5,226 research outputs found
Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of
large-scale linear systems that arise from the discretization of elliptic PDEs
amenable to rank compression. The preconditioner is based on hierarchical
low-rank approximations and the cyclic reduction method. The setup and
application phases of the preconditioner achieve log-linear complexity in
memory footprint and number of operations, and numerical experiments exhibit
good weak and strong scalability at large processor counts in a distributed
memory environment. Numerical experiments with linear systems that feature
symmetry and nonsymmetry, definiteness and indefiniteness, constant and
variable coefficients demonstrate the preconditioner applicability and
robustness. Furthermore, it is possible to control the number of iterations via
the accuracy threshold of the hierarchical matrix approximations and their
arithmetic operations, and the tuning of the admissibility condition parameter.
Together, these parameters allow for optimization of the memory requirements
and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics,
Dec 201
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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'
Linear solvers for power grid optimization problems: a review of GPU-accelerated linear solvers
The linear equations that arise in interior methods for constrained
optimization are sparse symmetric indefinite and become extremely
ill-conditioned as the interior method converges. These linear systems present
a challenge for existing solver frameworks based on sparse LU or LDL^T
decompositions. We benchmark five well known direct linear solver packages
using matrices extracted from power grid optimization problems. The achieved
solution accuracy varies greatly among the packages. None of the tested
packages delivers significant GPU acceleration for our test cases
Analysis of A Splitting Approach for the Parallel Solution of Linear Systems on GPU Cards
We discuss an approach for solving sparse or dense banded linear systems
on a Graphics Processing Unit (GPU) card. The
matrix is possibly nonsymmetric and
moderately large; i.e., . The ${\it split\ and\
parallelize}{\tt SaP}{\bf A}{\bf A}_ii=1,\ldots,P{\bf A}_i{\tt SaP::GPU}{\tt PARDISO}{\tt SuperLU}{\tt MUMPS}{\tt SaP::GPU}{\tt MKL}{\tt SaP::GPU}{\tt SaP::GPU}$ is publicly available and distributed as
open source under a permissive BSD3 license.Comment: 38 page
Hierarchical interpolative factorization for elliptic operators: differential equations
This paper introduces the hierarchical interpolative factorization for
elliptic partial differential equations (HIF-DE) in two (2D) and three
dimensions (3D). This factorization takes the form of an approximate
generalized LU/LDL decomposition that facilitates the efficient inversion of
the discretized operator. HIF-DE is based on the multifrontal method but uses
skeletonization on the separator fronts to sparsify the dense frontal matrices
and thus reduce the cost. We conjecture that this strategy yields linear
complexity in 2D and quasilinear complexity in 3D. Estimated linear complexity
in 3D can be achieved by skeletonizing the compressed fronts themselves, which
amounts geometrically to a recursive dimensional reduction scheme. Numerical
experiments support our claims and further demonstrate the performance of our
algorithm as a fast direct solver and preconditioner. MATLAB codes are freely
available.Comment: 37 pages, 13 figures, 12 tables; to appear, Comm. Pure Appl. Math.
arXiv admin note: substantial text overlap with arXiv:1307.266
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