Exploiting dynamic sparse matrices for performance portable linear algebra operations

This paper presents a low-overhead optimizer for the ubiquitous sparse matrix-vector multiplication (SpMV) kernel. Architectural diversity among different processors together with structural diversity among different sparse matrices lead to bottleneck diversity. This justifies an SpMV optimizer that is both matrix- and architecture-adaptive through runtime specialization. To this direction, we present an approach that first identifies the performance bottlenecks of SpMV for a given sparse matrix on the target platform either through profiling or by matrix property inspection, and then selects suitable optimizations to tackle those bottlenecks. Our optimization pool is based on the widely used Compressed Sparse Row (CSR) sparse matrix storage format and has low preprocessing overheads, making our overall approach practical even in cases where fast decision making and optimization setup is required. We evaluate our optimizer on three x86-based computing platforms and demonstrate that it is able to distinguish and appropriately optimize SpMV for the majority of matrices in a representative test suite, leading to significant speedups over the CSR and Inspector-Executor CSR SpMV kernels available in the latest release of the Intel MKL library.

Sparse Matrix-Vector Multiplication (SpMV) kernel dominates the computing cost in numerous scientific applications. Many implementations based on different sparse formats were proposed recently for this kernel on the GPU side. Since the performance of these sparse formats varies significantly according to the sparsity characteristics of the input matrix and the hardware specifications, no one of them can be considered as the best one to use for every sparse matrix. In this paper, we address the problem of selecting the best representation for a given sparse matrix on GPU by using a machine learning approach. First, we present some interesting and easy to compute features for characterizing the sparse matrices on GPU. Second, we use a multiclass Support Vector Machine (SVM) classifier to select the best format for each input matrix. We consider in this paper four popular formats (COO, CSR, ELL, and HYB), but our work can be extended to support more sparse representations. Experimental results on two different GPUs (Fermi GTX 580 and Maxwell GTX 980 Ti) show that we achieved more than 98% of the performance possible with a perfect selection.

Several manufacturers have already started to commercialize near-bank Processing-In-Memory (PIM) architectures, after decades of research efforts. Near-bank PIM architectures place simple cores close to DRAM banks. Recent research demonstrates that they can yield significant performance and energy improvements in parallel applications by alleviating data access costs. Real PIM systems can provide high levels of parallelism, large aggregate memory bandwidth and low memory access latency, thereby being a good fit to accelerate the Sparse Matrix Vector Multiplication (SpMV) kernel. SpMV has been characterized as one of the most significant and thoroughly studied scientific computation kernels. It is primarily a memory-bound kernel with intensive memory accesses due its algorithmic nature, the compressed matrix format used, and the sparsity patterns of the input matrices given. This paper provides the first comprehensive analysis of SpMV on a real-world PIM architecture, and presents SparseP, the first SpMV library for real PIM architectures. We make two key contributions. First, we design efficient SpMV algorithms to accelerate the SpMV kernel in current and future PIM systems, while covering a wide variety of sparse matrices with diverse sparsity patterns. Second, we provide the first comprehensive analysis of SpMV on a real PIM architecture. Specifically, we conduct our rigorous experimental analysis of SpMV kernels in the UPMEM PIM system, the first publicly-available real-world PIM architecture. Our extensive evaluation provides new insights and recommendations for software designers and hardware architects to efficiently accelerate the SpMV kernel on real PIM systems. For more information about our thorough characterization on the SpMV PIM execution, results, insights and the open-source SparseP software package [21], we refer the reader to the full version of the paper [3, 4]. The SparseP software package is publicly and freely available at https://github.com/CMU-SAFARI/SparseP.

Sparse linear algebra includes the fundamental and important operations in various large-scale scientific computing and real-world applications. There exists performance bottleneck for sparse linear algebra since it mainly contains the memory-bound computations with low arithmetic intensity. How to improve its performance has increasingly become a focus of research efforts. Using parallel computing techniques to accelerate sparse linear algebra is currently the most popular method, while facing various challenges, e.g., large-scale data brings difficulties in storage, and the sparsity of data leads to irregular memory accesses and parallel load imbalance. Therefore, this article provides a comprehensive overview on acceleration of sparse linear algebra operations using parallel computing platforms, where we focus on four main classifications: sparse matrix-vector multiplication (SpMV), sparse matrix-sparse vector multiplication (SpMSpV), sparse general matrix-matrix multiplication (SpGEMM), and sparse tensor algebra. The takeaways from this article include the following: understanding the challenges of accelerating linear sparse algebra on various hardware platforms; understanding how structured data sparsity can improve storage efficiency; understanding how to optimize parallel load balance; understanding how to improve the efficiency of memory accesses; understanding how do the adaptive frameworks automatically select the optimal algorithms; and understanding recent design trends for acceleration of parallel sparse linear algebra.

A row displacement method compresses efficiently a sparse matrix into a one-dimensional array. The access time with this method is O(l), but the application was restricted to the static matrices. In order to extend the use of the row displacement method to the dynamic matrices, the algorithms for insertion and deletion are proposed and the effectivity is confirmed by theoretical and empirical observations.

The development, analysis and production of algorithms in sparse linear algebra often requires the use of test problems to demonstrate the effectiveness and applicability of the algorithms. Many algorithms have been developed in the context of specific application areas and have been tested in the context of sets of test problems collected by the developers. Comparisons of algorithms across application areas and comparisons between algorithms has often been incomplete, due to the lack of a comprehensive set of test problems. Additionally we believe that a comprehensive set of test problems will lead to a better understanding of the range of structures in sparse matrix problems and thence to better classification and development of algorithms. We have agreed to sponsor and maintain a general library of sparse matrix test problems, available on request to anyone for a nominal fee to cover postal charges. Contributors to the library will, of course, receive a free copy.

MDS matrices are used in the design of diffusion layers in many block ciphers and hash functions due to their optimal branch number. But MDS matrices, in general, have costly implementations. So in search for efficiently implementable MDS matrices, there have been many proposals. In particular, circulant, Hadamard, and recursive MDS matrices from companion matrices have been widely studied. In a recent work, recursive MDS matrices from sparse DSI matrices are studied, which are of interest due to their low fixed cost in hardware implementation. In this paper, we present results on the exhaustive search for (recursive) MDS matrices over GL(4, F2). Specifically, circulant MDS matrices of order 4, 5, 6, 7, 8; Hadamard MDS matrices of order 4, 8; recursive MDS matrices from companion matrices of order 4; recursive MDS matrices from sparse DSI matrices of order 4, 5, 6, 7, 8 are considered. It is to be noted that the exhaustive search is impractical with a naive approach. We first use some linear algebra tools to restrict the search to a smaller domain and then apply some space-time trade-off techniques to get the solutions. From the set of solutions in the restricted domain, one can easily generate all the solutions in the full domain. From the experimental results, we can see the (non) existence of (involutory) MDS matrices for the choices mentioned above. In particular, over GL(4, F2), we provide companion matrices of order 4 that yield involutory MDS matrices, circulant MDS matrices of order 8, and establish the nonexistence of involutory circulant MDS matrices of order 6, 8, circulant MDS matrices of order 7, sparse DSI matrices of order 4 that yield involutory MDS matrices, and sparse DSI matrices of order 5, 6, 7, 8 that yield MDS matrices. To the best of our knowledge, these results were not known before. For the choices mentioned above, if such MDS matrices exist, we provide base sets of MDS matrices, from which all the MDS matrices with the least cost (with respect to d-XOR and s-XOR counts) can be obtained. We also take this opportunity to present some results on the search for sparse DSI matrices over finite fields that yield MDS matrices. We establish that there is no sparse DSI matrix S of order 8 over F28 such that S8 is MDS.

Particle loading as a design parameter for composite radiation shielding

Context Strategies for achieving greater ruminant livestock productivity are essential to meet the food demands of growing populations, but sustainable changes are difficult to identify given the inherent complexity of such systems. Systems models can address this issue by allowing the impact of potential changes to be explored. Aims To develop a holistic systems model for goat production in an extensive Mediterranean environment which could allow changes in key management factors influencing the system to be investigated. Methods Initially, a conceptual comprehensive stock-and-flow model of a representative Mediterranean goat production system was constructed. This was used to identify informative indicators that would represent the overall technical and economic performance of the system. Sub-models were then assembled to build the full systems model. The model was parameterised with data collected over 3 years for goat holdings in northern Morocco. Scenario analysis techniques are used to explore the strategies that optimise performance under climate and feed price challenges. Key results Meat production is particularly important during periods of drought when increased meat yields can counteract the expected reduction in milk yields, to protect human food security, prevent excessive rangeland degradation and preserve natural nutritional resources. Feed price shocks during drought can have significant negative impacts on the system and zero feed input is shown to be a more sustainable strategy than reliance on high price feed during drought. Any alternative feed sources need to have a high forage component to reduce grazing periods significantly and promote rangeland preservation. Implications A diverse management strategy with a mixed meat and dairy semi-intensive production is more stable than specialised dairy systems and allows goat production and financial viability of intensification to be maintained under climatic stress; maintaining meat production was necessary to optimise performance. Conclusions The model allows improved insight into management strategies which could optimise animal husbandry performance in goat subsistence systems. However, the work also demonstrates the difficulty of constructing a truly holistic model since, to be practical, such constructs must necessarily be bounded; parameter selection and the limits to the boundaries imposed are inevitably critical.

This article describes the R package mcglm implemented for fitting multivariate covariance generalized linear models (McGLMs). McGLMs provide a general statistical modeling framework for normal and non-normal multivariate data analysis, designed to handle multivariate response variables, along with a wide range of temporal and spatial correlation structures defined in terms of a covariance link function and a matrix linear predictor involving known symmetric matrices. The models take non-normality into account in the conventional way by means of a variance function, and the mean structure is modeled by means of a link function and a linear predictor. The models are fitted using an estimating function approach based on second-moment assumptions. This provides a unified approach to a wide variety of different types of response variables and covariance structures, including multivariate extensions of repeated measures, time series, longitudinal, genetic, spatial and spatio-temporal structures. The mcglm package allows a flexible specification of the mean and covariance structures, and explicitly deals with multivariate response variables, through a user friendly formula interface similar to the ordinary glm function. Illustrations in this article cover a wide range of applications from the traditional one response variable Gaussian mixed models to multivariate spatial models for areal data using the multivariate Tweedie distribution. Additional features, such as robust and bias-corrected standard errors for regression parameters, residual analysis, measures of goodness-of-fit and model selection using the score information criterion are discussed through six worked examples. The mcglm package is a full R implementation based on the Matrix package which provides efficient access to BLAS (basic linear algebra subroutines), Lapack (dense matrix), TAUCS (sparse matrix) and UMFPACK (sparse matrix) routines for efficient linear algebra in R.

Multiplying a sparse matrix with a vector, denoted spmv, is a fundamental operation in linear algebra with several applications. Hence, efficient and scalable implementation of spmv has been a topic of immense research. Recent efforts are aimed at implementations on GPUs, multicore architectures, and such emerging computational platforms. Owing to the highly irregular nature of spmv, it is observed that GPUs and CPUs can offer comparable performance.In this paper, we propose three heterogeneous algorithms for spmv that simultaneously utilize both the CPU and the GPU. This is shown to lead to better resource utilization apart from performance gains. Our experiments of the work division schemes on standard datasets indicate that it is not in general possible to choose the most appropriate scheme given a matrix. We therefore consider a class of sparse matrices that exhibit a scale-free nature and identify a scheme that works well for such matrices. Finally, we use simple and effective mechanisms to determine the appropriate amount of work to be alloted to the CPU and the GPU.

Linear algebra provides the building blocks for a wide variety of scientific and engineering simulation codes. Users of these codes face a world of continuously changing algorithms and high-performance implementations. In this paper, we describe new capabilities of our Lighthouse framework, whose goal is to match specific problems in the area of high-performance numerical computing with the best available solutions. Lighthouse's innovative strategy eliminates intensive reading of documents and automates the process for developing linear algebra software. Lighthouse provides a searchable taxonomy of popular but difficult to use numerical software for dense and sparse linear algebra while providing the user with the best algorithms for a given problem based on machine learning methods. We introduce the design of Lighthouse and show examples of its interface. We also present algorithm classification results for the preconditioned iterative linear solvers in the Parallel Extensible Toolkit for Scientific Computation (PETSc) and the Trilinos library.

The efficiency of a sparse linear algebra operation heavily relies on the ability of the sparse matrix storage format to exploit the computing power of the underlying hardware. Since no format is universally better than the others across all possible kinds of operations and computers, sparse linear algebra software packages should provide facilities to easily implement and integrate new storage formats within a sparse linear algebra application without the need to modify it; it should also allow to dynamically change a storage format at run-time depending on the specific operations to be performed. Aiming at these important features, we present an Object Oriented design model for a sparse linear algebra package which relies on Design Patterns. We show that an implementation of our model can be efficiently achieved through some of the unique features of the Fortran 2003 language. Experimental results show that the proposed software infrastructure improves the modularity and ease of use of the code at no performance loss.

Computation of absorbing boundary conditions at the discrete level for acoustic waves in the frequency domain

Random access schemes for efficient FPGA SpMV acceleration