DOCSLIB.ORG

Sparse Matrices and Optimized Parallel Implementations ______

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sparse Matrices and Optimized Parallel Implementations ______ Sparse Matrices and Optimized Parallel Implementations ________________________________________________ Stan Tomov Innovative Computing Laboratory Computer Science Department The University of Tennessee April 5, 2017 CS 594 Lecture Notes Slide 1 / 32 04/05/2017 Topics Projection in Scientific Computing Sparse matrices, PDEs, Numerical parallel implementations solution, Tools, etc. Iterative Methods Slide 2 / 32 Outline • Part I – Discussion • Part II – Sparse matrix computations • Part III – Reordering algorithms and parallelization Slide 3 / 32 Part I Discussion Slide 4 / 32 Orthogonalization • We can orthonormalize non-orthogonal basis. How? Other approaches: QR using Householder transformation (as in LAPACK), Cholesky, or/and SVD on normal equations (as in the homework) Hybrid CPU-GPU CPU computation NVIDIA Quadro FX 5600 AMD Opteron (tm), Processor 265 (1.8 Ghz, 1 GB cache)‏ 130 2 120 1.8 110 128 vectors 1.6 100 90 1.4 80 1.2 70 1 60 Gflop/s QR _it_svd Gflop/s MGS 50 QR _it_C hol/svd 0.8 QR _it_svd QR _it_C hol/svd 40 0.6 30 0.4 20 0.2 10 0 0 1 2 4 8 16 32 64 128 256 512 1 2 4 8 16 32 64 128 256 512 Vector size x 1,000 Vector size x 1,000 Slide 5 / 32 What if the basis is not orthonormal? • If we do not want to orthonormalize: / 'Multiply' by x , ..., x to get u ≈ P u = c1 x1 + c2 x2 + . + cm xm 1 m (u , x1) = c1 (x1 , x1)+ c2 (x2 , x1)+ . + cm (xm , x1) . (u, xm) = c1 (x1 , xm)+ c2 (x2 , xm)+ . + cm (xm , xm) • These are the so called Petrov-Galerkin conditions • We saw examples of their use in * optimization, and * PDE discretization, e.g. FEM Slide 6 / 32 What if the basis is not orthonormal? • If we do not want to orthonormalize, e.g. in FEM / 'Multiply' by ϕ , ..., ϕ to get u ≈ P u = c1 ϕ1 + c2 ϕ2 + . + c7 ϕ7 1 7 a 7x7 system a( c1 ϕ1 + c2 ϕ2 + . + c7 ϕ7 , ϕi) = F(ϕi) for i = 1, ... , 7 l Two examples of basis functions ϕi l The more ϕi overlap, the denser the resulting matrix l Spectral element methods (high-order FEM) (Image taken from http://www.urel.feec.vutbr.cz/~raida)‏ Slide 7 / 32 Stencil Computations • K. Datta, S. Kamil, S. Williams, L. Oliker, J. Shaft, K. Yelick, “Optimization and Performance Modeling of Stencil Computations on Modern Microprocessors”, SIAM Review, 2008. http://bebop.cs.berkeley.edu/pubs/datta2008-stencil-sirev.pdf Slide 8 / 32 Part II Sparse matrix computations Slide 9 / 32 Sparse matrices • Sparse matrix: substantial part of the coefficients is zero • Naturally arise from PDE discretizations – finite differences, FEM, etc.; we saw examples in the 5-point finite difference operator 1-D piece-wise linear FEM 10 Vertices are indexed, e.g. 5 6 7 2 Row 6 will have 5 non-zero elements: Row 3, for example, will have 3 non-zeros A6,2, A6,5, A6,6, A6,7, and A6,10 A3,2, A3,3, A3,4 Slide 10 / 32 Sparse matrices • In general : 100 115 * Degrees of freedom (DOF), associated for ex. with vertices (or edges, faces, etc.), are indexed 332 * A basis function is associated with every DOF 10 201 (unknown) * A Petrov-Galerkin condition (equation) is derived for every basis function, representing 35 a row in the resulting system * Only 'a few' elements per row will be nonzero as the basis functions have local support eg. what happens at '10' is described by - eg. row 10, using continuous piecewise linear FEM, will have 6 nonzeroes: the physical state at '10' and the A , A , A , A , A ,A 10,10 10,35 10,100 10,332 10,115 10,201 neighbouring 35, 201, 115, 100, and 332. - physical intuition behind: PDEs describe changes in physical processes; describing/discretizing these changes numerically, based only on local/neighbouring information, results in sparse matrices Slide 11 / 32 Sparse matrices • Can we take advantage of this sparse structure? – To solve for example very large problems – To solve them efficiently • Yes! There are algorithms – Linear solvers and preconditioners (to cover some in the last 2 lectures) – Efficient data storage and implementation (next ...) Slide 12 / 32 Sparse matrix formats • It pays to avoid storing the zeros! • Common sparse storage formats: – AIJ – Compressed row/column storage (CRS/CCS) – Compressed diagonal storage (CDS) * for more see the 'Templates' book http://www.netlib.org/linalg/html_templates/node90.html#SECTION00931000000000000000 – Blocked versions (why?) Slide 13 / 32 AIJ • Stored in 3 arrays – The same length – No order implied I J AIJ 1 1 1 1 2 2 2 1 3 2 3 4 3 2 5 3 4 6 4 3 7 4 5 8 Slide 14 / 32 CRS • Stored in 3 arrays – J and AIJ the same length – I (representing rows) is compressed I J AIJ 1 1 1 3 2 2 5 1 3 7 3 4 array I : think of it as pointers to 9 2 5 where next row starts 4 6 3 7 CCS : similar but J is compressed 5 8 Slide 15 / 32 CDS • For matrices with non-zeros along sub-diagonals Sub-diagonal index sub-diagonals -1 0 1 Slide 16 / 32 Performance (Mat-vec product) • Notoriously bad for running at just a fraction of the performance peak! • Why ? Consider mat-vec product for matrix in CRS: for i = 1, n for j = I[i], I[i+1]-1 y[i] += AIJ [j] * x [ J[j] ] Slide 17 / 32 Performance (Mat-vec product) • Notoriously bad for running at just a fraction of the performance peak! • Why ? Consider mat-vec product for matrix in CRS: for i = 1, n * Irregular indirect memory access for x for j = I[i], I[i+1]-1 - result in cache trashing * performance often y[i] += AIJ [j] * x [ J[j] ] <10% peak 100 115 332 10 201 35 Slide 18 / 32 Performance (Mat-vec product) * Performance of mat-vec products of various sizes on a 2.4 GHz Pentium 4 * An example from Gahvari et.al.: http://bebop.cs.berkeley.edu/pubs/gahvari2007-spmvbench-spec.pdf Slide 19 / 32 Performance (Mat-vec product) • How to improve the performance? – A common technique (as done for dense linear algebra) is blocking (register, cache: next ... ) – Index reordering (in Part II) – Exploit special matrix structure (e.g., symmetry, bands, other structures) Slide 20 / 32 Block Compressed Row Storage (BCRS) • Example of using 2x2 blocks BI BJ AIJ 1 1 1 3 2 0 4 3 0 4 2 3 * Reduced storage for indexes 5 0 * Drawback: add 0s 6 7 * What block size to choose? 8 9 * BCRS for register blocking * Discussion? Slide 21 / 32 BCRS Slide 22 / 32 Cache blocking • Improve cache reuse for x in Ax by splitting A into a set of sparse matrices, e.g. Sparse matrix and its splitting For more info check: SPARSITY: An Optimization Framework for Sparse Matrix Kernels (International Journal of High Performance Computing Applications, 18 (1), pp. 135-158, February 2004. Eun-Jin Im, K. Yelick, R. Vuduc Slide 23 / 32 Part III Reordering algorithms and Parallelization Slide 24 / 32 Reorder to preserve locality 100 115 332 10 201 35 e.g., Cuthill-McKee Ordering: start from arbitrary node, say '10' and reorder * '10' becomes 0 * neighbours are ordered next to become 1, 2, 3, 4, 5, denote this as level 1 * neighbours to level 1 nodes are next consecutively reordered, and so on until end Slide 25 / 32 Cuthill-McKee Ordering • Reversing the ordering (RCM) results in ordering that is better for sparse LU • Reduces matrix bandwidth (see example) • Improves cache performance • Can be used as partitioner (parallelization) p1 but in general does not reduce edge cut p2 p3 p4 Slide 26 / 32 Self-Avoiding Walks (SAW) • Enumeration of mesh elements through 'consecutive elements' (sharing face, edge, vertex, etc.) * similar to space-filling curves but for unstructured meshes * improves cache reuse * can be used as partitioner with good load balance but in general does not reduce edge cut Slide 27 / 32 Graph partitioning • Refer back to Lecture #9, Part II Mesh Generation and Load Balancing • Can be used for reordering • Metis/ParMetis: – multilevel partitioning – Good load balance and minimize edge cut Slide 28 / 32 Parallel Mat-Vec Product • Easiest way: p1 – 1D partitioning p2 p3 – May lead to load unbalance (why?) p4 – May need a lot of communication for x • Can use any of the just mentioned techniques • Most promising seems to be spectral multilevel methods (as in Metis/ParMetis) Slide 29 / 32 Possible optimizations • Block communication – To send the min. required part of x – e.g., pre-compute blocks of interfaces • Load balance, minimize edge cut – e.g., a good partitioner would do it • Reordering • Advantage of additional structure (symmetry, bands, etc) Slide 30 / 32 Comparison Distributed memory implementation (by X. Li, L. Oliker, G. Heber, R. Biswas) – ORIG ordering has large edge cut (interprocessor comm) and poor locality (high number of cache misses) – MeTiS minimizes edge cut, while SAW minimizes cache misses Slide 31 / 32 Learning Goals • Efficient sparse computations are challenging! • Computational challenges and issues related to sparse matrices – Data formats – Optimization • Blocking • Reordering • Other • Parallel sparse Mat-Vec product – Code optimization opportunities Slide 32 / 32 .
Load more

Recommended publications
  • Recursive Approach in Sparse Matrix LU Factorization
    51 Recursive approach in sparse matrix LU factorization Jack Dongarra, Victor Eijkhout and the resulting matrix is often guaranteed to be positive Piotr Łuszczek∗ definite or close to it. However, when the linear sys- University of Tennessee, Department of Computer tem matrix is strongly unsymmetric or indefinite, as Science, Knoxville, TN 37996-3450, USA is the case with matrices originating from systems of Tel.: +865 974 8295; Fax: +865 974 8296 ordinary differential equations or the indefinite matri- ces arising from shift-invert techniques in eigenvalue methods, one has to revert to direct methods which are This paper describes a recursive method for the LU factoriza- the focus of this paper. tion of sparse matrices. The recursive formulation of com- In direct methods, Gaussian elimination with partial mon linear algebra codes has been proven very successful in pivoting is performed to find a solution of Eq. (1). Most dense matrix computations. An extension of the recursive commonly, the factored form of A is given by means technique for sparse matrices is presented. Performance re- L U P Q sults given here show that the recursive approach may per- of matrices , , and such that: form comparable to leading software packages for sparse ma- LU = PAQ, (2) trix factorization in terms of execution time, memory usage, and error estimates of the solution. where: – L is a lower triangular matrix with unitary diago- nal, 1. Introduction – U is an upper triangular matrix with arbitrary di- agonal, Typically, a system of linear equations has the form: – P and Q are row and column permutation matri- Ax = b, (1) ces, respectively (each row and column of these matrices contains single a non-zero entry which is A n n A ∈ n×n x where is by real matrix ( R ), and 1, and the following holds: PPT = QQT = I, b n b, x ∈ n and are -dimensional real vectors ( R ).
    [Show full text]
  • Abstract 1 Introduction
    Implementation in ScaLAPACK of Divide-and-Conquer Algorithms for Banded and Tridiagonal Linear Systems A. Cleary Department of Computer Science University of Tennessee J. Dongarra Department of Computer Science University of Tennessee Mathematical Sciences Section Oak Ridge National Laboratory Abstract Described hereare the design and implementation of a family of algorithms for a variety of classes of narrow ly banded linear systems. The classes of matrices include symmetric and positive de - nite, nonsymmetric but diagonal ly dominant, and general nonsymmetric; and, al l these types are addressed for both general band and tridiagonal matrices. The family of algorithms captures the general avor of existing divide-and-conquer algorithms for banded matrices in that they have three distinct phases, the rst and last of which arecompletely paral lel, and the second of which is the par- al lel bottleneck. The algorithms have been modi ed so that they have the desirable property that they are the same mathematical ly as existing factorizations Cholesky, Gaussian elimination of suitably reordered matrices. This approach represents a departure in the nonsymmetric case from existing methods, but has the practical bene ts of a smal ler and more easily hand led reduced system. All codes implement a block odd-even reduction for the reduced system that al lows the algorithm to scale far better than existing codes that use variants of sequential solution methods for the reduced system. A cross section of results is displayed that supports the predicted performance results for the algo- rithms. Comparison with existing dense-type methods shows that for areas of the problem parameter space with low bandwidth and/or high number of processors, the family of algorithms described here is superior.
    [Show full text]
  • Overview of Iterative Linear System Solver Packages
    Overview of Iterative Linear System Solver Packages Victor Eijkhout July, 1998 Abstract Description and comparison of several packages for the iterative solu- tion of linear systems of equations. 1 1 Intro duction There are several freely available packages for the iterative solution of linear systems of equations, typically derived from partial di erential equation prob- lems. In this rep ort I will give a brief description of a numberofpackages, and giveaninventory of their features and de ning characteristics. The most imp ortant features of the packages are which iterative metho ds and preconditioners supply; the most relevant de ning characteristics are the interface they present to the user's data structures, and their implementation language. 2 2 Discussion Iterative metho ds are sub ject to several design decisions that a ect ease of use of the software and the resulting p erformance. In this section I will give a global discussion of the issues involved, and how certain p oints are addressed in the packages under review. 2.1 Preconditioners A go o d preconditioner is necessary for the convergence of iterative metho ds as the problem to b e solved b ecomes more dicult. Go o d preconditioners are hard to design, and this esp ecially holds true in the case of parallel pro cessing. Here is a short inventory of the various kinds of preconditioners found in the packages reviewed. 2.1.1 Ab out incomplete factorisation preconditioners Incomplete factorisations are among the most successful preconditioners devel- op ed for single-pro cessor computers. Unfortunately, since they are implicit in nature, they cannot immediately b e used on parallel architectures.
    [Show full text]
  • 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.
    [Show full text]
  • Over-Scalapack.Pdf
    1 2 The Situation: Parallel scienti c applications are typically Overview of ScaLAPACK written from scratch, or manually adapted from sequen- tial programs Jack Dongarra using simple SPMD mo del in Fortran or C Computer Science Department UniversityofTennessee with explicit message passing primitives and Mathematical Sciences Section Which means Oak Ridge National Lab oratory similar comp onents are co ded over and over again, co des are dicult to develop and maintain http://w ww .ne tl ib. org /ut k/p e opl e/J ackDo ngarra.html debugging particularly unpleasant... dicult to reuse comp onents not really designed for long-term software solution 3 4 What should math software lo ok like? The NetSolve Client Many more p ossibilities than shared memory Virtual Software Library Multiplicityofinterfaces ! Ease of use Possible approaches { Minimum change to status quo Programming interfaces Message passing and ob ject oriented { Reusable templates { requires sophisticated user Cinterface FORTRAN interface { Problem solving environments Integrated systems a la Matlab, Mathematica use with heterogeneous platform Interactiveinterfaces { Computational server, like netlib/xnetlib f2c Non-graphic interfaces Some users want high p erf., access to details {MATLAB interface Others sacri ce sp eed to hide details { UNIX-Shell interface Not enough p enetration of libraries into hardest appli- Graphic interfaces cations on vector machines { TK/TCL interface Need to lo ok at applications and rethink mathematical software { HotJava Browser
    [Show full text]
  • Prospectus for the Next LAPACK and Scalapack Libraries
    Prospectus for the Next LAPACK and ScaLAPACK Libraries James Demmel1, Jack Dongarra23, Beresford Parlett1, William Kahan1, Ming Gu1, David Bindel1, Yozo Hida1, Xiaoye Li1, Osni Marques1, E. Jason Riedy1, Christof Voemel1, Julien Langou2, Piotr Luszczek2, Jakub Kurzak2, Alfredo Buttari2, Julie Langou2, Stanimire Tomov2 1 University of California, Berkeley CA 94720, USA, 2 University of Tennessee, Knoxville TN 37996, USA, 3 Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA, 1 Introduction Dense linear algebra (DLA) forms the core of many scienti¯c computing appli- cations. Consequently, there is continuous interest and demand for the devel- opment of increasingly better algorithms in the ¯eld. Here 'better' has a broad meaning, and includes improved reliability, accuracy, robustness, ease of use, and most importantly new or improved algorithms that would more e±ciently use the available computational resources to speed up the computation. The rapidly evolving high end computing systems and the close dependence of DLA algo- rithms on the computational environment is what makes the ¯eld particularly dynamic. A typical example of the importance and impact of this dependence is the development of LAPACK [1] (and later ScaLAPACK [2]) as a successor to the well known and formerly widely used LINPACK [3] and EISPACK [3] libraries. Both LINPACK and EISPACK were based, and their e±ciency depended, on optimized Level 1 BLAS [4]. Hardware development trends though, and in par- ticular an increasing Processor-to-Memory speed gap of approximately 50% per year, started to increasingly show the ine±ciency of Level 1 BLAS vs Level 2 and 3 BLAS, which prompted e®orts to reorganize DLA algorithms to use block matrix operations in their innermost loops.
    [Show full text]
  • Jack Dongarra: Supercomputing Expert and Mathematical Software Specialist
    Biographies Jack Dongarra: Supercomputing Expert and Mathematical Software Specialist Thomas Haigh University of Wisconsin Editor: Thomas Haigh Jack J. Dongarra was born in Chicago in 1950 to a he applied for an internship at nearby Argonne National family of Sicilian immigrants. He remembers himself as Laboratory as part of a program that rewarded a small an undistinguished student during his time at a local group of undergraduates with college credit. Dongarra Catholic elementary school, burdened by undiagnosed credits his success here, against strong competition from dyslexia.Onlylater,inhighschool,didhebeginto students attending far more prestigious schools, to Leff’s connect material from science classes with his love of friendship with Jim Pool who was then associate director taking machines apart and tinkering with them. Inspired of the Applied Mathematics Division at Argonne.2 by his science teacher, he attended Chicago State University and majored in mathematics, thinking that EISPACK this would combine well with education courses to equip Dongarra was supervised by Brian Smith, a young him for a high school teaching career. The first person in researcher whose primary concern at the time was the his family to go to college, he lived at home and worked lab’s EISPACK project. Although budget cuts forced Pool to in a pizza restaurant to cover the cost of his education.1 make substantial layoffs during his time as acting In 1972, during his senior year, a series of chance director of the Applied Mathematics Division in 1970– events reshaped Dongarra’s planned career. On the 1971, he had made a special effort to find funds to suggestion of Harvey Leff, one of his physics professors, protect the project and hire Smith.
    [Show full text]
  • Distibuted Dense Numerical Linear Algebra Algorithms on Massively Parallel Architectures: DPLASMA
    Distibuted Dense Numerical Linear Algebra Algorithms on massively parallel architectures: DPLASMA † George Bosilca∗, Aurelien Bouteiller∗, Anthony Danalis∗ , Mathieu Faverge∗, Azzam Haidar∗, ‡ §¶ Thomas Herault∗ , Jakub Kurzak∗, Julien Langou , Pierre Lemarinier∗, Hatem Ltaief∗, † Piotr Luszczek∗, Asim YarKhan∗ and Jack Dongarra∗ ∗University of Tennessee Innovative Computing Laboratory †Oak Ridge National Laboratory ‡University Paris-XI §University of Colorado Denver ¶Research was supported by the National Science Foundation grant no. NSF CCF-811520 Abstract—We present DPLASMA, a new project related first two, combined with the economic necessity of using to PLASMA, that operates in the distributed memory many thousands of CPUs to scale up to petascale and regime. It uses a new generic distributed Direct Acyclic larger systems. Graph engine for high performance computing (DAGuE). Our work also takes advantage of some of the features of More transistors and slower clocks means multicore DAGuE, such as DAG representation that is independent of designs and more parallelism required. The modus problem-size, overlapping of communication and computa- operandi of traditional processor design, increase the tion, task prioritization, architecture-aware scheduling and transistor density, speed up the clock rate, raise the management of micro-tasks on distributed architectures voltage has now been blocked by a stubborn set of that feature heterogeneous many-core nodes. The origi- nality of this engine is that it is capable of translating physical barriers: excess heat produced, too much power a sequential nested-loop code into a concise and synthetic consumed, too much voltage leaked. Multicore designs format which it can be interpret and then execute in a are a natural response to this situation.
    [Show full text]
  • LAPACK WORKING NOTE 195: SCALAPACK's MRRR ALGORITHM 1. Introduction. Since 2005, the National Science Foundation Has Been Fund
    LAPACK WORKING NOTE 195: SCALAPACK’S MRRR ALGORITHM CHRISTOF VOMEL¨ ∗ Abstract. The sequential algorithm of Multiple Relatively Robust Representations, MRRR, can compute numerically orthogonal eigenvectors of an unreduced symmetric tridiagonal matrix T ∈ Rn×n with O(n2) cost. This paper describes the design of ScaLAPACK’s parallel MRRR algorithm. One emphasis is on the critical role of the representation tree in achieving both numerical accuracy and parallel scalability. A second point concerns the favorable properties of this code: subset computation, the use of static memory, and scalability. Unlike ScaLAPACK’s Divide & Conquer and QR, MRRR can compute subsets of eigenpairs at reduced cost. And in contrast to inverse iteration which can fail, it is guaranteed to produce a numerically satisfactory answer while maintaining memory scalability. ParEig, the parallel MRRR algorithm for PLAPACK, uses dynamic memory allocation. This is avoided by our code at marginal additional cost. We also use a different representation tree criterion that allows for more accurate computation of the eigenvectors but can make parallelization more difficult. AMS subject classifications. 65F15, 65Y15. Key words. Symmetric eigenproblem, multiple relatively robust representations, ScaLAPACK, numerical software. 1. Introduction. Since 2005, the National Science Foundation has been funding an initiative [19] to improve the LAPACK [3] and ScaLAPACK [14] libraries for Numerical Linear Algebra computations. One of the ambitious goals of this initiative is to put more of LAPACK into ScaLAPACK, recognizing the gap between available sequential and parallel software support. In particular, parallelization of the MRRR algorithm [24, 50, 51, 26, 27, 28, 29], the topic of this paper, is identified as one key improvement to ScaLAPACK.
    [Show full text]
  • Reciprocity Relations
    Reciprocity relations - May 11, 2006 Using the reciprocity theorem a relation between the seismic reflection response and transmission data can be derived. This relation is used to calculate transmission data from reflection data measured at the surface. In this presentation the steps involved in this calculation are explained in detail and possible pitfalls are discussed. At the end a simple example is given to illustrate the discussed procedure. 1 One-way reciprocity One-way reciprocity relation of the convolution type: + − − + 2x + − − + 2x {PA PB − PA PB }d H = {PA PB − PA PB }d H (1) Z∂D0 Z∂Dm One-way reciprocity relation of the correlation type: + ∗ + − ∗ − 2x + ∗ + − ∗ − 2x {(PA ) PB − (PA ) PB }d H = {(PA ) PB − (PA ) PB }d H (2) Z∂D0 Z∂Dm the medium parameters in both states A and B are identical, lossless and 3-D inhomogeneous and the domain D is source free. State A State B x xA x xB z0 ∂D0 ∂D0 R0(x, xA,ω) R0(x, xB ,ω) ∆z ∆z D T0(x, xA,ω) D T0(x, xB ,ω) x zm x ∂Dm ∂Dm 1 Surface ∂D0 Field State A State B + P δ(xH − xH,A)sA(ω) δ(xH − xH,B )sB(ω) − P R0(x, xA,ω)sA(ω) R0(x, xB ,ω)sB(ω) Surface ∂Dm + P T0(x, xA,ω)sA(ω) T0(x, xB ,ω)sB(ω) P − 0 0 One-way reciprocity relation of the convolution type: 2 sAsB {δ(xH − xH,A)R0(x, xB ) − R0(x, xA)δ(xH − xH,B )}d xH =0 (3a) Z∂D0 R0(xA, xB )= R0(xB , xA) (3b) One-way reciprocity relation of the correlation type: x x ∗ x x x x 2x ∗ x x x x 2x δ( H,A − H,B) − R0( , A)R0( , B )d H = T0 ( , A)T0( , B )d H (4) Z∂D0 Z∂Dm Equation (4) can be used to estimate transmission effects on wave propagation (coda) from reflection data at the surface.
    [Show full text]
  • On the Performance and Energy Efficiency of Sparse Linear Algebra on Gpus
    Original Article The International Journal of High Performance Computing Applications 1–16 On the performance and energy efficiency Ó The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav of sparse linear algebra on GPUs DOI: 10.1177/1094342016672081 hpc.sagepub.com Hartwig Anzt1, Stanimire Tomov1 and Jack Dongarra1,2,3 Abstract In this paper we unveil some performance and energy efficiency frontiers for sparse computations on GPU-based super- computers. We compare the resource efficiency of different sparse matrix–vector products (SpMV) taken from libraries such as cuSPARSE and MAGMA for GPU and Intel’s MKL for multicore CPUs, and develop a GPU sparse matrix–matrix product (SpMM) implementation that handles the simultaneous multiplication of a sparse matrix with a set of vectors in block-wise fashion. While a typical sparse computation such as the SpMV reaches only a fraction of the peak of current GPUs, we show that the SpMM succeeds in exceeding the memory-bound limitations of the SpMV. We integrate this kernel into a GPU-accelerated Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) eigensolver. LOBPCG is chosen as a benchmark algorithm for this study as it combines an interesting mix of sparse and dense linear algebra operations that is typical for complex simulation applications, and allows for hardware-aware optimizations. In a detailed analysis we compare the performance and energy efficiency against a multi-threaded CPU counterpart. The reported performance and energy efficiency results are indicative of sparse computations on supercomputers. Keywords sparse eigensolver, LOBPCG, GPU supercomputer, energy efficiency, blocked sparse matrix–vector product 1Introduction GPU-based supercomputers.
    [Show full text]
  • Best Practice Guide
    Best Practice Guide - Application porting and code-optimization activities for European HPC systems Sebastian Lührs, JSC, Forschungszentrum Jülich GmbH, Germany Björn Dick, HLRS, Germany Chris Johnson, EPCC, United Kingdom Luca Marsella, CSCS, Switzerland Gerald Mathias, LRZ, Germany Cristian Morales, BSC, Spain Lilit Axner, KTH, Sweden Anastasiia Shamakina, HLRS, Germany Hayk Shoukourian, LRZ, Germany 30-04-2020 1 Best Practice Guide - Application porting and code-optimization ac- tivities for European HPC systems Table of Contents 1. Introduction .............................................................................................................................. 3 2. EuroHPC Joint Undertaking ........................................................................................................ 3 3. Application porting and optimization activities in PRACE ................................................................ 4 3.1. Preparatory Access .......................................................................................................... 4 3.1.1. Overview ............................................................................................................ 4 3.1.2. Application and review processes ............................................................................ 4 3.1.3. Project white papers .............................................................................................. 4 3.2. SHAPE ........................................................................................................................
    [Show full text]