Various popular simulation algorithms in high performance computing hpc, such as computational dynamics for rigid bodies, rely on sparse matrix matrix multiplication spmmm as one of their computational kernels. Highperformance sparse matrixmatrix products on intel. Motivation for tuning sparse matrices sparse matrix kernels can dominate solver time sparse matrixvector multiply spmv spmv. Creating a diagonal matrix with every second diagonal element zero can be done as. Partitioning sparse matrices the sparse matrix corresponds to a graph. Demmel, chair this dissertation presents an automated system to generate highly e cient, platformadapted implementations of sparse matrix kernels. A sparse matrix stores only nonzero elements to save space 9.
Get the next nonzero element of the matrix, ai,j say. These two operations represent a number of challenges for the memory and computation. A key component of the hpmixed procedure is the use of sparse matrix techniques for computing and optimizing the likelihood expression given in the section model assumptions. In the past four years cloud computing has emerged as an alternative platform for high performance computing. There are a few reasons for an almost full sparse matrix being more computationally expensive than just using a full matrix. Keywords spmv, code optimization, simd, vectorization, hpc. Optimization of sparse matrixvector multiplication on. A sparse matrix contains a high percentage of zeros, as compared to dense matrices and to obtain high end performance, special data structures are needed as well as logic to avoid meaningless multiplies. The high performance computing hpc community has therefore continuously invested a lot of effort to provide an efficient spmv kernel on modern cpu architectures. The computational requirements for this kind of operation are enormous, especially. Formats are generally optimized for sparse matrixvector multiplication spmv.
B is a sparse matrix, as it contains only comparably few nonzero elements as does a. The most obvious, as you pointed out, is that sparse elements must be indexed for a general sparse matrix, i believe matlab uses a compressed row storage scheme another, less apparent slowdown, is due to vectorization and pipelining data into the processor. 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 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. Computational advantages of sparse matrices matlab. Improving performance of sparse matrixvector multiplication. Ahmedsameh computerscience,purdueuniversity 1 supportaro,intel,nsf.
If not, the geometric interpretation of pca shows that the first principal component will be close to the vector of means and all subsequent pcs will be orthogonal to it, which will prevent them from approximating any pcs that happen to be close to that first vector. The proposed solution capitalizes on the increased performance of sparse matrixvector multiplication and exploits. Increasingly, gpus are used to accelerate these sparse. First of all, you really do want to center the data. Using sparse matrices to store data that contains a large number of zerovalued elements can both save a significant amount of memory and speed up the processing of that data. Improving the performance of the symmetric sparse matrix. Evaluating hybrid memory cube infrastructure for support of highperformance sparse algorithms 10 base verilog design gups is modified to accept addresses, operations, and data from host software application two test cases. Serial issue of all memory commands to 1 queueport interacting with hmc 2. The sparse matrixvector product spmv is a fundamental operation in many scientific applications from various fields. This class specifies a sparse matrix microbenchmark. Matt j on 6 nov 20 i have a sparse matrix a dimension 4000000 x 000 and i want to calculate the matrix product. Efficient sparse matrixvector multiplication on cuda nathan bell and michael garland nvidia technical report nvr2008004, december 2008 modeldriven autotuning of sparse matrixvector multiply on gpus jee whan choi, amik singh and richard w.
Spmv is an important kernel in scientific computation vendors need to know how well their machines perform it. Heath department of computer science and center of simulation of advanced rockets university of illinois at urbanachampaign abstract sparse matrixvector multiplication spmxv is one of the most important computational kernels in scienti. Nonzero elements of the sparse matrix in information retrieval system. Nearmemory data transformation for efficient sparse matrix multi. Benchmarking sparse matrixvector multiply in just 5 minutes. The international journal of high performance computing applications. Computing the sparse matrix vector product using block. A high memory bandwidth fpga accelerator for sparse matrix.
Hpc algorithms and applications winter 17 sccswiki. Unfortunately, there is still confusion about the cloud model and its advantages and disadvantages. Initialise an iterator for the nonzero elements of the matrix, a. Cloud programming paradigms for technical computing. Lets say, the storage size of the elementtype of the matrix is a, and there are n empty elements and m nonempty elements. For a mesh partitioning, number of neighbors is bounded independent of the mesh size. For many programs in high performance computing, but also in every day. Modelguided performance analysis of the sparse matrix. Improving performance of sparse matrixvector multiplication ali p. Benchmarks for highperformance computing environments. The first vector nonzero vector stores nonzero elements of the sparse matrix. Vuduc proceedings of principles and practice of parallel programming ppopp 2010. I sometimes some zeros are stored explicit zeros to maintain block or symmetric sparsity patterns, for example. Keywordssparse matrix vector multiplication, fpga, accelerator, spmv, smvm, reconfigurable computing, hpc.
Proceedings of the symposium on high performance computingapril. Efficient manipulation of sparse matrices is critical to a wide range of hpc applications. 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 sparsematrix is implemented as an immutable object all operations create new matrix instead of changing the matrix the operation is called on. We can hope that most of the later pcs will be approximately correct, but.
How can i do a memory efficient sparse matrix multiplication. Sparse matrix computations and their io complexity mediatum. 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. All sparse matrixvector multiplication algorithms that i have ever seen boil down to the same steps. Sparse matrix matrix spmm multiplication is involved in a wide range of scientific and technical applications.
If we only store the nonempty elements of the matrix and construct the matrix by filling the rest empty elements in the matrix. Optimization framework for sparse matrix kernels eun. Performance evaluation of sparse matrix multiplication. Due to its central role in the applications and its computational complexity it is of.
Although it has been shown that blockbased kernels help to achieve high performance, they are. The simplest sparse matrix storage structure is coo. Partition the graph in a way such that communication is reduced. Sparse matrix data structures for high performance computing. Dimensionality reduction svd or pca on a large, sparse. Sparse matrixdensevector multiply multiply a dense vector by a sparse matrix one whose entries are mostly zeroes why do we need a benchmark. The spm compressed sparse extendedv kernel has been characterized as one. The sparse matrixvector product spmv is an important operation in. Number of columns in the matrix has to be the same as the size of the. Cloud programming paradigms for technical computing applications geoffrey fox, indiana university dennis gannon, microsoft. The index structure is stored in three sparse vectors in coo. There are two aspects to sparse matrix techniques, namely, sparse matrix storage and sparse matrix computations.
Outline 1 matrix operations importance dense and sparse matrices matrices and arrays 2 matrixvector multiplication rowsweep algorithm columnsweep algorithm 3 matrixmatrix multiplication \standard algorithm ijkforms cps343 parallel and hpc matrix multiplication spring 2020 232. The streamed storage format for sparse matrices showed good. In many simulation applications, the solving of the sparse matrixvector multiplication spmv is critical for high performing computing. For this reason, they have been extremely intriguing in the context of high performance computing hpc. Article information, pdf download for applications of the streamed storage format for. In many scientific problems, the sparse matrix is not arbitrary. Adaptive sparse matrixmatrix multiplication on the gpu people. Fast sparse matrix multiplication on gpu proceedings of the. Graph algorithms in sparse matrix computation many, many graph algorithms have been used, invented, implemented at large scale for sparse matrix computation. Diffusion matrix, source vector assembly scattering elementoperators into sparse matrix and vector sparse matrix vector product during cg solve vector operations level1 blas.