总结了一些合适的优化方法,并对其进行了定量的分析.实验表明,这些优化可以将二级BLAS函数单线程的性能提升20%以上,多线程下也可以得到2.5倍左右的加速比,这对今后多核龙芯上的系统软件优化工作有着一定的帮助.     

基于申威1621的通用矩阵向量乘法的性能分析与优化

通用矩阵向量乘法（GEMV）函数是整个二级基础线性代数子程序（BLAS）函数库的构建基础,BLAS作为关键基础计算软件之一,目前在申威处理器上却没有一个高性能实现的版本。针对上述问题,为充分发挥申威1621平台的高性能BLAS库计算优势,提出一种基于申威1621的通用矩阵向量乘法的性能分析与优化方法。首先对GEMV函数进行计算重排序、循环分块的改进;然后采取单指令多数据流（SIMD）以及指令重排的优化方式;最后对内存分配方式进行择优选择。测试结果表明,GEMV函数平均性能达到GotoBLAS版的2.17倍。在使用堆栈分配内存空间或增加对y向量步长的判断分支两种方案后,相较于GotoBLAS,小规模矩阵的平均性能由2.265倍提升至2.875倍。为提高大规模矩阵的性能,以及发挥申威1621多核处理器并行机制,在开启4线程后,平均性能达到单核的3.57倍。因此,优化后的GEMV函数在申威平台上较好的体现了并行效果。     

基于申威1600的3级BLAS GEMM函数优化

BLAS是当前科学计算领域重要的底层支持数学库之一,其中的3级BLAS函数应用最为广泛.本文基于国产申威1600平台,提出了一种基础线性代数库BLAS的三级函数通用矩阵乘GEMM的高性能实现方法.在单核上,使用乘加指令、循环展开、软件流水线指令重排、SIMD向量化运算、寄存器分块技术等与平台架构相关的技术手段,实现汇编级手工优化;在多核上,提出了适用于该平台的多线程加速方案.实验结果显示,在单核串行性能测试中,与知名开源数学库GotoBLAS相比,我们实现了平均4.72倍的加速效果;在多核并行扩展测试中,4线程版的性能则平均达到了单线程版性能的3.02倍.     

BLAS 库在多核处理器上的性能测试与分析

BLAS 库是高性能计算中最基本的数学库,它的性能对超级计算机的性能有着极大的影响.而且随着CPU多核化的发展,BLAS 的多核并行性能已经变得比与体系结构相关的单核性能更加重要.实验以流行于高性能计算的Xeon、Opteron 系列多核X86 处理器为例,全面测试了GotoBLAS、Atlas、MKL 和ACML 四种主流的BLAS 库的所有1,2,3 级函数,并覆盖了不同计算规模和多核并行方面的测试.通过测试结果,分析源代码、BLAS 库资料和论文的方式,分析BLAS 有效的优化和并行方法,以及它们所适合的平台.为BLAS 的优化、使用,甚至高性能处理器的发展上提供有益的建议.实验结果表明,比起一个逻辑处理强大但是复杂的处理器,一个cache 更大、性能更好,内存带宽更宽、延迟更小,主频更高的处理器往往能在高性能计算中取得更好的性能.同时,X86 平台上的状况对其他体系结构也有巨大的借鉴意义.     

异构HPL算法中CPU端高性能BLAS库优化

异构HPL（high-performance Linpack）效率的提高需要充分发挥加速部件和通用CPU计算能力,加速部件集成了更多的计算核心,负责主要的计算,通用CPU负责任务调度的同时也参与计算.在合理划分任务、平衡负载的前提下,优化CPU端计算性能对整体效率的提升尤为重要.针对具体平台体系结构特点对BLAS（basic linear algebra subprograms）函数进行优化往往可以更加充分地利用通用CPU计算能力,提高系统整体效率.BLIS（BLAS-like library instantiation software）算法库是开源的BLAS函数框架,具有易开发、易移植和模块化等优点.基于异构系统平台体系结构以及HPL算法特点,充分利用三级缓存、向量化指令和多线程并行等技术手段优化CPU端调用的各级BLAS函数,应用auto-tuning技术优化矩阵分块参数,从而形成了HygonBLIS算法库.与MKL相比,在异构环境下,HPL算法整体性能提高了11.8%.     

异构HPL算法中CPU端高性能BLAS库优化

异构HPL（High-performance Linpack）效率的提高需要充分发挥加速部件和通用CPU计算能力,加速部件集成了更多的计算核心,负责主要的计算,通用CPU负责任务调度的同时也参与计算.在合理划分任务,平衡负载的前提下,优化CPU端计算性能对整体效率的提升尤为重要.针对具体平台体系结构特点对BLAS（Basic linear Algebra Subprograms）函数进行优化往往可以更加充分的利用通用CPU计算能力,提高系统整体效率.BLIS（BLAS-like Library Instantiation Software）算法库是开源的BLAS函数框架,具有易开发、易移植和模块化等优点.本文基于异构系统平台体系结构以及HPL算法特点,充分利用三级缓存、向量化指令和多线程并行等技术手段优化CPU端调用的各级BLAS函数,应用auto-tuning技术优化矩阵分块参数,从而形成了HygonBLIS算法库,与MKL相比,异构环境下HPL整体性能提高了11.8%.     

基于申威1621的高精度点积算法实现与优化

点积函数是BLAS库中的一级基础函数,其被科学计算等领域广泛调用.由于浮点计算会引入舍入误差,现有BLAS库中双精度点积函数不足以满足某些应用领域的精度要求,因此需要高精度算法来实现更精确可靠的计算.在本文中,面向国产申威1621平台,在现有的BLAS库的基础上,新增高精度点积函数的实现接口,来满足应用的高精度需求.同时,对于高精度点积算法运用循环展开、访存优化、指令重排等优化策略,实现汇编级手工优化.实验结果显示,文中高精度点积算法的计算结果精度,近似达到了双精度点积的两倍,有效提升了原始算法精度.同时,在保证精度提升的基础上,文中优化后的高精度点积函数相比未优化前,平均性能加速比达到了1.61.     

Performance evaluation of kernel fusion BLAS routines on the GPU: iterative solvers as case study

Programmers usually implement iterative methods that solve partial differential equations by expressing them using a sequence of basic kernels from libraries optimized for the graphics processing unit (GPU). The global runtime of the resulting combination is often penalized by the smallest and most inefficient vector operations. To improve the GPU exploitation, we identify and analyze the potential kernels to be fused according to the data dependence, data type and size, and GPU resources. This paper provides an extensive analysis of the impact of fusing vector operations [level 1 of Basic Linear Algebra Subprograms (BLAS)] on the performance of the GPU. The experimental evaluation shows that this optimization provides noticeable improvement especially for kernels with lower memory requirements and on more modern GPUs. It is worth noting that the fused BLAS operations can be very useful to help programmers efficiently code iterative methods to solve large linear systems of equations for the GPU. Iterative methods such as biconjugate gradient method (BCG) are one of the examples that can benefit from this optimization strategy. Indeed, kernel fusion of vector routines makes the most efficient GPU implementation of BCG run between \(1.09\times \) and \(1.27\times \) faster on three GPUs of different characteristics.     

Computations with symmetric, positive definite and band matrices on a parallel vector processor

Computations involving symmetric, positive definite and band matrices are kernel operations in the numerical treatment of many models arising in science and engineering. It is desirable to achieve a high level of performance when such operations are to be carried out on a vector processor. If the operations are performed by rows or columns (as in the EXTENDED BLAS subroutines), then the loops are vectorized but the speed of computations, measured in Mflops, is not very high, because the arrays involved are normally short. Therefore the computations should be organized by diagonals. Furthermore, some special devices are to be applied in order to unrol the loops. Finally, one should be careful with the storage scheme. It is demonstrated that if (i) the computations are organized by diagonals, (ii) the main loops are unrolled and (iii) the storage scheme is such that the work with some zero-elements is avoided, then the speed of computations is nearly the same as that obtained in the computations with dense matrices. If a particular vector machine is in use (in our case a CRAY X-MP computer), then the speed can be increased further by (iv) coding some basic operations in machine language and (v) using the different processors of the vector computer in parallel. The efficiency of the exploitation of the special features of the particular computer that is to be used is also illustrated by numerical examples.

Kernel subroutines performing matrix-vector multiplications are described. Representative tests are used to demonstrate the efficiency of these kernels.     

Tridiagonalization of a dense symmetric matrix on multiple GPUs and its application to symmetric eigenvalue problems

For software to fully exploit the computing power of emerging heterogeneous computers, not only must the required computational kernels be optimized for the specific hardware architectures but also an effective scheduling scheme is needed to utilize the available heterogeneous computational units and to hide the communication between them. As a case study, we develop a static scheduling scheme for the tridiagonalization of a symmetric dense matrix on multicore CPUs with multiple graphics processing units (GPUs) on a single compute node. We then parallelize and optimize the Basic Linear Algebra Subroutines (BLAS)‐2 symmetric matrix‐vector multiplication, and the BLAS‐3 low rank symmetric matrix updates on the GPUs. We demonstrate the good scalability of these multi‐GPU BLAS kernels and the effectiveness of our scheduling scheme on twelve Intel Xeon processors and three NVIDIA GPUs. We then integrate our hybrid CPU‐GPU kernel into computational kernels at higher‐levels of software stacks, that is, a shared‐memory dense eigensolver and a distributed‐memory sparse eigensolver. Our experimental results show that our kernels greatly improve the performance of these higher‐level kernels, not only reducing the solution time but also enabling the solution of larger‐scale problems. Because such symmetric eigenvalue problems arise in many scientific and engineering simulations, our kernels could potentially lead to new scientific discoveries. Furthermore, these dense linear algebra algorithms present algorithmic characteristics that can be found in other algorithms. Hence, they are not only important computational kernels on their own but also useful testbeds to study the performance of the emerging computers and the effects of the various optimization techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Using Ginkgo's memory accessor for improving the accuracy of memory-bound low precision BLAS

The roofline model not only provides a powerful tool to relate an application's performance with the specific constraints imposed by the target hardware but also offers a graphic representation of the balance between memory access cost and compute throughput. In this work, we present a strategy to break up the tight coupling between the precision format used for arithmetic operations and the storage format employed for memory operations. (At a high level, this idea is equivalent to compressing/decompressing the data in registers before/after invoking store/load memory operations.) In practice, we demonstrate that a “memory accessor” that hides the data compression behind the memory access, can virtually push the bandwidth-induced roofline, yielding higher performance for memory-bound applications using high precision arithmetic that can handle the numerical effects associated with lossy compression. We also demonstrate that memory-bound applications operating on low precision data can increase the accuracy by relying on the memory accessor to perform all arithmetic operations in high precision. In particular, we demonstrate that memory-bound BLAS operations (including the sparse matrix-vector product) can be re-engineered with the memory accessor and that the resulting accessor-enabled BLAS routines achieve lower rounding errors while delivering the same performance as the fast low precision BLAS.     

Minimizing development and maintenance costs in supporting persistently optimized BLAS

The Basic Linear Algebra Subprograms (BLAS) define one of the most heavily used performance‐critical APIs in scientific computing today. It has long been understood that the most important of these routines, the dense Level 3 BLAS, may be written efficiently given a highly optimized general matrix multiply routine. In this paper, however, we show that an even larger set of operations can be efficiently maintained using a much simpler matrix multiply kernel. Indeed, this is how our own project, ATLAS (which provides one of the most widely used BLAS implementations in use today), supports a large variety of performance‐critical routines. Copyright © 2004 John Wiley & Sons, Ltd.