P1673R3: a Free Function Linear Algebra Interface Based on the BLAS
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Linpack Evaluation on a Supercomputer with Heterogeneous Accelerators
Linpack Evaluation on a Supercomputer with Heterogeneous Accelerators Toshio Endo Akira Nukada Graduate School of Information Science and Engineering Global Scientific Information and Computing Center Tokyo Institute of Technology Tokyo Institute of Technology Tokyo, Japan Tokyo, Japan [email protected] [email protected] Satoshi Matsuoka Naoya Maruyama Global Scientific Information and Computing Center Global Scientific Information and Computing Center Tokyo Institute of Technology/National Institute of Informatics Tokyo Institute of Technology Tokyo, Japan Tokyo, Japan [email protected] [email protected] Abstract—We report Linpack benchmark results on the Roadrunner or other systems described above, it includes TSUBAME supercomputer, a large scale heterogeneous system two types of accelerators. This is due to incremental upgrade equipped with NVIDIA Tesla GPUs and ClearSpeed SIMD of the system, which has been the case in commodity CPU accelerators. With all of 10,480 Opteron cores, 640 Xeon cores, 648 ClearSpeed accelerators and 624 NVIDIA Tesla GPUs, clusters; they may have processors with different speeds as we have achieved 87.01TFlops, which is the third record as a result of incremental upgrade. In this paper, we present a heterogeneous system in the world. This paper describes a Linpack implementation and evaluation results on TSUB- careful tuning and load balancing method required to achieve AME with 10,480 Opteron cores, 624 Tesla GPUs and 648 this performance. On the other hand, since the peak speed is ClearSpeed accelerators. In the evaluation, we also used a 163 TFlops, the efficiency is 53%, which is lower than other systems. -
2 Homework Solutions 18.335 " Fall 2004
2 Homework Solutions 18.335 - Fall 2004 2.1 Count the number of ‡oating point operations required to compute the QR decomposition of an m-by-n matrix using (a) Householder re‡ectors (b) Givens rotations. 2 (a) See Trefethen p. 74-75. Answer: 2mn2 n3 ‡ops. 3 (b) Following the same procedure as in part (a) we get the same ‘volume’, 1 1 namely mn2 n3: The only di¤erence we have here comes from the 2 6 number of ‡opsrequired for calculating the Givens matrix. This operation requires 6 ‡ops (instead of 4 for the Householder re‡ectors) and hence in total we need 3mn2 n3 ‡ops. 2.2 Trefethen 5.4 Let the SVD of A = UV : Denote with vi the columns of V , ui the columns of U and i the singular values of A: We want to …nd x = (x1; x2) and such that: 0 A x x 1 = 1 A 0 x2 x2 This gives Ax2 = x1 and Ax1 = x2: Multiplying the 1st equation with A 2 and substitution of the 2nd equation gives AAx2 = x2: From this we may conclude that x2 is a left singular vector of A: The same can be done to see that x1 is a right singular vector of A: From this the 2m eigenvectors are found to be: 1 v x = i ; i = 1:::m p2 ui corresponding to the eigenvalues = i. Therefore we get the eigenvalue decomposition: 1 0 A 1 VV 0 1 VV = A 0 p2 U U 0 p2 U U 3 2.3 If A = R + uv, where R is upper triangular matrix and u and v are (column) vectors, describe an algorithm to compute the QR decomposition of A in (n2) time. -
A Fast Algorithm for the Recursive Calculation of Dominant Singular Subspaces N
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 218 (2008) 238–246 www.elsevier.com/locate/cam A fast algorithm for the recursive calculation of dominant singular subspaces N. Mastronardia,1, M. Van Barelb,∗,2, R. Vandebrilb,2 aIstituto per le Applicazioni del Calcolo, CNR, via Amendola122/D, 70126, Bari, Italy bDepartment of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium Received 26 September 2006 Abstract In many engineering applications it is required to compute the dominant subspace of a matrix A of dimension m × n, with m?n. Often the matrix A is produced incrementally, so all the columns are not available simultaneously. This problem arises, e.g., in image processing, where each column of the matrix A represents an image of a given sequence leading to a singular value decomposition-based compression [S. Chandrasekaran, B.S. Manjunath, Y.F. Wang, J. Winkeler, H. Zhang, An eigenspace update algorithm for image analysis, Graphical Models and Image Process. 59 (5) (1997) 321–332]. Furthermore, the so-called proper orthogonal decomposition approximation uses the left dominant subspace of a matrix A where a column consists of a time instance of the solution of an evolution equation, e.g., the flow field from a fluid dynamics simulation. Since these flow fields tend to be very large, only a small number can be stored efficiently during the simulation, and therefore an incremental approach is useful [P. Van Dooren, Gramian based model reduction of large-scale dynamical systems, in: Numerical Analysis 1999, Chapman & Hall, CRC Press, London, Boca Raton, FL, 2000, pp. -
ARM HPC Ecosystem
ARM HPC Ecosystem Darren Cepulis HPC Segment Manager ARM Business Segment Group HPC Forum, Santa Fe, NM 19th April 2017 © ARM 2017 ARM Collaboration for Exascale Programs United States Japan ARM is currently a participant in Fujitsu and RIKEN announced that the two Department of Energy funded Post-K system targeted at Exascale will pre-Exascale projects: Data be based on ARMv8 with new Scalable Movement Dominates and Fast Vector Extensions. Forward 2. European Union China Through FP7 and Horizon 2020, James Lin, vice director for the Center ARM has been involved in several of HPC at Shanghai Jiao Tong University funded pre-Exascale projects claims China will build three pre- including the Mont Blanc program Exascale prototypes to select the which deployed one of the first architecture for their Exascale system. ARM prototype HPC systems. The three prototypes are based on AMD, SunWei TaihuLight, and ARMv8. 2 © ARM 2017 ARM HPC deployments starting in 2H2017 Tw o recent announcements about ARM in HPC in Europe: 3 © ARM 2017 Japan Exascale slides from Fujitsu at ISC’16 4 © ARM 2017 Foundational SW Ecosystem for HPC . Linux OS’s – RedHat, SUSE, CENTOS, UBUNTU,… Commercial . Compilers – ARM, GNU, LLVM,… . Libraries – ARM, OpenBLAS, BLIS, ATLAS, FFTW… . Parallelism – OpenMP, OpenMPI, MVAPICH2,… . Debugging – Allinea, RW Totalview, GDB,… Open-source . Analysis – ARM, Allinea, HPCToolkit, TAU,… . Job schedulers – LSF, PBS Pro, SLURM,… . Cluster mgmt – Bright, CMU, warewulf,… Predictable Baseline 5 © ARM 2017 – now on ARM Functional Supported packages / components Areas OpenHPC defines a baseline. It is a community effort to Base OS RHEL/CentOS 7.1, SLES 12 provide a common, verified set of open source packages for Administrative Conman, Ganglia, Lmod, LosF, ORCM, Nagios, pdsh, HPC deployments Tools prun ARM’s participation: Provisioning Warewulf Resource Mgmt. -
Numerical Linear Algebra Revised February 15, 2010 4.1 the LU
Numerical Linear Algebra Revised February 15, 2010 4.1 The LU Decomposition The Elementary Matrices and badgauss In the previous chapters we talked a bit about the solving systems of the form Lx = b and Ux = b where L is lower triangular and U is upper triangular. In the exercises for Chapter 2 you were asked to write a program x=lusolver(L,U,b) which solves LUx = b using forsub and backsub. We now address the problem of representing a matrix A as a product of a lower triangular L and an upper triangular U: Recall our old friend badgauss. function B=badgauss(A) m=size(A,1); B=A; for i=1:m-1 for j=i+1:m a=-B(j,i)/B(i,i); B(j,:)=a*B(i,:)+B(j,:); end end The heart of badgauss is the elementary row operation of type 3: B(j,:)=a*B(i,:)+B(j,:); where a=-B(j,i)/B(i,i); Note also that the index j is greater than i since the loop is for j=i+1:m As we know from linear algebra, an elementary opreration of type 3 can be viewed as matrix multipli- cation EA where E is an elementary matrix of type 3. E looks just like the identity matrix, except that E(j; i) = a where j; i and a are as in the MATLAB code above. In particular, E is a lower triangular matrix, moreover the entries on the diagonal are 1's. We call such a matrix unit lower triangular. -
Massively Parallel Poisson and QR Factorization Solvers
Computers Math. Applic. Vol. 31, No. 4/5, pp. 19-26, 1996 Pergamon Copyright~)1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/96 $15.00 + 0.00 0898-122 ! (95)00212-X Massively Parallel Poisson and QR Factorization Solvers M. LUCK£ Institute for Control Theory and Robotics, Slovak Academy of Sciences DdbravskA cesta 9, 842 37 Bratislava, Slovak Republik utrrluck@savba, sk M. VAJTERSIC Institute of Informatics, Slovak Academy of Sciences DdbravskA cesta 9, 840 00 Bratislava, P.O. Box 56, Slovak Republic kaifmava©savba, sk E. VIKTORINOVA Institute for Control Theoryand Robotics, Slovak Academy of Sciences DdbravskA cesta 9, 842 37 Bratislava, Slovak Republik utrrevka@savba, sk Abstract--The paper brings a massively parallel Poisson solver for rectangle domain and parallel algorithms for computation of QR factorization of a dense matrix A by means of Householder re- flections and Givens rotations. The computer model under consideration is a SIMD mesh-connected toroidal n x n processor array. The Dirichlet problem is replaced by its finite-difference analog on an M x N (M + 1, N are powers of two) grid. The algorithm is composed of parallel fast sine transform and cyclic odd-even reduction blocks and runs in a fully parallel fashion. Its computational complexity is O(MN log L/n2), where L = max(M + 1, N). A parallel proposal of QI~ factorization by the Householder method zeros all subdiagonal elements in each column and updates all elements of the given submatrix in parallel. For the second method with Givens rotations, the parallel scheme of the Sameh and Kuck was chosen where the disjoint rotations can be computed simultaneously. -
Supermatrix: a Multithreaded Runtime Scheduling System for Algorithms-By-Blocks
SuperMatrix: A Multithreaded Runtime Scheduling System for Algorithms-by-Blocks Ernie Chan Field G. Van Zee Paolo Bientinesi Enrique S. Quintana-Ort´ı Robert van de Geijn Department of Computer Science Gregorio Quintana-Ort´ı Department of Computer Sciences Duke University Departamento de Ingenier´ıa y Ciencia de The University of Texas at Austin Durham, NC 27708 Computadores Austin, Texas 78712 [email protected] Universidad Jaume I {echan,field,rvdg}@cs.utexas.edu 12.071–Castellon,´ Spain {quintana,gquintan}@icc.uji.es Abstract which led to the adoption of out-of-order execution in many com- This paper describes SuperMatrix, a runtime system that paral- puter microarchitectures [32]. For the dense linear algebra opera- lelizes matrix operations for SMP and/or multi-core architectures. tions on which we will concentrate in this paper, many researchers We use this system to demonstrate how code described at a high in the early days of distributed-memory computing recognized that level of abstraction can achieve high performance on such archi- “compute-ahead” techniques could be used to improve parallelism. tectures while completely hiding the parallelism from the library However, the coding complexity required of such an effort proved programmer. The key insight entails viewing matrices hierarchi- too great for these techniques to gain wide acceptance. In fact, cally, consisting of blocks that serve as units of data where oper- compute-ahead optimizations are still absent from linear algebra ations over those blocks are treated as units of computation. The packages such as ScaLAPACK [12] and PLAPACK [34]. implementation transparently enqueues the required operations, in- Recently, there has been a flurry of interest in reviving the idea ternally tracking dependencies, and then executes the operations of compute-ahead [1, 25, 31]. -
Benchmark of C++ Libraries for Sparse Matrix Computation
Benchmark of C++ Libraries for Sparse Matrix Computation Georg Holzmann http://grh.mur.at email: [email protected] August 2007 This report presents benchmarks of C++ scientific computing libraries for small and medium size sparse matrices. Be warned: these benchmarks are very special- ized on a neural network like algorithm I had to implement. However, also the initialization time of sparse matrices and a matrix-vector multiplication was mea- sured, which might be of general interest. WARNING At the time of its writing this document did not consider the eigen library (http://eigen.tuxfamily.org). Please evaluate also this very nice, fast and well maintained library before making any decisions! See http://grh.mur.at/blog/matrix-library-benchmark-follow for more information. Contents 1 Introduction 2 2 Benchmarks 2 2.1 Initialization.....................................3 2.2 Matrix-Vector Multiplication............................4 2.3 Neural Network like Operation...........................4 3 Results 4 4 Conclusion 12 A Libraries and Flags 12 B Implementations 13 1 1 Introduction Quite a lot open-source libraries exist for scientific computing, which makes it hard to choose between them. Therefore, after communication on various mailing lists, I decided to perform some benchmarks, specialized to the kind of algorithms I had to implement. Ideally algorithms should be implemented in an object oriented, reuseable way, but of course without a loss of performance. BLAS (Basic Linear Algebra Subprograms - see [1]) and LA- PACK (Linear Algebra Package - see [3]) are the standard building blocks for efficient scientific software. Highly optimized BLAS implementations for all relevant hardware architectures are available, traditionally expressed in Fortran routines for scalar, vector and matrix operations (called Level 1, 2 and 3 BLAS). -
Using Machine Learning to Improve Dense and Sparse Matrix Multiplication Kernels
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2019 Using machine learning to improve dense and sparse matrix multiplication kernels Brandon Groth Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Applied Mathematics Commons, and the Computer Sciences Commons Recommended Citation Groth, Brandon, "Using machine learning to improve dense and sparse matrix multiplication kernels" (2019). Graduate Theses and Dissertations. 17688. https://lib.dr.iastate.edu/etd/17688 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Using machine learning to improve dense and sparse matrix multiplication kernels by Brandon Micheal Groth A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Applied Mathematics Program of Study Committee: Glenn R. Luecke, Major Professor James Rossmanith Zhijun Wu Jin Tian Kris De Brabanter The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2019 Copyright c Brandon Micheal Groth, 2019. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my wife Maria and our puppy Tiger. -
0 BLIS: a Framework for Rapidly Instantiating BLAS Functionality
0 BLIS: A Framework for Rapidly Instantiating BLAS Functionality FIELD G. VAN ZEE and ROBERT A. VAN DE GEIJN, The University of Texas at Austin The BLAS-like Library Instantiation Software (BLIS) framework is a new infrastructure for rapidly in- stantiating Basic Linear Algebra Subprograms (BLAS) functionality. Its fundamental innovation is that virtually all computation within level-2 (matrix-vector) and level-3 (matrix-matrix) BLAS operations can be expressed and optimized in terms of very simple kernels. While others have had similar insights, BLIS reduces the necessary kernels to what we believe is the simplest set that still supports the high performance that the computational science community demands. Higher-level framework code is generalized and imple- mented in ISO C99 so that it can be reused and/or re-parameterized for different operations (and different architectures) with little to no modification. Inserting high-performance kernels into the framework facili- tates the immediate optimization of any BLAS-like operations which are cast in terms of these kernels, and thus the framework acts as a productivity multiplier. Users of BLAS-dependent applications are given a choice of using the the traditional Fortran-77 BLAS interface, a generalized C interface, or any other higher level interface that builds upon this latter API. Preliminary performance of level-2 and level-3 operations is observed to be competitive with two mature open source libraries (OpenBLAS and ATLAS) as well as an established commercial product (Intel MKL). Categories and Subject Descriptors: G.4 [Mathematical Software]: Efficiency General Terms: Algorithms, Performance Additional Key Words and Phrases: linear algebra, libraries, high-performance, matrix, BLAS ACM Reference Format: ACM Trans. -
Anatomy of High-Performance Matrix Multiplication
Anatomy of High-Performance Matrix Multiplication KAZUSHIGE GOTO The University of Texas at Austin and ROBERT A. VAN DE GEIJN The University of Texas at Austin We present the basic principles which underlie the high-performance implementation of the matrix- matrix multiplication that is part of the widely used GotoBLAS library. Design decisions are justified by successively refining a model of architectures with multilevel memories. A simple but effective algorithm for executing this operation results. Implementations on a broad selection of architectures are shown to achieve near-peak performance. Categories and Subject Descriptors: G.4 [Mathematical Software]: —Efficiency General Terms: Algorithms;Performance Additional Key Words and Phrases: linear algebra, matrix multiplication, basic linear algebra subprogrms 1. INTRODUCTION Implementing matrix multiplication so that near-optimal performance is attained requires a thorough understanding of how the operation must be layered at the macro level in combination with careful engineering of high-performance kernels at the micro level. This paper primarily addresses the macro issues, namely how to exploit a high-performance “inner-kernel”, more so than the the micro issues related to the design and engineering of that “inner-kernel”. In [Gunnels et al. 2001] a layered approach to the implementation of matrix multiplication was reported. The approach was shown to optimally amortize the required movement of data between two adjacent memory layers of an architecture with a complex multi-level memory. Like other work in the area [Agarwal et al. 1994; Whaley et al. 2001], that paper ([Gunnels et al. 2001]) casts computation in terms of an “inner-kernel” that computes C := AB˜ + C for some mc × kc matrix A˜ that is stored contiguously in some packed format and fits in cache memory. -
DD2358 – Introduction to HPC Linear Algebra Libraries & BLAS
DD2358 – Introduction to HPC Linear Algebra Libraries & BLAS Stefano Markidis, KTH Royal Institute of Technology After this lecture, you will be able to • Understand the importance of numerical libraries in HPC • List a series of key numerical libraries including BLAS • Describe which kind of operations BLAS supports • Experiment with OpenBLAS and perform a matrix-matrix multiply using BLAS 2021-02-22 2 Numerical Libraries are the Foundation for Application Developments • While these applications are used in a wide variety of very different disciplines, their underlying computational algorithms are very similar to one another. • Application developers do not have to waste time redeveloping supercomputing software that has already been developed elsewhere. • Libraries targeting numerical linear algebra operations are the most common, given the ubiquity of linear algebra in scientific computing algorithms. 2021-02-22 3 Libraries are Tuned for Performance • Numerical libraries have been highly tuned for performance, often for more than a decade – It makes it difficult for the application developer to match a library’s performance using a homemade equivalent. • Because they are relatively easy to use and their highly tuned performance across a wide range of HPC platforms – The use of scientific computing libraries as software dependencies in computational science applications has become widespread. 2021-02-22 4 HPC Community Standards • Apart from acting as a repository for software reuse, libraries serve the important role of providing a