Parallel Asynchronous Hybrid Krylov Methods for Minimization of Energy Consumption. Langshi CHEN 1,2,3 Supervised by Serge PETITON 2
|
|
- Aron Clarke
- 5 years ago
- Views:
Transcription
1 1 / 23 Parallel Asynchronous Hybrid Krylov Methods for Minimization of Energy Consumption Langshi CHEN 1,2,3 Supervised by Serge PETITON 2 Maison de la Simulation Lille 1 University CNRS March 18, 2013
2 2 / 23 Background Introduction Outline 1 Background Introduction 2 Energy efficiency in Krylov iterative methods 3 Current Work 4 Future Work
3 3 / 23 Background Introduction Petascale and Exascale Supercomputing High Performance Computing is shipping from Petaflops(10 15 ) era to Exaflops(10 18 ) era: Titan of Oak Ridge National Laborary Speed: PetaFLOPS (LINPACK) Power: 8.2MW Architecture:18,688 AMD Opteron core CPUs 18,688 Nvidia Tesla K20X GPUs Rank: 1 in Top500 November 2012
4 4 / 23 Background Introduction Problem of Energy Consumption Modern Supercomputer power: 4-6 Megawatts: electricity enough to supply 5000 homes Potential exascale computer: 1.5 Gigawatts: a nuclear sized power plant Green 500 List Evaluation of HPC via FLOPS per watt Energy efficiency: CPU, GPU, memory, disk, etc hardware design, algorithm, etc.
5 5 / 23 Background Introduction Energy efficieny in components of HPC Hardware: Processor, Memory, Disk,etc Algorithmic: float point computation data communication, etc Improve the energy efficiency of HPC in algorithmic aspects: Power aware programming (e.g. Dynamic Voltage Scale) Communication Avoiding (communicaiton consumes lots of energy). Auto-tuning methods, parameter optimization etc.
6 6 / 23 Energy efficiency in Krylov iterative methods Outline 1 Background Introduction 2 Energy efficiency in Krylov iterative methods 3 Current Work 4 Future Work
7 7 / 23 Energy efficiency in Krylov iterative methods Krylov iterative methods Iterative methods are widely used in solving: non linear equations large scale linear problems (order millions of variables) Krylov subspace K r (A, b) = span{b, Ab, A 2 b,..., A r 1 b} (1) Example: Conjugate Gradient, GMRES, etc
8 8 / 23 Energy efficiency in Krylov iterative methods Auto-tuning technology Optimization in runtime of parameters in Krylov methods Example: change the size r of Krylov subspace Smaller size: less time for orthogonalization K r (A, b) but more time for convergence Larger size: more time for orthogonalization, but faster of convergence Find the best r size dynamically to shorten the computation time. Also we will use the energy consumption as criterion of auto-tuning optimization
9 9 / 23 Energy efficiency in Krylov iterative methods Communication Avoiding Construction of Krylov space needs large part of sparse matrix dense vector multiplication (SpMV), which is heavily communication consumed. Especially when Large scale problem Parallel computing environment Communication Avoiding is a group of algorithms, which use redundant computation to reduce the communication data. Thus Shorten the total consumed time Improve the energy efficiency but it depends on structure of matrix like TSQR (Tall Skinny QR method), communication avoidng specially for dense matrix whose rows many more than columns.
10 10 / 23 Current Work Outline 1 Background Introduction 2 Energy efficiency in Krylov iterative methods 3 Current Work 4 Future Work
11 11 / 23 Current Work SpMV algorithms Sparse matrix-vector multiplication (SpMV) is a basic component in Krylov subspace construction. Choose different sparse matrix formats Evaluation of communication avoiding methods Energy consumption analysis Experimentation on MdS s machine Poincare: CPU, GPU mixed cluster (has 4 nodes) 2 Processor Sandy Bridge E per node 64 Go Memory per node 4 GPU Tesla K10 (Cuda Capability 3.0, 3.5 Go memory) per node
12 12 / 23 Current Work SpMV algorithms Codes originally written by Maxime Hugues and modified to test on Poincare. The input sparse matrix has two sources: 1 generated structured sparse matrix 1 Chosen number of continous diagonals above the main diagonal 2 Equidistributed diagonals 2 Unstructured sparse matrix from real industrial applicaitons.
13 13 / 23 Current Work SpMV algorithms The following steps are executed: 1 Generated sparse matrix A = [a ij ] n n 2 Doing Y = A r X, with iteration r 3 m MPI process divided A into m submatrix B = [b ij ] n/m n. 4 Each MPI process solvase its sub SpMV on its own binded GPU (CUDA 5.0). 5 MPI communication to forme the final Y (openmpi1.6.3)
14 Current Work Different Sparse matrix format Generated structured sparse matrix with single precision Gflops test on C-diagonal generated matrix CSR CSC Ell Ell-col Dimension row = col = Continous diagonal elements (15 diagonals above main diagonal) Gflops 8 Diagonal values = 1.(without perturbation). 4 Four different sparse matrix format: Nb of MPI Process CSR, CSC, Ellpack Row, Ellpack Col. 14 / 23
15 15 / 23 Current Work Different Sparse matrix format Both of them have bad scalabilities (communication cost) CSR is outperformed by the others when m is small, the difference is large when m augments, the difference is narrowing Results depends on Sparse matrix structure hardware environment
16 16 / 23 Current Work Communication Avoiding implementation Pre-compute the position of nonzero entries in Y 1 Record those column index of B where zero entries exists 2 These column index corresponds to the row index in Y whose entries values must be zero 3 These row index of Y is exclusived from communication Reduce the data movement between MPI process From O(n) to O(bnnz), where bnnz is the number of nonzero entries in Y.
17 17 / 23 Current Work Communication Avoiding implementation Gflops Gflops test on C-diagonal generated matrix Nb of MPI Process CSR CSR-avoid Generated structured sparse matrix with single precision Dimension row = col = Continous diagonal elements (15 diagonals above main diagonal) Diagonal values = 1.(without perturbation). Comparison between CSR and CSR communication avoiding in FLOPS
18 Current Work Communication Avoiding implementation MPITime Percent test on C-diagonal generated matrix Generated structured sparse matrix with single precision CSR CSR-avoid Dimension row = col = MPI/Total time (%) Continous diagonal elements (15 diagonals above main diagonal) Diagonal values = 1.(without perturbation). Comparison between CSR and CSR communication avoiding in percents of MPI time in Total Time Nb of MPI Process 18 / 23
19 19 / 23 Current Work Communication Avoiding implementation The results shows that CSR with communication avoiding Has a good scalability Has a low proportion of Communication Time For CSR standard Total time: O( nnz m ) + δ n(m 1) t O( ) + latency O(m 1) m For CSR communication avoiding Total time: bnnz << n O( nnz m ) + δ bnnz(m 1) t O( ) + latency O(m 1) m
20 20 / 23 Future Work Outline 1 Background Introduction 2 Energy efficiency in Krylov iterative methods 3 Current Work 4 Future Work
21 21 / 23 Future Work Energy evaluation on K20 Test on NVIDIA K20 Poincare has updated its GPU to K20 Redo the test for comparison Evaluation of energy using NVIDIA Management Library (NVML) nvmldevicegettemperature nvmldevicegetperformancestate... Add the energy variation as a criterion to evaluate various auto-tuning methods. E.g. Change Krylov subspace size and evaluate variation of energy consumption.
22 22 / 23 Future Work New way for Energy minimization 1 Find more parameters for Auto-tuning Parameters to control communication avoiding... 2 Machine learning for smart-tuning Semantics in optimization Supervised learning via history records 3 Stochastic way of communication avoiding
23 23 / 23 Future Work End Thank you Any Question?
Energy Consumption Evaluation for Krylov Methods on a Cluster of GPU Accelerators
PARIS-SACLAY, FRANCE Energy Consumption Evaluation for Krylov Methods on a Cluster of GPU Accelerators Serge G. Petiton a and Langshi Chen b April the 6 th, 2016 a Université de Lille, Sciences et Technologies
More informationPerformance and Energy Analysis of the Iterative Solution of Sparse Linear Systems on Multicore and Manycore Architectures
Performance and Energy Analysis of the Iterative Solution of Sparse Linear Systems on Multicore and Manycore Architectures José I. Aliaga Performance and Energy Analysis of the Iterative Solution of Sparse
More informationMinisymposia 9 and 34: Avoiding Communication in Linear Algebra. Jim Demmel UC Berkeley bebop.cs.berkeley.edu
Minisymposia 9 and 34: Avoiding Communication in Linear Algebra Jim Demmel UC Berkeley bebop.cs.berkeley.edu Motivation (1) Increasing parallelism to exploit From Top500 to multicores in your laptop Exponentially
More informationMARCH 24-27, 2014 SAN JOSE, CA
MARCH 24-27, 2014 SAN JOSE, CA Sparse HPC on modern architectures Important scientific applications rely on sparse linear algebra HPCG a new benchmark proposal to complement Top500 (HPL) To solve A x =
More informationQR Factorization of Tall and Skinny Matrices in a Grid Computing Environment
QR Factorization of Tall and Skinny Matrices in a Grid Computing Environment Emmanuel AGULLO (INRIA / LaBRI) Camille COTI (Iowa State University) Jack DONGARRA (University of Tennessee) Thomas HÉRAULT
More informationWelcome to MCS 572. content and organization expectations of the course. definition and classification
Welcome to MCS 572 1 About the Course content and organization expectations of the course 2 Supercomputing definition and classification 3 Measuring Performance speedup and efficiency Amdahl s Law Gustafson
More informationExploiting Low-Rank Structure in Computing Matrix Powers with Applications to Preconditioning
Exploiting Low-Rank Structure in Computing Matrix Powers with Applications to Preconditioning Erin C. Carson, Nicholas Knight, James Demmel, Ming Gu U.C. Berkeley SIAM PP 12, Savannah, Georgia, USA, February
More informationAccelerating linear algebra computations with hybrid GPU-multicore systems.
Accelerating linear algebra computations with hybrid GPU-multicore systems. Marc Baboulin INRIA/Université Paris-Sud joint work with Jack Dongarra (University of Tennessee and Oak Ridge National Laboratory)
More informationLattice Boltzmann simulations on heterogeneous CPU-GPU clusters
Lattice Boltzmann simulations on heterogeneous CPU-GPU clusters H. Köstler 2nd International Symposium Computer Simulations on GPU Freudenstadt, 29.05.2013 1 Contents Motivation walberla software concepts
More informationParallelization of Molecular Dynamics (with focus on Gromacs) SeSE 2014 p.1/29
Parallelization of Molecular Dynamics (with focus on Gromacs) SeSE 2014 p.1/29 Outline A few words on MD applications and the GROMACS package The main work in an MD simulation Parallelization Stream computing
More informationLeveraging Task-Parallelism in Energy-Efficient ILU Preconditioners
Leveraging Task-Parallelism in Energy-Efficient ILU Preconditioners José I. Aliaga Leveraging task-parallelism in energy-efficient ILU preconditioners Universidad Jaime I (Castellón, Spain) José I. Aliaga
More informationCME342 Parallel Methods in Numerical Analysis. Matrix Computation: Iterative Methods II. Sparse Matrix-vector Multiplication.
CME342 Parallel Methods in Numerical Analysis Matrix Computation: Iterative Methods II Outline: CG & its parallelization. Sparse Matrix-vector Multiplication. 1 Basic iterative methods: Ax = b r = b Ax
More informationA model leading to self-consistent iteration computation with need for HP LA (e.g, diagonalization and orthogonalization)
A model leading to self-consistent iteration computation with need for HP LA (e.g, diagonalization and orthogonalization) Schodinger equation: Hψ = Eψ Choose a basis set of wave functions Two cases: Orthonormal
More information4.8 Arnoldi Iteration, Krylov Subspaces and GMRES
48 Arnoldi Iteration, Krylov Subspaces and GMRES We start with the problem of using a similarity transformation to convert an n n matrix A to upper Hessenberg form H, ie, A = QHQ, (30) with an appropriate
More informationAccelerating Linear Algebra on Heterogeneous Architectures of Multicore and GPUs using MAGMA and DPLASMA and StarPU Schedulers
UT College of Engineering Tutorial Accelerating Linear Algebra on Heterogeneous Architectures of Multicore and GPUs using MAGMA and DPLASMA and StarPU Schedulers Stan Tomov 1, George Bosilca 1, and Cédric
More informationIntroduction to numerical computations on the GPU
Introduction to numerical computations on the GPU Lucian Covaci http://lucian.covaci.org/cuda.pdf Tuesday 1 November 11 1 2 Outline: NVIDIA Tesla and Geforce video cards: architecture CUDA - C: programming
More informationTR A Comparison of the Performance of SaP::GPU and Intel s Math Kernel Library (MKL) for Solving Dense Banded Linear Systems
TR-0-07 A Comparison of the Performance of ::GPU and Intel s Math Kernel Library (MKL) for Solving Dense Banded Linear Systems Ang Li, Omkar Deshmukh, Radu Serban, Dan Negrut May, 0 Abstract ::GPU is a
More informationCommunication-avoiding parallel and sequential QR factorizations
Communication-avoiding parallel and sequential QR factorizations James Demmel, Laura Grigori, Mark Hoemmen, and Julien Langou May 30, 2008 Abstract We present parallel and sequential dense QR factorization
More informationECS289: Scalable Machine Learning
ECS289: Scalable Machine Learning Cho-Jui Hsieh UC Davis Sept 27, 2015 Outline Linear regression Ridge regression and Lasso Time complexity (closed form solution) Iterative Solvers Regression Input: training
More informationClaude Tadonki. MINES ParisTech PSL Research University Centre de Recherche Informatique
Claude Tadonki MINES ParisTech PSL Research University Centre de Recherche Informatique claude.tadonki@mines-paristech.fr Monthly CRI Seminar MINES ParisTech - CRI June 06, 2016, Fontainebleau (France)
More informationFeeding of the Thousands. Leveraging the GPU's Computing Power for Sparse Linear Algebra
SPPEXA Annual Meeting 2016, January 25 th, 2016, Garching, Germany Feeding of the Thousands Leveraging the GPU's Computing Power for Sparse Linear Algebra Hartwig Anzt Sparse Linear Algebra on GPUs Inherently
More informationOpen-Source Parallel FE Software : FrontISTR -- Performance Considerations about B/F (Byte per Flop) of SpMV on K-Supercomputer and GPU-Clusters --
Parallel Processing for Energy Efficiency October 3, 2013 NTNU, Trondheim, Norway Open-Source Parallel FE Software : FrontISTR -- Performance Considerations about B/F (Byte per Flop) of SpMV on K-Supercomputer
More informationCommunication-avoiding parallel and sequential QR factorizations
Communication-avoiding parallel and sequential QR factorizations James Demmel Laura Grigori Mark Frederick Hoemmen Julien Langou Electrical Engineering and Computer Sciences University of California at
More informationJacobi-Davidson Eigensolver in Cusolver Library. Lung-Sheng Chien, NVIDIA
Jacobi-Davidson Eigensolver in Cusolver Library Lung-Sheng Chien, NVIDIA lchien@nvidia.com Outline CuSolver library - cusolverdn: dense LAPACK - cusolversp: sparse LAPACK - cusolverrf: refactorization
More informationComputers and Mathematics with Applications
Computers and Mathematics with Applications 68 (2014) 1151 1160 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A GPU
More informationBreaking Computational Barriers: Multi-GPU High-Order RBF Kernel Problems with Millions of Points
Breaking Computational Barriers: Multi-GPU High-Order RBF Kernel Problems with Millions of Points Michael Griebel Christian Rieger Peter Zaspel Institute for Numerical Simulation Rheinische Friedrich-Wilhelms-Universität
More informationJacobi-Based Eigenvalue Solver on GPU. Lung-Sheng Chien, NVIDIA
Jacobi-Based Eigenvalue Solver on GPU Lung-Sheng Chien, NVIDIA lchien@nvidia.com Outline Symmetric eigenvalue solver Experiment Applications Conclusions Symmetric eigenvalue solver The standard form is
More informationNuclear Physics and Computing: Exascale Partnerships. Juan Meza Senior Scientist Lawrence Berkeley National Laboratory
Nuclear Physics and Computing: Exascale Partnerships Juan Meza Senior Scientist Lawrence Berkeley National Laboratory Nuclear Science and Exascale i Workshop held in DC to identify scientific challenges
More informationEnhancing Performance of Tall-Skinny QR Factorization using FPGAs
Enhancing Performance of Tall-Skinny QR Factorization using FPGAs Abid Rafique Imperial College London August 31, 212 Enhancing Performance of Tall-Skinny QR Factorization using FPGAs 1/18 Our Claim Common
More informationIntroduction to communication avoiding algorithms for direct methods of factorization in Linear Algebra
Introduction to communication avoiding algorithms for direct methods of factorization in Linear Algebra Laura Grigori Abstract Modern, massively parallel computers play a fundamental role in a large and
More informationMS4: Minimizing Communication in Numerical Algorithms Part I of II
MS4: Minimizing Communication in Numerical Algorithms Part I of II Organizers: Oded Schwartz (Hebrew University of Jerusalem) and Erin Carson (New York University) Talks: 1. Communication-Avoiding Krylov
More informationJulian Merten. GPU Computing and Alternative Architecture
Future Directions of Cosmological Simulations / Edinburgh 1 / 16 Julian Merten GPU Computing and Alternative Architecture Institut für Theoretische Astrophysik Zentrum für Astronomie Universität Heidelberg
More informationPractical Combustion Kinetics with CUDA
Funded by: U.S. Department of Energy Vehicle Technologies Program Program Manager: Gurpreet Singh & Leo Breton Practical Combustion Kinetics with CUDA GPU Technology Conference March 20, 2015 Russell Whitesides
More informationOn the design of parallel linear solvers for large scale problems
On the design of parallel linear solvers for large scale problems ICIAM - August 2015 - Mini-Symposium on Recent advances in matrix computations for extreme-scale computers M. Faverge, X. Lacoste, G. Pichon,
More information- Part 4 - Multicore and Manycore Technology: Chances and Challenges. Vincent Heuveline
- Part 4 - Multicore and Manycore Technology: Chances and Challenges Vincent Heuveline 1 Numerical Simulation of Tropical Cyclones Goal oriented adaptivity for tropical cyclones ~10⁴km ~1500km ~100km 2
More informationc 2015 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 37, No. 3, pp. C307 C330 c 2015 Society for Industrial and Applied Mathematics MIXED-PRECISION CHOLESKY QR FACTORIZATION AND ITS CASE STUDIES ON MULTICORE CPU WITH MULTIPLE GPUS
More informationDynamic Scheduling within MAGMA
Dynamic Scheduling within MAGMA Emmanuel Agullo, Cedric Augonnet, Jack Dongarra, Mathieu Faverge, Julien Langou, Hatem Ltaief, Samuel Thibault and Stanimire Tomov April 5, 2012 Innovative and Computing
More informationMS 28: Scalable Communication-Avoiding and -Hiding Krylov Subspace Methods I
MS 28: Scalable Communication-Avoiding and -Hiding Krylov Subspace Methods I Organizers: Siegfried Cools, University of Antwerp, Belgium Erin C. Carson, New York University, USA 10:50-11:10 High Performance
More informationSP-CNN: A Scalable and Programmable CNN-based Accelerator. Dilan Manatunga Dr. Hyesoon Kim Dr. Saibal Mukhopadhyay
SP-CNN: A Scalable and Programmable CNN-based Accelerator Dilan Manatunga Dr. Hyesoon Kim Dr. Saibal Mukhopadhyay Motivation Power is a first-order design constraint, especially for embedded devices. Certain
More informationImplementing QR Factorization Updating Algorithms on GPUs. Andrew, Robert and Dingle, Nicholas J. MIMS EPrint:
Implementing QR Factorization Updating Algorithms on GPUs Andrew, Robert and Dingle, Nicholas J. 214 MIMS EPrint: 212.114 Manchester Institute for Mathematical Sciences School of Mathematics The University
More informationPiz Daint & Piz Kesch : from general purpose supercomputing to an appliance for weather forecasting. Thomas C. Schulthess
Piz Daint & Piz Kesch : from general purpose supercomputing to an appliance for weather forecasting Thomas C. Schulthess 1 Cray XC30 with 5272 hybrid, GPU accelerated compute nodes Piz Daint Compute node:
More informationPerformance of Random Sampling for Computing Low-rank Approximations of a Dense Matrix on GPUs
Performance of Random Sampling for Computing Low-rank Approximations of a Dense Matrix on GPUs Théo Mary, Ichitaro Yamazaki, Jakub Kurzak, Piotr Luszczek, Stanimire Tomov, Jack Dongarra presenter 1 Low-Rank
More informationCommunication-avoiding LU and QR factorizations for multicore architectures
Communication-avoiding LU and QR factorizations for multicore architectures DONFACK Simplice INRIA Saclay Joint work with Laura Grigori INRIA Saclay Alok Kumar Gupta BCCS,Norway-5075 16th April 2010 Communication-avoiding
More informationCourse Notes: Week 1
Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues
More informationSupercomputing: Why, What, and Where (are we)?
Supercomputing: Why, What, and Where (are we)? R. Govindarajan Indian Institute of Science, Bangalore, INDIA govind@serc.iisc.ernet.in (C)RG@SERC,IISc Why Supercomputer? Third and Fourth Legs RG@SERC,IISc
More informationEfficient implementation of the overlap operator on multi-gpus
Efficient implementation of the overlap operator on multi-gpus Andrei Alexandru Mike Lujan, Craig Pelissier, Ben Gamari, Frank Lee SAAHPC 2011 - University of Tennessee Outline Motivation Overlap operator
More informationDeflation Strategies to Improve the Convergence of Communication-Avoiding GMRES
Deflation Strategies to Improve the Convergence of Communication-Avoiding GMRES Ichitaro Yamazaki, Stanimire Tomov, and Jack Dongarra Department of Electrical Engineering and Computer Science University
More informationAccelerating computation of eigenvectors in the nonsymmetric eigenvalue problem
Accelerating computation of eigenvectors in the nonsymmetric eigenvalue problem Mark Gates 1, Azzam Haidar 1, and Jack Dongarra 1,2,3 1 University of Tennessee, Knoxville, TN, USA 2 Oak Ridge National
More informationThe Green Index (TGI): A Metric for Evalua:ng Energy Efficiency in HPC Systems
The Green Index (TGI): A Metric for Evalua:ng Energy Efficiency in HPC Systems Wu Feng and Balaji Subramaniam Metrics for Energy Efficiency Energy- Delay Product (EDP) Used primarily in circuit design
More informationHPC and High-end Data Science
HPC and High-end Data Science for the Power Grid Alex Pothen August 3, 2018 Outline High-end Data Science 1 3 Data Anonymization Contingency Analysis 2 4 Parallel Oscillation Monitoring 2 / 22 PMUs in
More informationError Bounds for Iterative Refinement in Three Precisions
Error Bounds for Iterative Refinement in Three Precisions Erin C. Carson, New York University Nicholas J. Higham, University of Manchester SIAM Annual Meeting Portland, Oregon July 13, 018 Hardware Support
More informationThe new challenges to Krylov subspace methods Yousef Saad Department of Computer Science and Engineering University of Minnesota
The new challenges to Krylov subspace methods Yousef Saad Department of Computer Science and Engineering University of Minnesota SIAM Applied Linear Algebra Valencia, June 18-22, 2012 Introduction Krylov
More informationHybrid static/dynamic scheduling for already optimized dense matrix factorization. Joint Laboratory for Petascale Computing, INRIA-UIUC
Hybrid static/dynamic scheduling for already optimized dense matrix factorization Simplice Donfack, Laura Grigori, INRIA, France Bill Gropp, Vivek Kale UIUC, USA Joint Laboratory for Petascale Computing,
More informationComputing least squares condition numbers on hybrid multicore/gpu systems
Computing least squares condition numbers on hybrid multicore/gpu systems M. Baboulin and J. Dongarra and R. Lacroix Abstract This paper presents an efficient computation for least squares conditioning
More informationCommunication avoiding parallel algorithms for dense matrix factorizations
Communication avoiding parallel dense matrix factorizations 1/ 44 Communication avoiding parallel algorithms for dense matrix factorizations Edgar Solomonik Department of EECS, UC Berkeley October 2013
More informationIntroduction to communication avoiding linear algebra algorithms in high performance computing
Introduction to communication avoiding linear algebra algorithms in high performance computing Laura Grigori Inria Rocquencourt/UPMC Contents 1 Introduction............................ 2 2 The need for
More informationSparse Matrices and Iterative Methods
Sparse Matrices and Iterative Methods K. 1 1 Department of Mathematics 2018 Iterative Methods Consider the problem of solving Ax = b, where A is n n. Why would we use an iterative method? Avoid direct
More informationLecture 8: Fast Linear Solvers (Part 7)
Lecture 8: Fast Linear Solvers (Part 7) 1 Modified Gram-Schmidt Process with Reorthogonalization Test Reorthogonalization If Av k 2 + δ v k+1 2 = Av k 2 to working precision. δ = 10 3 2 Householder Arnoldi
More informationContents. Preface... xi. Introduction...
Contents Preface... xi Introduction... xv Chapter 1. Computer Architectures... 1 1.1. Different types of parallelism... 1 1.1.1. Overlap, concurrency and parallelism... 1 1.1.2. Temporal and spatial parallelism
More informationCase Study: Quantum Chromodynamics
Case Study: Quantum Chromodynamics Michael Clark Harvard University with R. Babich, K. Barros, R. Brower, J. Chen and C. Rebbi Outline Primer to QCD QCD on a GPU Mixed Precision Solvers Multigrid solver
More informationParallel Sparse Tensor Decompositions using HiCOO Format
Figure sources: A brief survey of tensors by Berton Earnshaw and NVIDIA Tensor Cores Parallel Sparse Tensor Decompositions using HiCOO Format Jiajia Li, Jee Choi, Richard Vuduc May 8, 8 @ SIAM ALA 8 Outline
More informationarxiv: v1 [hep-lat] 31 Oct 2015
and Code Optimization arxiv:1511.00088v1 [hep-lat] 31 Oct 2015 Hwancheol Jeong, Sangbaek Lee, Weonjong Lee, Lattice Gauge Theory Research Center, CTP, and FPRD, Department of Physics and Astronomy, Seoul
More informationParallel Eigensolver Performance on High Performance Computers 1
Parallel Eigensolver Performance on High Performance Computers 1 Andrew Sunderland STFC Daresbury Laboratory, Warrington, UK Abstract Eigenvalue and eigenvector computations arise in a wide range of scientific
More informationSolution to Laplace Equation using Preconditioned Conjugate Gradient Method with Compressed Row Storage using MPI
Solution to Laplace Equation using Preconditioned Conjugate Gradient Method with Compressed Row Storage using MPI Sagar Bhatt Person Number: 50170651 Department of Mechanical and Aerospace Engineering,
More informationArchitecture-Aware Algorithms and Software for Peta and Exascale Computing
Architecture-Aware Algorithms and Software for Peta and Exascale Computing Jack Dongarra University of Tennessee Oak Ridge National Laboratory University of Manchester 4/25/2011 1 H. Meuer, H. Simon, E.
More informationAccelerating computation of eigenvectors in the dense nonsymmetric eigenvalue problem
Accelerating computation of eigenvectors in the dense nonsymmetric eigenvalue problem Mark Gates 1, Azzam Haidar 1, and Jack Dongarra 1,2,3 1 University of Tennessee, Knoxville, TN, USA 2 Oak Ridge National
More informationAccelerating Model Reduction of Large Linear Systems with Graphics Processors
Accelerating Model Reduction of Large Linear Systems with Graphics Processors P. Benner 1, P. Ezzatti 2, D. Kressner 3, E.S. Quintana-Ortí 4, Alfredo Remón 4 1 Max-Plank-Institute for Dynamics of Complex
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 21: Sensitivity of Eigenvalues and Eigenvectors; Conjugate Gradient Method Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis
More informationSOLVING SPARSE LINEAR SYSTEMS OF EQUATIONS. Chao Yang Computational Research Division Lawrence Berkeley National Laboratory Berkeley, CA, USA
1 SOLVING SPARSE LINEAR SYSTEMS OF EQUATIONS Chao Yang Computational Research Division Lawrence Berkeley National Laboratory Berkeley, CA, USA 2 OUTLINE Sparse matrix storage format Basic factorization
More informationCommunication-avoiding Krylov subspace methods
Motivation Communication-avoiding Krylov subspace methods Mark mhoemmen@cs.berkeley.edu University of California Berkeley EECS MS Numerical Libraries Group visit: 28 April 2008 Overview Motivation Current
More informationAlgorithm for Sparse Approximate Inverse Preconditioners in the Conjugate Gradient Method
Algorithm for Sparse Approximate Inverse Preconditioners in the Conjugate Gradient Method Ilya B. Labutin A.A. Trofimuk Institute of Petroleum Geology and Geophysics SB RAS, 3, acad. Koptyug Ave., Novosibirsk
More informationMeasuring freeze-out parameters on the Bielefeld GPU cluster
Measuring freeze-out parameters on the Bielefeld GPU cluster Outline Fluctuations and the QCD phase diagram Fluctuations from Lattice QCD The Bielefeld hybrid GPU cluster Freeze-out conditions from QCD
More informationA CPU-GPU Hybrid Implementation and Model-Driven Scheduling of the Fast Multipole Method
A CPU-GPU Hybrid Implementation and Model-Driven Scheduling of the Fast Multipole Method Jee Choi 1, Aparna Chandramowlishwaran 3, Kamesh Madduri 4, and Richard Vuduc 2 1 ECE, Georgia Tech 2 CSE, Georgia
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)
AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 19: Computing the SVD; Sparse Linear Systems Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical
More informationJ.I. Aliaga 1 M. Bollhöfer 2 A.F. Martín 1 E.S. Quintana-Ortí 1. March, 2009
Parallel Preconditioning of Linear Systems based on ILUPACK for Multithreaded Architectures J.I. Aliaga M. Bollhöfer 2 A.F. Martín E.S. Quintana-Ortí Deparment of Computer Science and Engineering, Univ.
More informationUniversität Dortmund UCHPC. Performance. Computing for Finite Element Simulations
technische universität dortmund Universität Dortmund fakultät für mathematik LS III (IAM) UCHPC UnConventional High Performance Computing for Finite Element Simulations S. Turek, Chr. Becker, S. Buijssen,
More informationAccelerating Three-Body Potentials using GPUs NVIDIA Tesla K20X
Using a Hybrid Cray Supercomputer to Model Non-Icing Surfaces for Cold- Climate Wind Turbines Accelerating Three-Body Potentials using GPUs NVIDIA Tesla K20X GE Global Research Masako Yamada Opportunity
More informationScalable Hybrid Programming and Performance for SuperLU Sparse Direct Solver
Scalable Hybrid Programming and Performance for SuperLU Sparse Direct Solver Sherry Li Lawrence Berkeley National Laboratory Piyush Sao Rich Vuduc Georgia Institute of Technology CUG 14, May 4-8, 14, Lugano,
More informationOn Portability, Performance and Scalability of a MPI OpenCL Lattice Boltzmann Code
On Portability, Performance and Scalability of a MPI OpenCL Lattice Boltzmann Code E Calore, S F Schifano, R Tripiccione Enrico Calore INFN Ferrara, Italy 7 th Workshop on UnConventional High Performance
More informationSparse LU Factorization on GPUs for Accelerating SPICE Simulation
Nano-scale Integrated Circuit and System (NICS) Laboratory Sparse LU Factorization on GPUs for Accelerating SPICE Simulation Xiaoming Chen PhD Candidate Department of Electronic Engineering Tsinghua University,
More informationRegression. Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning)
Linear Regression Regression Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning) Example: Height, Gender, Weight Shoe Size Audio features
More informationRegression. Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning)
Linear Regression Regression Goal: Learn a mapping from observations (features) to continuous labels given a training set (supervised learning) Example: Height, Gender, Weight Shoe Size Audio features
More informationTall and Skinny QR Matrix Factorization Using Tile Algorithms on Multicore Architectures LAPACK Working Note - 222
Tall and Skinny QR Matrix Factorization Using Tile Algorithms on Multicore Architectures LAPACK Working Note - 222 Bilel Hadri 1, Hatem Ltaief 1, Emmanuel Agullo 1, and Jack Dongarra 1,2,3 1 Department
More informationSome thoughts about energy efficient application execution on NEC LX Series compute clusters
Some thoughts about energy efficient application execution on NEC LX Series compute clusters G. Wellein, G. Hager, J. Treibig, M. Wittmann Erlangen Regional Computing Center & Department of Computer Science
More informationNVIDIA MPI-enabled Iterative Solvers for Large Scale Problems. Joe Eaton Manager, AmgX CUDA Library NVIDIA
NVIDIA MPI-enabled Iterative Solvers for Large Scale Problems Joe Eaton Manager, AmgX CUDA Library NVIDIA ANSYS Fluent Fluent control flow Accelerate this first Non-linear iterations Assemble Linear System
More informationResearch on GPU-accelerated algorithm in 3D finite difference neutron diffusion calculation method
NUCLEAR SCIENCE AND TECHNIQUES 25, 0501 (14) Research on GPU-accelerated algorithm in 3D finite difference neutron diffusion calculation method XU Qi ( 徐琪 ), 1, YU Gang-Lin ( 余纲林 ), 1 WANG Kan ( 王侃 ),
More informationPower-Aware Execution of Sparse and Dense Linear Algebra Libraries
Power-Aware Execution of Sparse and Dense Linear Algebra Libraries Enrique S. Quintana-Ortí quintana@icc.uji.es Power-aware execution of linear algebra libraries 1 CECAM Lausanne, Sept. 2011 Motivation
More informationA microsecond a day keeps the doctor away: Efficient GPU Molecular Dynamics with GROMACS
GTC 20130319 A microsecond a day keeps the doctor away: Efficient GPU Molecular Dynamics with GROMACS Erik Lindahl erik.lindahl@scilifelab.se Molecular Dynamics Understand biology We re comfortably on
More informationIncomplete Cholesky preconditioners that exploit the low-rank property
anapov@ulb.ac.be ; http://homepages.ulb.ac.be/ anapov/ 1 / 35 Incomplete Cholesky preconditioners that exploit the low-rank property (theory and practice) Artem Napov Service de Métrologie Nucléaire, Université
More informationMAGMA MIC 1.0: Linear Algebra Library for Intel Xeon Phi Coprocessors
MAGMA MIC 1.0: Linear Algebra Library for Intel Xeon Phi Coprocessors J. Dongarra, M. Gates, A. Haidar, Y. Jia, K. Kabir, P. Luszczek, and S. Tomov University of Tennessee, Knoxville 05 / 03 / 2013 MAGMA:
More informationAntti-Pekka Hynninen, 5/10/2017, GTC2017, San Jose CA
S7255: CUTT: A HIGH- PERFORMANCE TENSOR TRANSPOSE LIBRARY FOR GPUS Antti-Pekka Hynninen, 5/10/2017, GTC2017, San Jose CA MOTIVATION Tensor contractions are the most computationally intensive part of quantum
More informationEnhancing Scalability of Sparse Direct Methods
Journal of Physics: Conference Series 78 (007) 0 doi:0.088/7-6596/78//0 Enhancing Scalability of Sparse Direct Methods X.S. Li, J. Demmel, L. Grigori, M. Gu, J. Xia 5, S. Jardin 6, C. Sovinec 7, L.-Q.
More informationEfficient Serial and Parallel Coordinate Descent Methods for Huge-Scale Convex Optimization
Efficient Serial and Parallel Coordinate Descent Methods for Huge-Scale Convex Optimization Martin Takáč The University of Edinburgh Based on: P. Richtárik and M. Takáč. Iteration complexity of randomized
More informationParallel Multivariate SpatioTemporal Clustering of. Large Ecological Datasets on Hybrid Supercomputers
Parallel Multivariate SpatioTemporal Clustering of Large Ecological Datasets on Hybrid Supercomputers Sarat Sreepathi1, Jitendra Kumar1, Richard T. Mills2, Forrest M. Hoffman1, Vamsi Sripathi3, William
More informationPetascale Quantum Simulations of Nano Systems and Biomolecules
Petascale Quantum Simulations of Nano Systems and Biomolecules Emil Briggs North Carolina State University 1. Outline of real-space Multigrid (RMG) 2. Scalability and hybrid/threaded models 3. GPU acceleration
More informationQuantum Chemical Calculations by Parallel Computer from Commodity PC Components
Nonlinear Analysis: Modelling and Control, 2007, Vol. 12, No. 4, 461 468 Quantum Chemical Calculations by Parallel Computer from Commodity PC Components S. Bekešienė 1, S. Sėrikovienė 2 1 Institute of
More informationBinding Performance and Power of Dense Linear Algebra Operations
10th IEEE International Symposium on Parallel and Distributed Processing with Applications Binding Performance and Power of Dense Linear Algebra Operations Maria Barreda, Manuel F. Dolz, Rafael Mayo, Enrique
More informationHigh-Performance Scientific Computing
High-Performance Scientific Computing Instructor: Randy LeVeque TA: Grady Lemoine Applied Mathematics 483/583, Spring 2011 http://www.amath.washington.edu/~rjl/am583 World s fastest computers http://top500.org
More informationParallel Eigensolver Performance on High Performance Computers
Parallel Eigensolver Performance on High Performance Computers Andrew Sunderland Advanced Research Computing Group STFC Daresbury Laboratory CUG 2008 Helsinki 1 Summary (Briefly) Introduce parallel diagonalization
More informationImprovements for Implicit Linear Equation Solvers
Improvements for Implicit Linear Equation Solvers Roger Grimes, Bob Lucas, Clement Weisbecker Livermore Software Technology Corporation Abstract Solving large sparse linear systems of equations is often
More information