1,359 research outputs found
Performance of random sampling for computing low-rank approximations of a dense matrix on GPUs
International audienceA low-rank approximation of a dense matrix plays an important role in many applications. To compute such an approximation , a common approach uses the QR factorization with column pivoting (QRCP). Though the reliability and efficiency of QRCP have been demonstrated, this determin-istic approach requires costly communication at each step of the factorization. Since such communication is becoming increasingly expensive on modern computers, an alternative approach based on random sampling, which can be implemented using communication-optimal kernels, is becoming attractive. To study its potential, in this paper, we compare the performance of random sampling with that of QRCP on an NVIDIA Kepler GPU. Our performance results demonstrate that random sampling can be up to 12.8Ă faster than the deterministic approach for computing the approximation of the same accuracy. We also present the parallel scaling of the random sampling over multiple GPUs on a single compute node, showing a speedup of 3.8Ă over three Kepler GPUs. These results demonstrate the potential of the random sampling as an excellent computational tool for many applications, and its potential is likely to grow on the emerging computers with the increasing communication costs
Geometry-Oblivious FMM for Compressing Dense SPD Matrices
We present GOFMM (geometry-oblivious FMM), a novel method that creates a
hierarchical low-rank approximation, "compression," of an arbitrary dense
symmetric positive definite (SPD) matrix. For many applications, GOFMM enables
an approximate matrix-vector multiplication in or even time,
where is the matrix size. Compression requires storage and work.
In general, our scheme belongs to the family of hierarchical matrix
approximation methods. In particular, it generalizes the fast multipole method
(FMM) to a purely algebraic setting by only requiring the ability to sample
matrix entries. Neither geometric information (i.e., point coordinates) nor
knowledge of how the matrix entries have been generated is required, thus the
term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme
for hierarchical matrix computations that reduces synchronization barriers. We
present results on the Intel Knights Landing and Haswell architectures, and on
the NVIDIA Pascal architecture for a variety of matrices.Comment: 13 pages, accepted by SC'1
Algorithmic patterns for -matrices on many-core processors
In this work, we consider the reformulation of hierarchical ()
matrix algorithms for many-core processors with a model implementation on
graphics processing units (GPUs). matrices approximate specific
dense matrices, e.g., from discretized integral equations or kernel ridge
regression, leading to log-linear time complexity in dense matrix-vector
products. The parallelization of matrix operations on many-core
processors is difficult due to the complex nature of the underlying algorithms.
While previous algorithmic advances for many-core hardware focused on
accelerating existing matrix CPU implementations by many-core
processors, we here aim at totally relying on that processor type. As main
contribution, we introduce the necessary parallel algorithmic patterns allowing
to map the full matrix construction and the fast matrix-vector
product to many-core hardware. Here, crucial ingredients are space filling
curves, parallel tree traversal and batching of linear algebra operations. The
resulting model GPU implementation hmglib is the, to the best of the authors
knowledge, first entirely GPU-based Open Source matrix library of
this kind. We conclude this work by an in-depth performance analysis and a
comparative performance study against a standard matrix library,
highlighting profound speedups of our many-core parallel approach
Lecture 03: Hierarchically Low Rank Methods and Applications
As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to âsolveâ a computational problem, which suggest that we have often been âoversolvingâ them at the smaller scales of the past because we could afford to do so. We present innovations that allow to approach lin-log complexity in storage and operation count in many important algorithmic kernels and thus create an opportunity for full applications with optimal scalability
randUTV: A blocked randomized algorithm for computing a rank-revealing UTV factorization
This manuscript describes the randomized algorithm randUTV for computing a so
called UTV factorization efficiently. Given a matrix , the algorithm
computes a factorization , where and have orthonormal
columns, and is triangular (either upper or lower, whichever is preferred).
The algorithm randUTV is developed primarily to be a fast and easily
parallelized alternative to algorithms for computing the Singular Value
Decomposition (SVD). randUTV provides accuracy very close to that of the SVD
for problems such as low-rank approximation, solving ill-conditioned linear
systems, determining bases for various subspaces associated with the matrix,
etc. Moreover, randUTV produces highly accurate approximations to the singular
values of . Unlike the SVD, the randomized algorithm proposed builds a UTV
factorization in an incremental, single-stage, and non-iterative way, making it
possible to halt the factorization process once a specified tolerance has been
met. Numerical experiments comparing the accuracy and speed of randUTV to the
SVD are presented. These experiments demonstrate that in comparison to column
pivoted QR, which is another factorization that is often used as a relatively
economic alternative to the SVD, randUTV compares favorably in terms of speed
while providing far higher accuracy
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Preparing sparse solvers for exascale computing.
Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'
Massively parallel approximate Gaussian process regression
We explore how the big-three computing paradigms -- symmetric multi-processor
(SMC), graphical processing units (GPUs), and cluster computing -- can together
be brought to bare on large-data Gaussian processes (GP) regression problems
via a careful implementation of a newly developed local approximation scheme.
Our methodological contribution focuses primarily on GPU computation, as this
requires the most care and also provides the largest performance boost.
However, in our empirical work we study the relative merits of all three
paradigms to determine how best to combine them. The paper concludes with two
case studies. One is a real data fluid-dynamics computer experiment which
benefits from the local nature of our approximation; the second is a synthetic
data example designed to find the largest design for which (accurate) GP
emulation can performed on a commensurate predictive set under an hour.Comment: 24 pages, 6 figures, 1 tabl
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