QR Factorization of a Dense Matrix on a Hypercube Multiprocessor

Chu, Eleanor; George, Alan

doi:10.1137/0911057

Cited by 29 publications

(5 citation statements)

References 17 publications

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“…In this case, Q is m × n with orthonormal columns and R is n × n and upper-triangular. Parallel QR algorithms have received much study [4], [26]- [31], but focus has predominantly been on 2D blocked algorithms. 3D algorithms for QR have been proposed [7], [8].…”

Section: E Qr Factorizationmentioning

confidence: 99%

Communication-Avoiding Cholesky-QR2 for Rectangular Matrices

Hutter

Solomonik

2019

2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS)

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Scalable QR factorization algorithms for solving least squares and eigenvalue problems are critical given the increasing parallelism within modern machines. We introduce a more general parallelization of the CholeskyQR2 algorithm and show its effectiveness for a wide range of matrix sizes. Our algorithm executes over a 3D processor grid, the dimensions of which can be tuned to trade-off costs in synchronization, interprocessor communication, computational work, and memory footprint. We implement this algorithm, yielding a code that can achieve a factor of Θ(P 1/6 ) less interprocessor communication on P processors than any previous parallel QR implementation. Our performance study on Intel Knights-Landing and Cray XE supercomputers demonstrates the effectiveness of this CholeskyQR2 parallelization on a large number of nodes. Specifically, relative to ScaLAPACK's QR, on 1024 nodes of Stampede2, our CholeskyQR2 implementation is faster by 2.6x-3.3x in strong scaling tests and by 1.1x-1.9x in weak scaling tests.

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Section: E Qr Factorizationmentioning

confidence: 99%

Communication-Avoiding Cholesky-QR2 for Rectangular Matrices

Hutter

Solomonik

2019

2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS)

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“…We can interpret these sequences of algorithms as scalar implementations using a flat tree of the algorithm in Demmel et al [17]. In the late 1980s, the research shifted gears and presented algorithms based on Householder reflections [35], [14]. The motivation was to use vector computer capabilities.…”

Section: Communication Avoiding Qr (Caqr) Factorizationmentioning

confidence: 99%

QR factorization of tall and skinny matrices in a grid computing environment

Agullo

Coti

Dongarra

et al. 2010

2010 IEEE International Symposium on Parallel &Amp; Distributed Processing (IPDPS)

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Previous studies have reported that common dense linear algebra operations do not achieve speed up by using multiple geographical sites of a computational grid. Because such operations are the building blocks of most scientific applications, conventional supercomputers are still strongly predominant in high-performance computing and the use of grids for speeding up large-scale scientific problems is limited to applications exhibiting parallelism at a higher level. We have identified two performance bottlenecks in the distributed memory algorithms implemented in ScaLAPACK, a state-of-the-art dense linear algebra library. First, because ScaLAPACK assumes a homogeneous communication network, the implementations of ScaLAPACK algorithms lack locality in their communication pattern. Second, the number of messages sent in the ScaLAPACK algorithms is significantly greater than other algorithms that trade flops for communication. In this paper, we present a new approach for computing a QR factorization -one of the main dense linear algebra kernels -of tall and skinny matrices in a grid computing environment that overcomes these two bottlenecks. Our contribution is to articulate a recently proposed algorithm (Communication Avoiding QR) with a topology-aware middleware (QCG-OMPI) in order to confine intensive communications (ScaLAPACK calls) within the different geographical sites. An experimental study conducted on the Grid'5000 platform shows that the resulting performance increases linearly with the number of geographical sites on large-scale problems (and is in particular consistently higher than ScaLAPACK's).

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“…The optimization of the above formulation for computing eigenvalue is tightly related to improving the QR factorization part. Tiling the QR computations can significantly improve the cache performance [37,40,41].…”

Section: Literature Survey Of Optimiationsmentioning

confidence: 99%

TORCH Computational Reference Kernels - A Testbed for Computer Science Research

Kaiser¹,

Williams²,

Madduri³

et al. 2010

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For decades, computer scientists have sought guidance on how to evolve architectures, languages, and programming models in order to improve application performance, efficiency, and productivity. Unfortunately, without overarching advice about future directions in these areas, individual guidance is inferred from the existing software/hardware ecosystem, and each discipline often conducts their research independently assuming all other technologies remain fixed. In today's rapidly evolving world of on-chip parallelism, isolated and iterative improvements to performance may miss superior solutions in the same way gradient descent optimization techniques may get stuck in local minima. To combat this, we present TORCH: A Testbed for Optimization ResearCH. These computational reference kernels define the core problems of interest in scientific computing without mandating a specific language, algorithm, programming model, or implementation. To compliment the kernel (problem) definitions, we provide a set of algorithmically-expressed verification tests that can be used to verify a hardware/software co-designed solution produces an acceptable answer. Finally, to provide some illumination as to how researchers have implemented solutions to these problems in the past, we provide a set of reference implementations in C and MATLAB.

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QR Factorization of a Dense Matrix on a Hypercube Multiprocessor

Cited by 29 publications

References 17 publications

Communication-Avoiding Cholesky-QR2 for Rectangular Matrices

Communication-Avoiding Cholesky-QR2 for Rectangular Matrices

QR factorization of tall and skinny matrices in a grid computing environment

TORCH Computational Reference Kernels - A Testbed for Computer Science Research

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