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Mathematical Software

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preprint2017arXiv

The Unum Number Format: Mathematical Foundations, Implementation and Comparison to IEEE 754 Floating-Point Numbers

This thesis examines a modern concept for machine numbers based on interval arithmetic called 'Unums' and compares it to IEEE 754 floating-point arithmetic, evaluating possible uses of this format where floating-point numbers are inadequate. In the course of this examination, this thesis builds theoretical foundations for IEEE 754 floating-point numbers, interval arithmetic based on the projectively extended real numbers and Unums.

0 citations0 reviewsNo code linkedOpen access
preprint2014arXiv

An implementation of a randomized algorithm for principal component analysis

Recent years have witnessed intense development of randomized methods for low-rank approximation. These methods target principal component analysis (PCA) and the calculation of truncated singular value decompositions (SVD). The present paper presents an essentially black-box, fool-proof implementation for Mathworks' MATLAB, a popular software platform for numerical computation. As illustrated via several tests, the randomized algorithms for low-rank approximation outperform or at least match the classical techniques (such as Lanczos iterations) in basically all respects: accuracy, computational efficiency (both speed and memory usage), ease-of-use, parallelizability, and reliability. However, the classical procedures remain the methods of choice for estimating spectral norms, and are far superior for calculating the least singular values and corresponding singular vectors (or singular subspaces).

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preprint2016arXiv

An efficient hybrid tridiagonal divide-and-conquer algorithm on distributed memory architectures

In this paper, an efficient divide-and-conquer (DC) algorithm is proposed for the symmetric tridiagonal matrices based on ScaLAPACK and the hierarchically semiseparable (HSS) matrices. HSS is an important type of rank-structured matrices.Most time of the DC algorithm is cost by computing the eigenvectors via the matrix-matrix multiplications (MMM). In our parallel hybrid DC (PHDC) algorithm, MMM is accelerated by using the HSS matrix techniques when the intermediate matrix is large. All the HSS algorithms are done via the package STRUMPACK. PHDC has been tested by using many different matrices. Compared with the DC implementation in MKL, PHDC can be faster for some matrices with few deflations when using hundreds of processes. However, the gains decrease as the number of processes increases. The comparisons of PHDC with ELPA (the Eigenvalue soLvers for Petascale Applications library) are similar. PHDC is usually slower than MKL and ELPA when using 300 or more processes on Tianhe-2 supercomputer.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

jInv -- a flexible Julia package for PDE parameter estimation

Estimating parameters of Partial Differential Equations (PDEs) from noisy and indirect measurements often requires solving ill-posed inverse problems. These so called parameter estimation or inverse medium problems arise in a variety of applications such as geophysical, medical imaging, and nondestructive testing. Their solution is computationally intense since the underlying PDEs need to be solved numerous times until the reconstruction of the parameters is sufficiently accurate. Typically, the computational demand grows significantly when more measurements are available, which poses severe challenges to inversion algorithms as measurement devices become more powerful. In this paper we present jInv, a flexible framework and open source software that provides parallel algorithms for solving parameter estimation problems with many measurements. Being written in the expressive programming language Julia, jInv is portable, easy to understand and extend, cross-platform tested, and well-documented. It provides novel parallelization schemes that exploit the inherent structure of many parameter estimation problems and can be used to solve multiphysics inversion problems as is demonstrated

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preprint2016arXiv

SimTensor: A synthetic tensor data generator

SimTensor is a multi-platform, open-source software for generating artificial tensor data (either with CP/PARAFAC or Tucker structure) for reproducible research on tensor factorization algorithms. SimTensor is a stand-alone application based on MATALB. It provides a wide range of facilities for generating tensor data with various configurations. It comes with a user-friendly graphical user interface, which enables the user to generate tensors with complicated settings in an easy way. It also has this facility to export generated data to universal formats such as CSV and HDF5, which can be imported via a wide range of programming languages (C, C++, Java, R, Fortran, MATLAB, Perl, Python, and many more). The most innovative part of SimTensor is this that can generate temporal tensors with periodic waves, seasonal effects and streaming structure. it can apply constraints such as non-negativity and different kinds of sparsity to the data. SimTensor also provides this facility to simulate different kinds of change-points and inject various types of anomalies. The source code and binary versions of SimTensor is available for download in http://www.simtensor.org.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Julia Implementation of the Dynamic Distributed Dimensional Data Model

Julia is a new language for writing data analysis programs that are easy to implement and run at high performance. Similarly, the Dynamic Distributed Dimensional Data Model (D4M) aims to clarify data analysis operations while retaining strong performance. D4M accomplishes these goals through a composable, unified data model on associative arrays. In this work, we present an implementation of D4M in Julia and describe how it enables and facilitates data analysis. Several experiments showcase scalable performance in our new Julia version as compared to the original Matlab implementation.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

From NoSQL Accumulo to NewSQL Graphulo: Design and Utility of Graph Algorithms inside a BigTable Database

Google BigTable's scale-out design for distributed key-value storage inspired a generation of NoSQL databases. Recently the NewSQL paradigm emerged in response to analytic workloads that demand distributed computation local to data storage. Many such analytics take the form of graph algorithms, a trend that motivated the GraphBLAS initiative to standardize a set of matrix math kernels for building graph algorithms. In this article we show how it is possible to implement the GraphBLAS kernels in a BigTable database by presenting the design of Graphulo, a library for executing graph algorithms inside the Apache Accumulo database. We detail the Graphulo implementation of two graph algorithms and conduct experiments comparing their performance to two main-memory matrix math systems. Our results shed insight into the conditions that determine when executing a graph algorithm is faster inside a database versus an external system---in short, that memory requirements and relative I/O are critical factors.

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preprint2016arXiv

Benchmarking the Graphulo Processing Framework

Graph algorithms have wide applicablity to a variety of domains and are often used on massive datasets. Recent standardization efforts such as the GraphBLAS specify a set of key computational kernels that hardware and software developers can adhere to. Graphulo is a processing framework that enables GraphBLAS kernels in the Apache Accumulo database. In our previous work, we have demonstrated a core Graphulo operation called \textit{TableMult} that performs large-scale multiplication operations of database tables. In this article, we present the results of scaling the Graphulo engine to larger problems and scalablity when a greater number of resources is used. Specifically, we present two experiments that demonstrate Graphulo scaling performance is linear with the number of available resources. The first experiment demonstrates cluster processing rates through Graphulo's TableMult operator on two large graphs, scaled between $2^{17}$ and $2^{19}$ vertices. The second experiment uses TableMult to extract a random set of rows from a large graph ($2^{19}$ nodes) to simulate a cued graph analytic. These benchmarking results are of relevance to Graphulo users who wish to apply Graphu

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

OpenSBLI: A framework for the automated derivation and parallel execution of finite difference solvers on a range of computer architectures

Exascale computing will feature novel and potentially disruptive hardware architectures. Exploiting these to their full potential is non-trivial. Numerical modelling frameworks involving finite difference methods are currently limited by the 'static' nature of the hand-coded discretisation schemes and repeatedly may have to be re-written to run efficiently on new hardware. In contrast, OpenSBLI uses code generation to derive the model's code from a high-level specification. Users focus on the equations to solve, whilst not concerning themselves with the detailed implementation. Source-to-source translation is used to tailor the code and enable its execution on a variety of hardware.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Implementation and evaluation of data-compression algorithms for irregular-grid iterative methods on the PEZY-SC processor

Iterative methods on irregular grids have been used widely in all areas of comptational science and engineering for solving partial differential equations with complex geometry. They provide the flexibility to express complex shapes with relatively low computational cost. However, the direction of the evolution of high-performance processors in the last two decades have caused serious degradation of the computational efficiency of iterative methods on irregular grids, because of relatively low memory bandwidth. Data compression can in principle reduce the necessary memory memory bandwidth of iterative methods and thus improve the efficiency. We have implemented several data compression algorithms on the PEZY-SC processor, using the matrix generated for the HPCG benchmark as an example. For the SpMV (Sparse Matrix-Vector multiplication) part of the HPCG benchmark, the best implementation without data compression achieved 11.6Gflops/chip, close to the theoretical limit due to the memory bandwidth. Our implementation with data compression has achieved 32.4Gflops. This is of course rather extreme case, since the grid used in HPCG is geometrically regular and thus its compression effici

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Accelerating BLAS on Custom Architecture through Algorithm-Architecture Co-design

Basic Linear Algebra Subprograms (BLAS) play key role in high performance and scientific computing applications. Experimentally, yesteryear multicore and General Purpose Graphics Processing Units (GPGPUs) are capable of achieving up to 15 to 57% of the theoretical peak performance at 65W to 240W respectively for compute bound operations like Double/Single Precision General Matrix Multiplication (XGEMM). For bandwidth bound operations like Single/Double precision Matrix-vector Multiplication (XGEMV) the performance is merely 5 to 7% of the theoretical peak performance in multicores and GPGPUs respectively. Achieving performance in BLAS requires moving away from conventional wisdom and evolving towards customized accelerator tailored for BLAS through algorithm-architecture co-design. In this paper, we present acceleration of Level-1 (vector operations), Level-2 (matrix-vector operations), and Level-3 (matrix-matrix operations) BLAS through algorithm architecture co-design on a Coarse-grained Reconfigurable Architecture (CGRA). We choose REDEFINE CGRA as a platform for our experiments since REDEFINE can be adapted to support domain of interest through tailor-made Custom Function Units

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

A computer algebra user interface manifesto

Many computer algebra systems have more than 1000 built-in functions, making expertise difficult. Using mock dialog boxes, this article describes a proposed interactive general-purpose wizard for organizing optional transformations and allowing easy fine grain control over the form of the result even by amateurs. This wizard integrates ideas including: * flexible subexpression selection; * complete control over the ordering of variables and commutative operands, with well-chosen defaults; * interleaving the choice of successively less main variables with applicable function choices to provide detailed control without incurring a combinatorial number of applicable alternatives at any one level; * quick applicability tests to reduce the listing of inapplicable transformations; * using an organizing principle to order the alternatives in a helpful manner; * labeling quickly-computed alternatives in dialog boxes with a preview of their results, * using ellipsis elisions if necessary or helpful; * allowing the user to retreat from a sequence of choices to explore other branches of the tree of alternatives or to return quickly to branches already visited; * allowing the user to accumulat

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

dMath: Distributed Linear Algebra for DL

The paper presents a parallel math library, dMath, that demonstrates leading scaling when using intranode, internode, and hybrid-parallelism for deep learning (DL). dMath provides easy-to-use distributed primitives and a variety of domain-specific algorithms including matrix multiplication, convolutions, and others allowing for rapid development of scalable applications like deep neural networks (DNNs). Persistent data stored in GPU memory and advanced memory management techniques avoid costly transfers between host and device. dMath delivers performance, portability, and productivity to its specific domain of support.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Generalized Sampling in Julia

Generalized sampling is a numerically stable framework for obtaining reconstructions of signals in different bases and frames from their samples. In this paper, we will introduce a carefully documented toolbox for performing generalized sampling in Julia. Julia is a new language for technical computing with focus on performance, which is ideally suited to handle the large size problems often encountered in generalized sampling. The toolbox provides specialized solutions for the setup of Fourier bases and wavelets. The performance of the toolbox is compared to existing implementations of generalized sampling in MATLAB.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A Metaprogramming and Autotuning Framework for Deploying Deep Learning Applications

In recent years, deep neural networks (DNNs), have yielded strong results on a wide range of applications. Graphics Processing Units (GPUs) have been one key enabling factor leading to the current popularity of DNNs. However, despite increasing hardware flexibility and software programming toolchain maturity, high efficiency GPU programming remains difficult: it suffers from high complexity, low productivity, and low portability. GPU vendors such as NVIDIA have spent enormous effort to write special-purpose DNN libraries. However, on other hardware targets, especially mobile GPUs, such vendor libraries are not generally available. Thus, the development of portable, open, high-performance, energy-efficient GPU code for DNN operations would enable broader deployment of DNN-based algorithms. Toward this end, this work presents a framework to enable productive, high-efficiency GPU programming for DNN computations across hardware platforms and programming models. In particular, the framework provides specific support for metaprogramming, autotuning, and DNN-tailored data types. Using our framework, we explore implementing DNN operations on three different hardware targets: NVIDIA, AMD,

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preprint2016arXiv

Bidiagonalization with Parallel Tiled Algorithms

We consider algorithms for going from a "full" matrix to a condensed "band bidiagonal" form using orthogonal transformations. We use the framework of "algorithms by tiles". Within this framework, we study: (i) the tiled bidiagonalization algorithm BiDiag, which is a tiled version of the standard scalar bidiagonalization algorithm; and (ii) the R-bidiagonalization algorithm R-BiDiag, which is a tiled version of the algorithm which consists in first performing the QR factorization of the initial matrix, then performing the band-bidiagonalization of the R-factor. For both bidiagonalization algorithms BiDiag and R-BiDiag, we use four main types of reduction trees, namely FlatTS, FlatTT, Greedy, and a newly introduced auto-adaptive tree, Auto. We provide a study of critical path lengths for these tiled algorithms, which shows that (i) R-BiDiag has a shorter critical path length than BiDiag for tall and skinny matrices, and (ii) Greedy based schemes are much better than earlier proposed variants with unbounded resources. We provide experiments on a single multicore node, and on a few multicore nodes of a parallel distributed shared-memory system, to show the super

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preprint2016arXiv

A Case for Malleable Thread-Level Linear Algebra Libraries: The LU Factorization with Partial Pivoting

We propose two novel techniques for overcoming load-imbalance encountered when implementing so-called look-ahead mechanisms in relevant dense matrix factorizations for the solution of linear systems. Both techniques target the scenario where two thread teams are created/activated during the factorization, with each team in charge of performing an independent task/branch of execution. The first technique promotes worker sharing (WS) between the two tasks, allowing the threads of the task that completes first to be reallocated for use by the costlier task. The second technique allows a fast task to alert the slower task of completion, enforcing the early termination (ET) of the second task, and a smooth transition of the factorization procedure into the next iteration. The two mechanisms are instantiated via a new malleable thread-level implementation of the Basic Linear Algebra Subprograms (BLAS), and their benefits are illustrated via an implementation of the LU factorization with partial pivoting enhanced with look-ahead. Concretely, our experimental results on a six core Intel-Xeon processor show the benefits of combining WS+ET, reporting competitive performance in comparison wit

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preprint2015arXiv

A Collection of Challenging Optimization Problems in Science, Engineering and Economics

Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.

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preprint2015arXiv

A parallel edge orientation algorithm for quadrilateral meshes

One approach to achieving correct finite element assembly is to ensure that the local orientation of facets relative to each cell in the mesh is consistent with the global orientation of that facet. Rognes et al. have shown how to achieve this for any mesh composed of simplex elements, and deal.II contains a serial algorithm to construct a consistent orientation of any quadrilateral mesh of an orientable manifold. The core contribution of this paper is the extension of this algorithm for distributed memory parallel computers, which facilitates its seamless application as part of a parallel simulation system. Furthermore, our analysis establishes a link between the well-known Union-Find algorithm and the construction of a consistent orientation of a quadrilateral mesh. As a result, existing work on the parallelisation of the Union-Find algorithm can be easily adapted to construct further parallel algorithms for mesh orientations.

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preprint2016arXiv

GPU-Based Parallel Integration of Large Numbers of Independent ODE Systems

The task of integrating a large number of independent ODE systems arises in various scientific and engineering areas. For nonstiff systems, common explicit integration algorithms can be used on GPUs, where individual GPU threads concurrently integrate independent ODEs with different initial conditions or parameters. One example is the fifth-order adaptive Runge-Kutta-Cash-Karp (RKCK) algorithm. In the case of stiff ODEs, standard explicit algorithms require impractically small time-step sizes for stability reasons, and implicit algorithms are therefore commonly used instead to allow larger time steps and reduce the computational expense. However, typical high-order implicit algorithms based on backwards differentiation formulae (e.g., VODE, LSODE) involve complex logical flow that causes severe thread divergence when implemented on GPUs, limiting the performance. Therefore, alternate algorithms are needed. A GPU-based Runge-Kutta-Chebyshev (RKC) algorithm can handle moderate levels of stiffness and performs significantly faster than not only an equivalent CPU version but also a CPU-based implicit algorithm (VODE) based on results shown in the literature. In this chapter, we present

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preprint2016arXiv

DiffSharp: An AD Library for .NET Languages

DiffSharp is an algorithmic differentiation or automatic differentiation (AD) library for the .NET ecosystem, which is targeted by the C# and F# languages, among others. The library has been designed with machine learning applications in mind, allowing very succinct implementations of models and optimization routines. DiffSharp is implemented in F# and exposes forward and reverse AD operators as general nestable higher-order functions, usable by any .NET language. It provides high-performance linear algebra primitives---scalars, vectors, and matrices, with a generalization to tensors underway---that are fully supported by all the AD operators, and which use a BLAS/LAPACK backend via the highly optimized OpenBLAS library. DiffSharp currently uses operator overloading, but we are developing a transformation-based version of the library using F#'s "code quotation" metaprogramming facility. Work on a CUDA-based GPU backend is also underway.

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