Sphynx: a Parallel Multi-GPU Graph Partitioner for Distributed-Memory Systems
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Accelerating the LOBPCG Method on Gpus Using a Blocked Sparse Matrix Vector Product
Accelerating the LOBPCG method on GPUs using a blocked Sparse Matrix Vector Product Hartwig Anzt and Stanimire Tomov and Jack Dongarra Innovative Computing Lab University of Tennessee Knoxville, USA Email: [email protected], [email protected], [email protected] Abstract— the computing power of today’s supercomputers, often accel- erated by coprocessors like graphics processing units (GPUs), This paper presents a heterogeneous CPU-GPU algorithm design and optimized implementation for an entire sparse iter- becomes challenging. ative eigensolver – the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) – starting from low-level GPU While there exist numerous efforts to adapt iterative lin- data structures and kernels to the higher-level algorithmic choices ear solvers like Krylov subspace methods to coprocessor and overall heterogeneous design. Most notably, the eigensolver technology, sparse eigensolvers have so far remained out- leverages the high-performance of a new GPU kernel developed side the main focus. A possible explanation is that many for the simultaneous multiplication of a sparse matrix and a of those combine sparse and dense linear algebra routines, set of vectors (SpMM). This is a building block that serves which makes porting them to accelerators more difficult. Aside as a backbone for not only block-Krylov, but also for other from the power method, algorithms based on the Krylov methods relying on blocking for acceleration in general. The subspace idea are among the most commonly used general heterogeneous LOBPCG developed here reveals the potential of eigensolvers [1]. When targeting symmetric positive definite this type of eigensolver by highly optimizing all of its components, eigenvalue problems, the recently developed Locally Optimal and can be viewed as a benchmark for other SpMM-dependent applications. -
Present and Future Leadership Computers at OLCF
Present and Future Leadership Computers at OLCF Buddy Bland OLCF Project Director Presented at: SC’14 November 17-21, 2014 New Orleans ORNL is managed by UT-Battelle for the US Department of Energy Oak Ridge Leadership Computing Facility (OLCF) Mission: Deploy and operate the computational resources required to tackle global challenges Providing world-leading computational and data resources and specialized services for the most computationally intensive problems Providing stable hardware/software path of increasing scale to maximize productive applications development Providing the resources to investigate otherwise inaccessible systems at every scale: from galaxy formation to supernovae to earth systems to automobiles to nanomaterials With our partners, deliver transforming discoveries in materials, biology, climate, energy technologies, and basic science SC’14 Summit - Bland 2 Our Science requires that we continue to advance OLCF’s computational capability over the next decade on the roadmap to Exascale. Since clock-rate scaling ended in 2003, HPC Titan and beyond deliver hierarchical parallelism with performance has been achieved through increased very powerful nodes. MPI plus thread level parallelism parallelism. Jaguar scaled to 300,000 cores. through OpenACC or OpenMP plus vectors OLCF5: 5-10x Summit Summit: 5-10x Titan ~20 MW Titan: 27 PF Hybrid GPU/CPU Jaguar: 2.3 PF Hybrid GPU/CPU 10 MW Multi-core CPU 9 MW CORAL System 7 MW 2010 2012 2017 2022 3 SC’14 Summit - Bland Today’s Leadership System - Titan Hybrid CPU/GPU architecture, Hierarchical Parallelism Vendors: Cray™ / NVIDIA™ • 27 PF peak • 18,688 Compute nodes, each with – 1.45 TF peak – NVIDIA Kepler™ GPU - 1,311 GF • 6 GB GDDR5 memory – AMD Opteron™- 141 GF • 32 GB DDR3 memory – PCIe2 link between GPU and CPU • Cray Gemini 3-D Torus Interconnect • 32 PB / 1 TB/s Lustre® file system 4 SC’14 Summit - Bland Scientific Progress at all Scales Fusion Energy Liquid Crystal Film Stability A Princeton Plasma Physics Laboratory ORNL Postdoctoral fellow Trung Nguyen team led by C.S. -
16.6 Graph Partition 541
16.6 GRAPH PARTITION 541 INPUT OUTPUT 16.6 Graph Partition Input description: A (weighted) graph G =(V,E) and integers k and m. Problem description: Partition the vertices into m roughly equal-sized subsets such that the total edge cost spanning the subsets is at most k. Discussion: Graph partitioning arises in many divide-and-conquer algorithms, which gain their efficiency by breaking problems into equal-sized pieces such that the respective solutions can easily be reassembled. Minimizing the number of edges cut in the partition usually simplifies the task of merging. Graph partition also arises when we need to cluster the vertices into logical components. If edges link “similar” pairs of objects, the clusters remaining after partition should reflect coherent groupings. Large graphs are often partitioned into reasonable-sized pieces to improve data locality or make less cluttered drawings. Finally, graph partition is a critical step in many parallel algorithms. Consider the finite element method, which is used to compute the physical properties (such as stress and heat transfer) of geometric models. Parallelizing such calculations requires partitioning the models into equal-sized pieces whose interface is small. This is a graph-partitioning problem, since the topology of a geometric model is usually represented using a graph. Several different flavors of graph partitioning arise depending on the desired objective function: 542 16. GRAPH PROBLEMS: HARD PROBLEMS Figure 16.1: The maximum cut of a graph • Minimum cut set –Thesmallest set of edges to cut that will disconnect a graph can be efficiently found using network flow or randomized algorithms. -
Exploring Capabilities Within Fortrilinos by Solving the 3D Burgers Equation
Scientific Programming 20 (2012) 275–292 275 DOI 10.3233/SPR-2012-0353 IOS Press Exploring capabilities within ForTrilinos by solving the 3D Burgers equation Karla Morris a,∗, Damian W.I. Rouson a, M. Nicole Lemaster a and Salvatore Filippone b a Sandia National Laboratories, Livermore, CA, USA b Università di Roma “Tor Vergata”, Roma, Italy Abstract. We present the first three-dimensional, partial differential equation solver to be built atop the recently released, open-source ForTrilinos package (http://trilinos.sandia.gov/packages/fortrilinos). ForTrilinos currently provides portable, object- oriented Fortran 2003 interfaces to the C++ packages Epetra, AztecOO and Pliris in the Trilinos library and framework [ACM Trans. Math. Softw. 31(3) (2005), 397–423]. Epetra provides distributed matrix and vector storage and basic linear algebra cal- culations. Pliris provides direct solvers for dense linear systems. AztecOO provides iterative sparse linear solvers. We demon- strate how to build a parallel application that encapsulates the Message Passing Interface (MPI) without requiring the user to make direct calls to MPI except for startup and shutdown. The presented example demonstrates the level of effort required to set up a high-order, finite-difference solution on a Cartesian grid. The example employs an abstract data type (ADT) calculus [Sci. Program. 16(4) (2008), 329–339] that empowers programmers to write serial code that lower-level abstractions resolve into distributed-memory, parallel implementations. The ADT calculus uses compilable Fortran constructs that resemble the mathe- matical formulation of the partial differential equation of interest. Keywords: ForTrilinos, Trilinos, Fortran 2003/2008, object oriented programming 1. Introduction Burgers [4]. -
Tensor Spectral Clustering for Partitioning Higher-Order Network Structures
Tensor Spectral Clustering for Partitioning Higher-order Network Structures Austin R. Benson∗ David F. Gleichy Jure Leskovecz Abstract normalized) number of first-order structures (i.e., edges) that Spectral graph theory-based methods represent an important need to be cut in order to split the graph into two parts. In a class of tools for studying the structure of networks. Spec- similar spirit, a higher-order generalization of spectral clus- tral methods are based on a first-order Markov chain de- tering would try to minimize cutting higher-order structures rived from a random walk on the graph and thus they cannot that involve multiple nodes (e.g., triangles). take advantage of important higher-order network substruc- Incorporating higher-order graph information (that is, tures such as triangles, cycles, and feed-forward loops. Here network motifs/graphlets) into the partitioning process can we propose a Tensor Spectral Clustering (TSC) algorithm significantly improve our understanding of the underlying that allows for modeling higher-order network structures in a network. For example, triangles (three-dimensional network graph partitioning framework. Our TSC algorithm allows the structures involving three nodes) have proven fundamental to user to specify which higher-order network structures (cy- understanding social networks [14, 21] and their community cles, feed-forward loops, etc.) should be preserved by the structure [10, 26, 29]. Most importantly, higher-order spec- network clustering. Higher-order network structures of in- tral clustering would allow for greater modeling flexibility terest are represented using a tensor, which we then partition as the application would drive which higher-order network by developing a multilinear spectral method. -
The Latest in Tpetra: Trilinos' Parallel Sparse Linear Algebra
Photos placed in horizontal position with even amount of white space between photos The latest in Tpetra: Trilinos’ and header parallel sparse linear algebra Photos placed in horizontal position with even amount of white space Mark Hoemmen between photos and header Sandia National Laboratories 23 Apr 2019 SAND2019-4556 C (UUR) Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. Tpetra: parallel sparse linear algebra § “Parallel”: MPI + Kokkos § Tpetra implements § Sparse graphs & matrices, & dense vecs § Parallel kernels for solving Ax=b & Ax=λx § MPI communication & (re)distribution § Tpetra supports many linear solvers § Key Tpetra features § Can manage > 2 billion (10^9) unknowns § Can pick the type of values: § Real, complex, extra precision § Automatic differentiation § Types for stochastic PDE discretizations 2 Over 15 years of Tpetra 2004-5: Stage 1 rewrite of Tpetra to use Paul 2005-10: Mid 2010: Assemble Deprecate & Sexton ill-tested I start Kokkos 2.0 new Tpetra purge for starts research- work on Kokkos team; gather Trilinos 13 Tpetra ware Tpetra 2.0 requirements 2005 2007 2009 2011 2013 2015 2017 2019 Trilinos 10.0 Trilinos 11.0 Trilinos 12.0 Trilinos 13.0 ??? Trilinos 9.0 2006 2008 2010 2012 2014 2016 2018 2020 2008: Chris Baker Fix bugs; rewrite Stage 2: purge Improve GPU+MPI -
Warthog: a MOOSE-Based Application for the Direct Code Coupling of BISON and PROTEUS (MS-15OR04010310)
ORNL/TM-2015/532 Warthog: A MOOSE-Based Application for the Direct Code Coupling of BISON and PROTEUS (MS-15OR04010310) Alexander J. McCaskey Approved for public release. Stuart Slattery Distribution is unlimited. Jay Jay Billings September 2015 DOCUMENT AVAILABILITY Reports produced after January 1, 1996, are generally available free via US Department of Energy (DOE) SciTech Connect. Website: http://www.osti.gov/scitech/ Reports produced before January 1, 1996, may be purchased by members of the public from the following source: National Technical Information Service 5285 Port Royal Road Springfield, VA 22161 Telephone: 703-605-6000 (1-800-553-6847) TDD: 703-487-4639 Fax: 703-605-6900 E-mail: [email protected] Website: http://www.ntis.gov/help/ordermethods.aspx Reports are available to DOE employees, DOE contractors, Energy Technology Data Ex- change representatives, and International Nuclear Information System representatives from the following source: Office of Scientific and Technical Information PO Box 62 Oak Ridge, TN 37831 Telephone: 865-576-8401 Fax: 865-576-5728 E-mail: [email protected] Website: http://www.osti.gov/contact.html This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal lia- bility or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or rep- resents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not nec- essarily constitute or imply its endorsement, recommendation, or fa- voring by the United States Government or any agency thereof. -
Geeng Started with Trilinos
Geng Started with Trilinos Michael A. Heroux Photos placed in horizontal position Sandia National with even amount of white space Laboratories between photos and header USA Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND NO. 2011-XXXXP Outline - Some (Very) Basics - Overview of Trilinos: What it can do for you. - Trilinos Software Organization. - Overview of Packages. - Package Management. - Documentation, Building, Using Trilinos. 1 Online Resource § Trilinos Website: https://trilinos.org § Portal to Trilinos resources. § Documentation. § Mail lists. § Downloads. § WebTrilinos § Build & run Trilinos examples in your web browser! § Need username & password (will give these out later) § https://code.google.com/p/trilinos/wiki/TrilinosHandsOnTutorial § Example codes: https://code.google.com/p/trilinos/w/list 2 WHY USE MATHEMATICAL LIBRARIES? 3 § A farmer had chickens and pigs. There was a total of 60 heads and 200 feet. How many chickens and how many pigs did the farmer have? § Let x be the number of chickens, y be the number of pigs. § Then: x + y = 60 2x + 4y = 200 § From first equaon x = 60 – y, so replace x in second equaon: 2(60 – y) + 4y = 200 § Solve for y: 120 – 2y + 4y = 200 2y = 80 y = 40 § Solve for x: x = 60 – 40 = 20. § The farmer has 20 chickens and 40 pigs. 4 § A restaurant owner purchased one box of frozen chicken and another box of frozen pork for $60. Later the owner purchased 2 boxes of chicken and 4 boxes of pork for $200. -
Preconditioned Spectral Clustering for Stochastic Block Partition
Preconditioned Spectral Clustering for Stochastic Block Partition Streaming Graph Challenge (Preliminary version at arXiv.) David Zhuzhunashvili Andrew Knyazev University of Colorado Mitsubishi Electric Research Laboratories (MERL) Boulder, Colorado 201 Broadway, 8th Floor, Cambridge, MA 02139-1955 Email: [email protected] Email: [email protected], WWW: http://www.merl.com/people/knyazev Abstract—Locally Optimal Block Preconditioned Conjugate The graph partitioning problem can be formulated in terms Gradient (LOBPCG) is demonstrated to efficiently solve eigen- of spectral graph theory, e.g., using a spectral decomposition value problems for graph Laplacians that appear in spectral of a graph Laplacian matrix, obtained from a graph adjacency clustering. For static graph partitioning, 10–20 iterations of LOBPCG without preconditioning result in ˜10x error reduction, matrix with non-negative entries that represent positive graph enough to achieve 100% correctness for all Challenge datasets edge weights describing similarities of graph vertices. Most with known truth partitions, e.g., for graphs with 5K/.1M commonly, a multi-way graph partitioning is obtained from (50K/1M) Vertices/Edges in 2 (7) seconds, compared to over approximated “low frequency eigenmodes,” i.e. eigenvectors 5,000 (30,000) seconds needed by the baseline Python code. Our corresponding to the smallest eigenvalues, of the graph Lapla- Python code 100% correctly determines 98 (160) clusters from the Challenge static graphs with 0.5M (2M) vertices in 270 (1,700) cian matrix. Alternatively and, in some cases, e.g., normalized seconds using 10GB (50GB) of memory. Our single-precision cuts, equivalently, one can operate with a properly scaled graph MATLAB code calculates the same clusters at half time and adjacency matrix, turning it into a row-stochastic matrix that memory. -
Block Locally Optimal Preconditioned Eigenvalue Xolvers BLOPEX
Block Locally Optimal Preconditioned Eigenvalue Xolvers I.Lashuk, E.Ovtchinnikov, M.Argentati, A.Knyazev Block Locally Optimal Preconditioned Eigenvalue Xolvers BLOPEX Ilya Lashuk, Merico Argentati, Evgenii Ovtchinnikov, Andrew Knyazev (speaker) Department of Mathematical Sciences and Center for Computational Mathematics University of Colorado at Denver and Health Sciences Center Supported by the Lawrence Livermore National Laboratory and the National Science Foundation Center for Computational Mathematics, University of Colorado at Denver Block Locally Optimal Preconditioned Eigenvalue Xolvers I.Lashuk, E.Ovtchinnikov, M.Argentati, A.Knyazev Abstract Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) is a package, written in C, that at present includes only one eigenxolver, Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG). BLOPEX supports parallel computations through an abstract layer. BLOPEX is incorporated in the HYPRE package from LLNL and is availabe as an external block to the PETSc package from ANL as well as a stand-alone serial library. Center for Computational Mathematics, University of Colorado at Denver Block Locally Optimal Preconditioned Eigenvalue Xolvers I.Lashuk, E.Ovtchinnikov, M.Argentati, A.Knyazev Acknowledgements Supported by the Lawrence Livermore National Laboratory, Center for Applied Scientific Computing (LLNL–CASC) and the National Science Foundation. We thank Rob Falgout, Charles Tong, Panayot Vassilevski, and other members of the Hypre Scalable Linear Solvers project team for their help and support. We thank Jose E. Roman, a member of SLEPc team, for writing the SLEPc interface to our Hypre LOBPCG solver. The PETSc team has been very helpful in adding our BLOPEX code as an external package to PETSc. Center for Computational Mathematics, University of Colorado at Denver Block Locally Optimal Preconditioned Eigenvalue Xolvers I.Lashuk, E.Ovtchinnikov, M.Argentati, A.Knyazev CONTENTS 1. -
A New Overview of the Trilinos Project
Scientific Programming 20 (2012) 83–88 83 DOI 10.3233/SPR-2012-0355 IOS Press A new overview of the Trilinos project Michael A. Heroux ∗ and James M. Willenbring Sandia National Laboratories, Albuquerque, NM, USA Abstract. Since An Overview of the Trilinos Project [ACM Trans. Math. Softw. 31(3) (2005), 397–423] was published in 2005, Trilinos has grown significantly. It now supports the development of a broad collection of libraries for scalable computational science and engineering applications, and a full-featured software infrastructure for rigorous lean/agile software engineering. This growth has created significant opportunities and challenges. This paper focuses on some of the most notable changes to the Trilinos project in the last few years. At the time of the writing of this article, the current release version of Trilinos was 10.12.2. Keywords: Software libraries, software frameworks 1. Introduction teams. Scientific libraries have a natural scope such that one or a few people work closely together on a The Trilinos project is a community of develop- cohesive collection of concerns. In 1999 the Acceler- ers and a collection of reusable software components ated Strategic Computing Initiative (ASCI) was fund- called packages. Trilinos provides a large and grow- ing many new algorithmic and software efforts. These ing collection of open-source software libraries for efforts included an explicit requirement to increase the scalable parallel computational science and engineer- level of formal software engineering practices and pro- ing applications, as well as a software infrastructure cesses, including efforts of small development teams. that supports a rigorous lean/agile software lifecycle Even before this time, there were numerous disjoint model [13]. -
Big Graph Processing: Partitioning and Aggregated Querying
Big Graph Processing : Partitioning and Aggregated Querying Ghizlane Echbarthi To cite this version: Ghizlane Echbarthi. Big Graph Processing : Partitioning and Aggregated Querying. Databases [cs.DB]. Université de Lyon, 2017. English. NNT : 2017LYSE1225. tel-01707153 HAL Id: tel-01707153 https://tel.archives-ouvertes.fr/tel-01707153 Submitted on 12 Feb 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. No d’ordre NNT : 2017LYSE1225 THESEDEDOCTORATDEL’UNIVERSIT` EDELYON´ op´er´ee au sein de l’Universit´e Claude Bernard Lyon 1 Ecole´ Doctorale ED512 Infomath Sp´ecialit´e de doctorat : Informatique Soutenue publiquement le 23/10/2017, par : Ghizlane ECHBARTHI Big Graph Processing: Partitioning and Aggregated Querying Devant le jury compos´ede: Nom Pr´enom, grade/qualit´e, ´etablissement/entreprise Pr´esident(e) Karine Zeitouni, Professeur, Universit´e de Versaille Rapporteure Raphael Couturier, Professeur, Universit´e de Franche-Comt´e Rapporteur Angela Bonifati, Professeur, Universit´e Lyon 1 Examinatrice Mohand Boughanem, Professeur, Universit´e Paul Sabatier Examinateur Hamamache Kheddouci, Professeur, Universit´e Lyon 1 Directeur de th`ese Remerciements Tout d’abord je tiens `a remercier mon directeur de th`ese Hamamache KHEDDOUCI, de m’avoir acceuilli au sein de l’´equipe GOAL, d’avoir constamment veill´e sur la qualit´e de mon travail et de m’avoir apport´e un soutien scientifique et morale d’une grande importance.