User Interface Design With Matrix Algebra •

It is usually very hard, both for designers and users, to reason reliably about user interfaces. This article shows that 'push button' and 'point and click' user interfaces are algebraic structures. Users effectively do algebra when they interact, and therefore we can be precise about some important design issues and issues of usability. Matrix algebra, in particular, is useful for explicit calculation and for proof of various user interface properties. With matrix algebra, we are able to undertake with ease unusally thorough reviews of real user interfaces: this article examines a mobile phone, a handheld calculator and a digital multimeter as case studies, and draws general conclusions about the approach and its relevance to design

Format Abstraction for Sparse Tensor Algebra Compilers

This paper shows how to build a sparse tensor algebra compiler that is agnostic to tensor formats (data layouts). We develop an interface that describes formats in terms of their capabilities and properties, and show how to build a modular code generator where new formats can be added as plugins. We then describe six implementations of the interface that compose to form the dense, CSR/CSF, COO, DIA, ELL, and HASH tensor formats and countless variants thereof. With these implementations at hand, our code generator can generate code to compute any tensor algebra expression on any combination of the aforementioned formats. To demonstrate our technique, we have implemented it in the taco tensor algebra compiler. Our modular code generator design makes it simple to add support for new tensor formats, and the performance of the generated code is competitive with hand-optimized implementations. Furthermore, by extending taco to support a wider range of formats specialized for different application and data characteristics, we can improve end-user application performance. For example, if input data is provided in the COO format, our technique allows computing a single matrix-vector multiplication directly with the data in COO, which is up to 3.6$\times$ faster than by first converting the data to CSR.Comment: Presented at OOPSLA 201

DOLFIN: Automated Finite Element Computing

We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled and seamlessly integrated with efficient implementations of computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This paper discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code

GraphBLAST: A High-Performance Linear Algebra-based Graph Framework on the GPU

High-performance implementations of graph algorithms are challenging to implement on new parallel hardware such as GPUs because of three challenges: (1) the difficulty of coming up with graph building blocks, (2) load imbalance on parallel hardware, and (3) graph problems having low arithmetic intensity. To address some of these challenges, GraphBLAS is an innovative, on-going effort by the graph analytics community to propose building blocks based on sparse linear algebra, which will allow graph algorithms to be expressed in a performant, succinct, composable and portable manner. In this paper, we examine the performance challenges of a linear-algebra-based approach to building graph frameworks and describe new design principles for overcoming these bottlenecks. Among the new design principles is exploiting input sparsity, which allows users to write graph algorithms without specifying push and pull direction. Exploiting output sparsity allows users to tell the backend which values of the output in a single vectorized computation they do not want computed. Load-balancing is an important feature for balancing work amongst parallel workers. We describe the important load-balancing features for handling graphs with different characteristics. The design principles described in this paper have been implemented in "GraphBLAST", the first high-performance linear algebra-based graph framework on NVIDIA GPUs that is open-source. The results show that on a single GPU, GraphBLAST has on average at least an order of magnitude speedup over previous GraphBLAS implementations SuiteSparse and GBTL, comparable performance to the fastest GPU hardwired primitives and shared-memory graph frameworks Ligra and Gunrock, and better performance than any other GPU graph framework, while offering a simpler and more concise programming model.Comment: 50 pages, 14 figures, 14 table

Elements of Design for Containers and Solutions in the LinBox Library

We describe in this paper new design techniques used in the \cpp exact linear algebra library \linbox, intended to make the library safer and easier to use, while keeping it generic and efficient. First, we review the new simplified structure for containers, based on our \emph{founding scope allocation} model. We explain design choices and their impact on coding: unification of our matrix classes, clearer model for matrices and submatrices, \etc Then we present a variation of the \emph{strategy} design pattern that is comprised of a controller--plugin system: the controller (solution) chooses among plug-ins (algorithms) that always call back the controllers for subtasks. We give examples using the solution \mul. Finally we present a benchmark architecture that serves two purposes: Providing the user with easier ways to produce graphs; Creating a framework for automatically tuning the library and supporting regression testing.Comment: 8 pages, 4th International Congress on Mathematical Software, Seoul : Korea, Republic Of (2014

Accelerating Large-Scale Data Analysis by Offloading to High-Performance Computing Libraries using Alchemist

Apache Spark is a popular system aimed at the analysis of large data sets, but recent studies have shown that certain computations---in particular, many linear algebra computations that are the basis for solving common machine learning problems---are significantly slower in Spark than when done using libraries written in a high-performance computing framework such as the Message-Passing Interface (MPI). To remedy this, we introduce Alchemist, a system designed to call MPI-based libraries from Apache Spark. Using Alchemist with Spark helps accelerate linear algebra, machine learning, and related computations, while still retaining the benefits of working within the Spark environment. We discuss the motivation behind the development of Alchemist, and we provide a brief overview of its design and implementation. We also compare the performances of pure Spark implementations with those of Spark implementations that leverage MPI-based codes via Alchemist. To do so, we use data science case studies: a large-scale application of the conjugate gradient method to solve very large linear systems arising in a speech classification problem, where we see an improvement of an order of magnitude; and the truncated singular value decomposition (SVD) of a 400GB three-dimensional ocean temperature data set, where we see a speedup of up to 7.9x. We also illustrate that the truncated SVD computation is easily scalable to terabyte-sized data by applying it to data sets of sizes up to 17.6TB.Comment: Accepted for publication in Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, London, UK, 201