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  • 1.
    Barthels, Henrik
    et al.
    AICES, RWTH Aachen University, Schinkelstr. 2, Aachen, Germany.
    Psarras, Christos
    AICES, RWTH Aachen University, Schinkelstr. 2, Aachen, Germany.
    Bientinesi, Paolo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Linnea: Automatic Generation of Efficient Linear Algebra Programs2021In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 47, no 3, article id 22Article in journal (Refereed)
    Abstract [en]

    The translation of linear algebra computations into efficient sequences of library calls is a non-trivial task that requires expertise in both linear algebra and high-performance computing. Almost all high-level languages and libraries for matrix computations (e.g., Matlab, Eigen) internally use optimized kernels such as those provided by BLAS and LAPACK; however, their translation algorithms are often too simplistic and thus lead to a suboptimal use of said kernels, resulting in significant performance losses. To combine the productivity offered by high-level languages, and the performance of low-level kernels, we are developing Linnea, a code generator for linear algebra problems. As input, Linnea takes a high-level description of a linear algebra problem; as output, it returns an efficient sequence of calls to high-performance kernels. Linnea uses a custom best-first search algorithm to find a first solution in less than a second, and increasingly better solutions when given more time. In 125 test problems, the code generated by Linnea almost always outperforms Matlab, Julia, Eigen, and Armadillo, with speedups up to and exceeding 10×.

  • 2.
    Bartlett, Roscoe A.
    et al.
    Sandia National Laboratories.
    Van Bloemen Waanders, Bart
    Sandia National Laboratories.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    Hybrid Differentiation Strategies for Simulation and Analysis of Applications in C++2008In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 35, no 1Article in journal (Refereed)
  • 3.
    Granat, Robert
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kagstrom, Bo
    Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N). Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kressner, Daniel
    Shao, Meiyue
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Algorithm 953: Parallel Library Software for the Multishift QR Algorithm with Aggressive Early Deflation2015In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 41, no 4, article id 29Article in journal (Refereed)
    Abstract [en]

    Library software implementing a parallel small-bulge multishift QR algorithm with Aggressive Early Deflation (AED) targeting distributed memory high-performance computing systems is presented. Starting from recent developments of the parallel multishift QR algorithm [Granat et al., SIAM J. Sci. Comput. 32(4), 2010], we describe a number of algorithmic and implementation improvements. These include communication avoiding algorithms via data redistribution and a refined strategy for balancing between multishift QR sweeps and AED. Guidelines concerning several important tunable algorithmic parameters are provided. As a result of these improvements, a computational bottleneck within AED has been removed in the parallel multishift QR algorithm. A performance model is established to explain the scalability behavior of the new parallel multishift QR algorithm. Numerous computational experiments confirm that our new implementation significantly outperforms previous parallel implementations of the QR algorithm.

  • 4.
    Granat, Robert
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Algorithm 904: the SCASY library - parallel solvers for Sylvester-type matrix equations with applications in condition estimation, part II2010In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 37, no 3, article id 33Article in journal (Refereed)
    Abstract [en]

    We continue our presentation of parallel ScaLAPACK-style algorithms for solving Sylvester-type matrix equations. In Part II, we present SCASY (SCAlable SYlvester solvers), a state-of-the-art HPC software library for solving 44 sign and transpose variants of eight common standard and generalized Sylvester-type matrix equations.

  • 5.
    Granat, Robert
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Parallel Solvers for Sylvester-type Matrix Equations with Applications in Condition Estimation, Part I: Theory and Algorithms2010In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 37, no 3, p. 32:1-32:32Article in journal (Refereed)
    Abstract [en]

    Parallel ScaLAPACK-style algorithms for solving eight common standard and generalized Sylvester-type matrix equations and various sign and transposed variants are presented. All algorithms are blocked variants based on the Bartels--Stewart method and involve four major steps: reduction to triangular form, updating the right-hand side with respect to the reduction, computing the solution to the reduced triangular problem, and transforming the solution back to the original coordinate system. Novel parallel algorithms for solving reduced triangular matrix equations based on wavefront-like traversal of the right-hand side matrices are presented together with a generic scalability analysis. These algorithms are used in condition estimation and new robust parallel sep − 1-estimators are developed. Experimental results from three parallel platforms, including results from a mixed OpenMP/MPI platform, are presented and analyzed using several performance and accuracy metrics. The analysis includes results regarding general and triangular parallel solvers as well as parallel condition estimators.

  • 6.
    Gustavson, Fred G.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wasniewski, Jerzy
    Dongarra, Jack J.
    Herrero, Jose R.
    Langou, Julien
    Level-3 Cholesky Factorization Routines Improve Performance of Many Cholesky Algorithms2013In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 39, no 2, p. 9-Article in journal (Refereed)
    Abstract [en]

    Four routines called DPOTF3i, i = a, b, c, d, are presented. DPOTF3i are a novel type of level-3 BLAS for use by BPF (Blocked Packed Format) Cholesky factorization and LAPACK routine DPOTRF. Performance of routines DPOTF3i are still increasing when the performance of Level-2 routine DPOTF2 of LAPACK starts decreasing. This is our main result and it implies, due to the use of larger block size nb, that DGEMM, DSYRK, and DTRSM performance also increases! The four DPOTF3i routines use simple register blocking. Different platforms have different numbers of registers. Thus, our four routines have different register blocking sizes. BPF is introduced. LAPACK routines for POTRF and PPTRF using BPF instead of full and packed format are shown to be trivial modifications of LAPACK POTRF source codes. We call these codes BPTRF. There are two variants of BPF: lower and upper. Upper BPF is "identical" to Square Block Packed Format (SBPF). "LAPACK" implementations on multicore processors use SBPF. Lower BPF is less efficient than upper BPF. Vector inplace transposition converts lower BPF to upper BPF very efficiently. Corroborating performance results for DPOTF3i versus DPOTF2 on a variety of common platforms are given for n approximate to nb as well as results for large n comparing DBPTRF versus DPOTRF.

  • 7.
    Gustavson, Fred G.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Wasniewski, Jerzy
    Dongarra, Jack J.
    Langou, Julien
    Rectangular Full Packed Format for Cholesky's Algorithm: Factorization, Solution, and Inversion2010In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 37, no 2, p. 1-21, article id 18Article in journal (Refereed)
    Abstract [en]

    We describe a new data format for storing triangular, symmetric, and Hermitian matrices called Rectangular Full Packed Format (RFPF). The standard two-dimensional arrays of Fortran and C (also known as full format) that are used to represent triangular and symmetric matrices waste nearly half of the storage space but provide high performance via the use of Level 3 BLAS. Standard packed format arrays fully utilize storage (array space) but provide low performance as there is no Level 3 packed BLAS. We combine the good features of packed and full storage using RFPF to obtain high performance via using Level 3 BLAS as RFPF is a standard full-format representation. Also, RFPF requires exactly the same minimal storage as packed the format. Each LAPACK full and/or packed triangular, symmetric, and Hermitian routine becomes a single new RFPF routine based on eight possible data layouts of RFPF. This new RFPF routine usually consists of two calls to the corresponding LAPACK full-format routine and two calls to Level 3 BLAS routines. This means no new software is required. As examples, we present LAPACK routines for Cholesky factorization, Cholesky solution, and Cholesky inverse computation in RFPF to illustrate this new work and to describe its performance on several commonly used computer platforms. Performance of LAPACK full routines using RFPF versus LAPACK full routines using the standard format for both serial and SMP parallel processing is about the same while using half the storage. Performance gains are roughly one to a factor of 43 for serial and one to a factor of 97 for SMP parallel times faster using vendor LAPACK full routines with RFPF than with using vendor and/or reference packed routines.

  • 8.
    Gustavson, Fred
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Distributed SBP Cholesky factorization algorithms with near-optimal scheduling2009In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 36, no 2, p. 11:1-11:25Article in journal (Refereed)
  • 9.
    Gustavson, Fred
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Parallel and Cache-Efficient In-Place Matrix Storage Format Conversion2012In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 38, no 3, p. 17:1-17:32Article in journal (Refereed)
    Abstract [en]

    Techniques and algorithms for efficient in-place conversion to and from standard and blocked matrix storage formats are described. Such functionality is required by numerical libraries that use different data layouts internally. Parallel algorithms and a software package for in-place matrix storage format conversion based on in-place matrix transposition are presented and evaluated. A new algorithm for in-place transposition which efficiently determines the structure of the transposition permutation a priori is one of the key ingredients. It enables effective load balancing in a parallel environment.

  • 10.
    Karlsson, Lars
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kressner, Daniel
    Lang, Bruno
    Optimally Packed Chains of Bulges in Multishift QR Algorithms2014In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 40, no 2, p. 12-Article in journal (Refereed)
    Abstract [en]

    The QR algorithm is the method of choice for computing all eigenvalues of a dense nonsymmetric matrix A. After an initial reduction to Hessenberg form, a QR iteration can be viewed as chasing a small bulge from the top left to the bottom right corner along the subdiagonal of A. To increase data locality and create potential for parallelism, modern variants of the QR algorithm perform several iterations simultaneously, which amounts to chasing a chain of several bulges instead of a single bulge. To make effective use of level 3 BLAS, it is important to pack these bulges as tightly as possible within the chain. In this work, we show that the tightness of the packing in existing approaches is not optimal and can be increased. This directly translates into a reduced chain length by 33% compared to the state-of-the-art LAPACK implementation of the QR algorithm. To demonstrate the impact of our idea, we have modified the LAPACK implementation to make use of the optimal packing. Numerical experiments reveal a uniform reduction of the execution time, without affecting stability or robustness.

  • 11.
    Myllykoski, Mirko
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation2022In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 48, no 1, p. 1-36, article id 11Article in journal (Refereed)
    Abstract [en]

    The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED), and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

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  • 12.
    Psarras, Christos
    et al.
    RWTH Aachen University, Aachen, Germany.
    Barthels, Henrik
    RWTH Aachen University, Aachen, Germany.
    Bientinesi, Paolo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    The linear algebra mapping problem: Current state of linear algebra languages and libraries2022In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 48, no 3, article id 3549935Article in journal (Refereed)
    Abstract [en]

    We observe a disconnect between developers and end-users of linear algebra libraries. On the one hand, developers invest significant effort in creating sophisticated numerical kernels. On the other hand, end-users are progressively less likely to go through the time consuming process of directly using said kernels; instead, languages and libraries, which offer a higher level of abstraction, are becoming increasingly popular. These languages offer mechanisms that internally map the input program to lower level kernels. Unfortunately, our experience suggests that, in terms of performance, this translation is typically suboptimal.

    In this paper, we define the problem of mapping a linear algebra expression to a set of available building blocks as the "Linear Algebra Mapping Problem"(LAMP); we discuss its NP-complete nature, and investigate how effectively a benchmark of test problems is solved by popular high-level programming languages and libraries. Specifically, we consider Matlab, Octave, Julia, R, Armadillo (C++), Eigen (C++), and NumPy (Python); the benchmark is meant to test both compiler optimizations, as well as linear algebra specific optimizations, such as the optimal parenthesization of matrix products. The aim of this study is to facilitate the development of languages and libraries that support linear algebra computations.

  • 13.
    Psarras, Christos
    et al.
    RWTH Aachen University, Aachen, Germany.
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Bro, Rasmus
    University of Copenhagen, Copenhagen, Frederiksberg, Denmark.
    Bientinesi, Paolo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Algorithm 1026: Concurrent Alternating Least Squares for Multiple Simultaneous Canonical Polyadic Decompositions2022In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 48, no 3, article id 3519383Article in journal (Refereed)
    Abstract [en]

    Tensor decompositions, such as CANDECOMP/PARAFAC (CP), are widely used in a variety of applications, such as chemometrics, signal processing, and machine learning. A broadly used method for computing such decompositions relies on the Alternating Least Squares (ALS) algorithm. When the number of components is small, regardless of its implementation, ALS exhibits low arithmetic intensity, which severely hinders its performance and makes GPU offloading ineffective. We observe that, in practice, experts often have to compute multiple decompositions of the same tensor, each with a small number of components (typically fewer than 20), to ultimately find the best ones to use for the application at hand. In this article, we illustrate how multiple decompositions of the same tensor can be fused together at the algorithmic level to increase the arithmetic intensity. Therefore, it becomes possible to make efficient use of GPUs for further speedups; at the same time, the technique is compatible with many enhancements typically used in ALS, such as line search, extrapolation, and non-negativity constraints. We introduce the Concurrent ALS algorithm and library, which offers an interface to MATLAB, and a mechanism to effectively deal with the issue that decompositions complete at different times. Experimental results on artificial and real datasets demonstrate a shorter time to completion due to increased arithmetic intensity.

  • 14.
    Schwarz, Angelika Beatrix
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Robust level-3 blas inverse iteration from the Hessenberg matrix2022In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 48, no 3, article id 3544789Article in journal (Other academic)
    Abstract [en]

    Inverse iteration is known to be an effective method for com-puting eigenvectors corresponding to simple and well-separated eigenvalues. In the non-symmetric case, the solution of shifted Hessenberg systems is a central step. Existing inverse iteration solvers approach the solution of the shiftedHessenberg systems with either RQ or LU factorizations and, once factored, solve the corresponding systems. This approach has limited level-3 BLAS potential since distinct shifts have distinct factorizations. This paper rearranges the RQ approach such that data shared between distinct shifts is exposed. Thereby the backward substitution with the triangular R factor can be expressed mostly with matrix–matrix multiplications (level-3 BLAS). The resulting algorithm computes eigenvectors in a tiled, overflow-free, and task-parallel fashion. The numerical experiments show that the new algorithm outperforms existing inverse iteration solvers for the computation of both real and complex eigenvectors.

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