A Framework for Efficient Execution of Matrix Computations
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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. -
Fast Singular Value Thresholding Without Singular Value Decomposition∗
METHODS AND APPLICATIONS OF ANALYSIS. c 2013 International Press Vol. 20, No. 4, pp. 335–352, December 2013 002 FAST SINGULAR VALUE THRESHOLDING WITHOUT SINGULAR VALUE DECOMPOSITION∗ JIAN-FENG CAI† AND STANLEY OSHER‡ Abstract. Singular value thresholding (SVT) is a basic subroutine in many popular numerical schemes for solving nuclear norm minimization that arises from low-rank matrix recovery prob- lems such as matrix completion. The conventional approach for SVT is first to find the singular value decomposition (SVD) and then to shrink the singular values. However, such an approach is time-consuming under some circumstances, especially when the rank of the resulting matrix is not significantly low compared to its dimension. In this paper, we propose a fast algorithm for directly computing SVT for general dense matrices without using SVDs. Our algorithm is based on matrix Newton iteration for matrix functions, and the convergence is theoretically guaranteed. Numerical experiments show that our proposed algorithm is more efficient than the SVD-based approaches for general dense matrices. Key words. Low rank matrix, nuclear norm minimization, matrix Newton iteration. AMS subject classifications. 65F30, 65K99. 1. Introduction. Singular value thresholding (SVT) introduced in [7] is a key subroutine in many popular numerical schemes (e.g. [7, 12, 13, 52, 54, 66]) for solving nuclear norm minimization that arises from low-rank matrix recovery problems such as matrix completion [13–15,60]. Let Y Rm×n be a given matrix, and Y = UΣV T be its singular value decomposition (SVD),∈ where U and V are orthonormal matrices and Σ = diag(σ1, σ2,...,σs) is the diagonal matrix with diagonals being the singular values of Y . -
A Simplex-Type Voronoi Algorithm Based on Short Vector Computations of Copositive Quadratic Forms
Simons workshop Lattices Geometry, Algorithms and Hardness Berkeley, February 21st 2020 A simplex-type Voronoi algorithm based on short vector computations of copositive quadratic forms Achill Schürmann (Universität Rostock) based on work with Mathieu Dutour Sikirić and Frank Vallentin ( for Q n positive definite ) Perfect Forms 2 S>0 DEF: min(Q)= min Q[x] is the arithmetical minimum • x n 0 2Z \{ } Q is uniquely determined by min(Q) and Q perfect • , n MinQ = x Z : Q[x]=min(Q) { 2 } V(Q)=cone xxt : x MinQ is Voronoi cone of Q • { 2 } (Voronoi cones are full dimensional if and only if Q is perfect!) THM: Voronoi cones give a polyhedral tessellation of n S>0 and there are only finitely many up to GL n ( Z ) -equivalence. Voronoi’s Reduction Theory n t GLn(Z) acts on >0 by Q U QU S 7! Georgy Voronoi (1868 – 1908) Task of a reduction theory is to provide a fundamental domain Voronoi’s algorithm gives a recipe for the construction of a complete list of such polyhedral cones up to GL n ( Z ) -equivalence Ryshkov Polyhedron The set of all positive definite quadratic forms / matrices with arithmeticalRyshkov minimum Polyhedra at least 1 is called Ryshkov polyhedron = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 ≥ 2 \{ } is a locally finite polyhedron • R is a locally finite polyhedron • R Vertices of are perfect forms Vertices of are• perfect R • R 1 n n ↵ (det(Q + ↵Q0)) is strictly concave on • 7! S>0 Voronoi’s algorithm Voronoi’s Algorithm Start with a perfect form Q 1. -
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. -
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 -
Combinatorial Optimization, Packing and Covering
Combinatorial Optimization: Packing and Covering G´erard Cornu´ejols Carnegie Mellon University July 2000 1 Preface The integer programming models known as set packing and set cov- ering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the spe- cial structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integer, thus solving the problem. Sometimes, both the linear programming relaxation and its dual have integer optimal solutions. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. Min-max theorems, polyhedral combinatorics and graph theory all come together in this rich area of discrete mathemat- ics. In addition to min-max and polyhedral results, some of the deepest results in this area come in two flavors: “excluded minor” results and “decomposition” results. In these notes, we present several of these beautiful results. Three chapters cover min-max and polyhedral re- sults. The next four cover excluded minor results. In the last three, we present decomposition results. We hope that these notes will encourage research on the many intriguing open questions that still remain. In particular, we state 18 conjectures. For each of these conjectures, we offer $5000 as an incentive for the first correct solution or refutation before December 2020. 2 Contents 1Clutters 7 1.1 MFMC Property and Idealness . 9 1.2 Blocker . 13 1.3 Examples .......................... 15 1.3.1 st-Cuts and st-Paths . 15 1.3.2 Two-Commodity Flows . 17 1.3.3 r-Cuts and r-Arborescences . -
Speeding up Spmv for Power-Law Graph Analytics by Enhancing Locality & Vectorization
Speeding Up SpMV for Power-Law Graph Analytics by Enhancing Locality & Vectorization Serif Yesil Azin Heidarshenas Adam Morrison Josep Torrellas Dept. of Computer Science Dept. of Computer Science Blavatnik School of Dept. of Computer Science University of Illinois at University of Illinois at Computer Science University of Illinois at Urbana-Champaign Urbana-Champaign Tel Aviv University Urbana-Champaign [email protected] [email protected] [email protected] [email protected] Abstract—Graph analytics applications often target large-scale data-dependent behavior of some accesses makes them hard web and social networks, which are typically power-law graphs. to predict and optimize for. As a result, SpMV on large power- Graph algorithms can often be recast as generalized Sparse law graphs becomes memory bound. Matrix-Vector multiplication (SpMV) operations, making SpMV optimization important for graph analytics. However, executing To address this challenge, previous work has focused on SpMV on large-scale power-law graphs results in highly irregular increasing SpMV’s Memory-Level Parallelism (MLP) using memory access patterns with poor cache utilization. Worse, we vectorization [9], [10] and/or on improving memory access find that existing SpMV locality and vectorization optimiza- locality by rearranging the order of computation. The main tions are largely ineffective on modern out-of-order (OOO) techniques for improving locality are binning [11], [12], which processors—they are not faster (or only marginally so) than the standard Compressed Sparse Row (CSR) SpMV implementation. translates indirect memory accesses into efficient sequential To improve performance for power-law graphs on modern accesses, and cache blocking [13], which processes the ma- OOO processors, we propose Locality-Aware Vectorization (LAV). -
Level-3 BLAS on Myriad Multi-Core Media-Processor
Level-3 BLAS on Myriad Multi-Core Media-Processor SoC Tomasz Szydzik1 Marius Farcas2 Valeriu Ohan2 David Moloney3 1Insititute for Applied Microelectronics, University of Las Palmas of Gran Canaria, 2Codecart, 3Movidius Ltd. BLAS-like Library Instantiation Subprograms (BLIS) The Myriad media-processor SoCs The Basic Linear Algebra Subprograms (BLAS) are a set of low-level subroutines that perform common linear algebra Myriad architecture prioritises power-efficient operation and area efficiency. In order to guarantee sustained high operations. BLIS is a software framework for instantiating high-performance BLAS-like dense linear algebra libraries. performance and minimise power the proprietary SHAVE (Streaming Hybrid Architecture Vector Engine) processor was BLIS[1] was chosen over GoTOBLAS, ATLAS, etc. due to its portable micro-kernel architecture and active user-base. developed. Data and Instructions reside in a shared Connection MatriX (CMX) memory block shared by all Shave processors. Data is moved between peripherals, processors and memory via a bank of software-controlled DMA engines. BLIS features Linear Algebra Myriad Acceleration DDR Controller 128/256MB LPDDR2/3 Stacked Die 128-bit AXI 128-bit AHB High-level LAPACK EIGEN In-house 128kB 2-way L2 cache (SHAVE) IISO C99 code with flexible BSD license. routines DDR Controller 1MB CMX SRAM ISupport for BLAS API calling conventions. 128-bit AXI 128-bit AHB 128-bit AHB BLIS 128-bit 64-bit 64-bit 256kB 2-way L2 cache (SHAVE) 32-bit CMX Instr CMX CMX Competitive performance [2]. Level Level Level Level APB x 8 SHAVEs I f Port Port Port 32kB LRAM 2MB CMX SRAM 0 1m 2 3 Subroutines IMulti-core friendly. -
Quadratic Forms and Their Applications
Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999.