Petsc, a large c library, containing many different matrix solvers for a variety of matrix storage formats. Pdf general sparse matrix matrix multiplication spgemm is an essential building block in a number of applications. Hpcc, volume lncs 3726, pages 807816, sorrento, italy, september 2005. Read about how singularity is designed as a container solution for high performance computing hpc and is currently one of the most popular container implementation used on many hpc platforms. Smdm computations are au, and va, multiplication of a large sparse m x n matrix a by a matrix v of k rows of length m or a matrix u of k columns of length n, k matrix matrix multiplications with the tall u and wide v are also needed. Pdf on jan 1, 1984, sergio pissanetzky and others published sparse matrix technology find, read and cite all the research you need on researchgate. Implementing sparse matrix vector multiplication on throughputoriented processors nathan bell and michael garland proceedings of supercomputing 09 efficient sparse matrix vector multiplication on cuda nathan bell and michael garland nvidia technical report nvr2008004, december 2008 iterative methods for sparse linear systems yousef saad.
Nearmemory data transformation for efficient sparse matrix. Implementing sparse matrixvector multiplication on throughputoriented processors. Lets agree on computing flops for the symmetric sparse. Sparse matrix vector multiplication spmv is the dominant kernel in scientific simulations. Parallel sparse matrix matrix multiplication and indexing. The ideal hpc programming language communications of the. Matrix benchmark suite from uf sparse matrix collection for instance, in 8 the authors state that they are computing y ax and they are not considering symmetry, the data was chosen from the uf sparse matrix collection, the performance is calculated using 2.
Analyzing the performance of a sparse matrix vector multiply. Sparse di rect solvers based on the multifrontal method or the general sparse method now outperform band or envelope solvers on vector supercomputers such as the cray xmp. From the igure, we can ind that all existing solutions own higher l2 cache miss ratios on the scalefree sparse matrices comparing to the hpc sparse matrices. Implementing sparse matrix vector multiplication on throughputoriented processors. Sparse matrixvector multiplication spmv kernel dominates the computing cost in. An effort to create a more relevant metric for ranking hpc systems potential replacement for the high performance linpack hpl benchmark currently hpl is used by the top500 benchmark hpcg high performance conjugate gradient standalone code that measures the performance of basic operations sparse matrix vector. Pdf performance evaluation of algorithms for sparse. Matrix reorderings have shown potential to improve performance, but can incur substantial cost 2. Modeling the execution time of the sparse matrix vector multiplication spmv on a current cpu architecture is especially complex due to i irregular memory accesses.
Sparse matrix data structures i only nonzero elements are stored in sparse matrix data structures, which makes possible the storage of sparse matrices of large dimension. We demonstrate how to use design patterns to implement an interface for sparse matrix computations on nvidia gpus starting from psblas, an existing sparse matrix library, and from existing sets of. B is a sparse matrix, as it contains only comparably few nonzero elements as does a. The ideal hpc programming language communications of the acm. Nearmemory data transformation for efficient sparse matrix multivector multiplication. We consider the spmv operation y abstractwe present the design and implementation of. Computing sparse reducedrank approximations to sparse matrices michael w. In this paper, we implement sparse matrix vector multiplication spmv for. We choose spmv as it is a common operation in scientific and hpc applications. We investigate the use of the multistep successive preconditioning strategies msp to construct a class of parallel multilevel sparse approximate inverse sai preconditioners.
Accelerating the lobpcg method on gpus using a blocked. Sparse matrix computations proceedings of the 1992 acm. This paper summarizes progress in the use of direct methods for solving very large sparse symmetric positive definite systems of linear equations on vector supercomputers. When cache blocking sparse matrix vector multiply works and why applicable algebra in engineering, communication and computing, march 2007 rajesh nishtala, richard w. I sparse matrix computations have alarger integer overhead associated with each oatingpoint operation. Novel hpc techniques to batch execution of many variable size blas computations on gpus ahmad abdelfa. To handle the stringent performance requirements of future exascale high performance computing hpc applications, hpc systems need ultraefficient heterogeneous. The high performance computing hpc community has therefore continuously invested a lot of effort to provide an efficient spmv kernel on modern cpu architectures. Lets agree on computing flops for the symmetric sparse matrix. International journal of high performance computing applications vol. The implementations of these kernels in hypre and the code optimizations will be discussed.
In proceedings of the 28th acm international conference on supercomputing. Spmm is a generalization of spmv in which a sparse nbym matrix a is multiplied by a tall and narrow dense nbyk matrix b k sparse matrix computations on nvidia gpus starting from psblas, an existing sparse matrix library, and from existing sets of. Inconsistent library availability, whether a result of licensing or installation and bundling issues, also is an issue. Outline 1 matrix operations importance dense and sparse matrices matrices and arrays 2 matrix vector multiplication rowsweep algorithm columnsweep algorithm 3 matrix matrix multiplication \standard algorithm ijkforms cps343 parallel and hpc matrix multiplication spring 2020 232. Spmm is a generalization of spmv in which a sparse nbym matrix a is multiplied by a tall and narrow dense nbyk matrix b k sparse matrix 307 lemma 2.
Optimization of sparse matrixvector multiplication on. New sparse matrix support for bsr block sparse row format. Many software libraries support sparse matrices, and provide solvers for sparse matrix equations. I still, using sparsity can save large amounts of cpu time and also memory space. The sparse matrixvector product spmv is an important operation in. International conference on high performance computing in. Towards a universal fpga matrix vector multiplication architecture srinidhi kestury, john d. Optimizing and autotuning scalefree sparse matrixvector.
One of the best paper candidates also is a finalist for the acm gordon bell prize, which will be presented at sc16. Principles of programming languages popl 2012, philadelpha. This need for optimization and tuning at runtime is a major distinction from the dense case. Equally, in order to meet these workload demands, hpc systems continue to grow in size and computational power. In proceedings of the conference on high performance computing networking, storage and analysis sc 09. High performance computing for mechanical simulations. Stewart university of maryland, college park in many applicationslatent semantic indexing, for exampleit is required to obtain a reduced rank approximation to a sparse matrix. Pdf sparse matrix vector multiplication spmv is an important ker nel in both traditional high performance computing and emerging dataintensive. Novel hpc techniques to batch execution of many variable size. Sparse grids fundamentals not a solver, not multigrid not a sparse matrix approach not next.
Sparse matrix data structures for high performance computing. The number of zerovalued elements divided by the total number of elements e. Proceedings of the international conference on high performance. Keywords spmv, code optimization, simd, vectorization, hpc.
Depiction of the fubiniradon transform 1, based on the fourier slice theorem. As hpc resources continue to play a role in scientific discoveries, simulations are increasing in size and complexity. Towards a universal fpga matrixvector multiplication. For large and small k, the structure of the algorithm does not need to depend on the structure of the sparse matrix a, whereas for intermediate densities it is possible and necessary to find. In the international conference for high performance computing, networking, storage, and analysis sc 19, november 17. These sharedmemory kernels for single gpu are the building blocks of distributed matrix operations required by the solver across multiple gpus and compute nodes. Progress in sparse matrix methods for large linear systems. Algorithms and data structures for matrixfree finite element. Proceedings of the symposium on high performance computing accelerating the lobpcg method on gpus using a blocked sparse matrix vector product. This work presents a systematic exploration on the promise and special challenges of deep learning for sparse matrix format selectiona problem of determining the best storage format for a. The international journal of high performance computing applications. Pdffront matter title, copyright, welcome message from chair of the. Abstractsparse matrixvector multiplication smvm is a crucial primitive.
We group them by application domains and present their average l2 cache miss ratios in figure 1 details are shown in figure 7. It is well known that the sparse matrix vector product ax requires two. Bebop berkeley benchmarking and optimization home page. Optimization of gpu kernels for sparse matrix computations in. Grey ballard, christopher siefert, and jonathan hu. Performance of sparse matrixmultiple vectors multiplication. In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. A high memory bandwidth fpga accelerator for sparse matrix. Steps toward simplifying sparse matrix data structures. Sparse matrix data structures summary i sparse matrix algorithms aremore complicatedthan their dense equivalents, as we saw for sparse vector addition. An input adaptive sparse matrixvector multiplication. Sparse matrices for highperformance graph computation. Sparse matrix multiplication is an important algorithm in a wide variety of problems, including graph algorithms, simulations and linear solving to name a few. Acm conference on international conference on supercomputing.
Sparse matrix partitioning for optimizing spmv on cpugpu. Themajorapplication of sparse matrix techniques in optimization up to the present has been in the implementation of the simplex method for linear programming lp see, e. We do not use independent set ordering, but a diagonal dominance based matrix permutation to build a multilevel structure. Daichi fujiki, niladrish chatterjee, donghyuk lee, and mike oconnor. Highperformance computing hpc systems are crucial to scientific simulation and analysis. Sparse matrix vector multiplication on multicore and accelerator systems. Cpus and gpus to the forefront of high performance computing hpc.
Sparse grids higher dimensionalities and hpc aspects. Sparse matrices for highperformance graph computation john r. Fast sparse matrix vector multiplication by exploiting variable block structure. Seven technical papers selected as best paper nominees out of 442 technical papers submitted to sc16, only 81 were accepted and of these, seven have been nominated for the conferences best paper award. It is well known that the sparse matrix vector product ax requires two floatingpoint operations per each non zero element in a. Efficient sparse matrixvector multiplication on cuda.
While analytical models may yield accurate estimates for the total number of cache hitsmisses, they often fail to predict accurately the total execution. In 2015 ieee 22nd international conference on high performance computing hipc, pages 6474, dec 2015. Reducing interprocess communication overhead in parallel. This makes scalability analysis increasingly important. By contrast, if most of the elements are nonzero, then the matrix is considered dense. In fact, commercial codes for large lp problems seem to have predated codes for sparse linear equations even though solving a sparse lpproblem requires. However, it is performing double pass to compute sparsesparse matrix product. Sparse matrix based hpc tomography 7 fourier transforms and multiplication of s2cm n with a sinogram in vec tor form 1 n, n n np, sparse matrix vector multiplication or spmv produces a tomogram of dimension 1 m. Application finetuning performance optimization highperformance interconnect algorithmic cleverness to trade compute and io overlap compute and io with programming model graph analytics matrix methods deep learning handle x bigger datasets with a 100x better speedup with queries. Nov 15, 2019 this makes scalability analysis increasingly important. Proceedings of the 24th high performance computing.
Fast sparse matrix multiplication on gpu acm digital library. Suitesparse, a suite of sparse matrix algorithms, geared toward the direct solution of sparse linear systems. Computing the block triangular form of a sparse matrix. Sparse matrixvector multiplication spmv is an important computation kernel widely used in hpc and data centers. Sparse matrix vector multiplication we consider sparse matrix vector multiplication y ax with vectors y and x and a sparse n n matrix a. Efficient sparse matrix vector multiplication on cuda. Progress in sparse matrix methods for large linear systems on. It describes how to effectively bridge the gap between deep learning and the special needs of the pillar hpc problem through. Bridging the gap between deep learning and sparse matrix. If you recommend, please tell me the advantages and disadvantages of it, and the reason why you recommend it. Generating fast sparse matrix vector multiplication from a high. Pdf registerbased implementation of the sparse general. Optimization of gpu kernels for sparse matrix computations.
Adaptive multilevel blocking optimization for sparse matrix vector. Michael baderjhpc algorithms and applicationsjparallel spmvjws 2017 2. The sparse matrix vector multiply spmv is fundamental to a large class of hpc applications. Reducing communication costs for sparse matrix multiplication within algebraic multigrid. Performance computing for mechanical simulations using ansys jeff beisheim ansys, inc. Acm transactions on intelligent systems and technology tist 23. Analyzing the performance of a sparse matrix vector. Article pdf available in acm sigplan notices 486 october 2012 with 127 reads. Scalable directiterative hybrid solver for sparse matrices on multicore and. Computing the sparse matrix vector product using block. High order seismic simulations on the intel xeon phi processor knights landing.
The communication required for each spmv is dependent on the matrix being multiplied. Article information, pdf download for sparse matrix partitioning for. The international journal of high performance computing applications, online. I formats are generally optimized for sparse matrix vector. Leveraging nodelevel parallelism and architectural features 3. Performance modeling of the sparse matrixvector product. Sparse matrix dense matrix smdm multiplications are useful in block krylov or block lanczos methods. Yet, there are but a few works related to acceleration of sparse matrix multiplication on a gpu. However, when computing the number of flops for the symmetric sparse matrix vector product symspmv some subtleties should be considered because the number of non zero nnz elements reported on symmetric sparse matrices varies from one research work to another. Computing the sparse matrix vector product using blockbased. The objective of the project ticoh is to address the issue of currently unsatisfactory utilization of heterogeneous computing for irregular problems such as graph and sparse matrix processing.
Pdf adaptive sparse tiling for sparse matrix multiplication. Spmm is a generalization of spmv in which a sparse nbym matrix a is multiplied by a tall and narrow dense nbyk matrix b k sparse matrix multiplication is an important algorithm in a wide variety of problems, including graph algorithms, simulations and linear solving to name a few. In proceedings of the 2016 acm ieee 43rd annual international symposium on computer architecture, pp. The sparse matrix vector product spmv is a fundamental operation in many scientific applications from various fields. For small size problems, this is not a problem, however for large size problems e.
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