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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'
An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling
We present a sparse linear system solver that is based on a multifrontal
variant of Gaussian elimination, and exploits low-rank approximation of the
resulting dense frontal matrices. We use hierarchically semiseparable (HSS)
matrices, which have low-rank off-diagonal blocks, to approximate the frontal
matrices. For HSS matrix construction, a randomized sampling algorithm is used
together with interpolative decompositions. The combination of the randomized
compression with a fast ULV HSS factorization leads to a solver with lower
computational complexity than the standard multifrontal method for many
applications, resulting in speedups up to 7 fold for problems in our test
suite. The implementation targets many-core systems by using task parallelism
with dynamic runtime scheduling. Numerical experiments show performance
improvements over state-of-the-art sparse direct solvers. The implementation
achieves high performance and good scalability on a range of modern shared
memory parallel systems, including the Intel Xeon Phi (MIC). The code is part
of a software package called STRUMPACK -- STRUctured Matrices PACKage, which
also has a distributed memory component for dense rank-structured matrices
Multilevel quasiseparable matrices in PDE-constrained optimization
Optimization problems with constraints in the form of a partial differential
equation arise frequently in the process of engineering design. The
discretization of PDE-constrained optimization problems results in large-scale
linear systems of saddle-point type. In this paper we propose and develop a
novel approach to solving such systems by exploiting so-called quasiseparable
matrices. One may think of a usual quasiseparable matrix as of a discrete
analog of the Green's function of a one-dimensional differential operator. Nice
feature of such matrices is that almost every algorithm which employs them has
linear complexity. We extend the application of quasiseparable matrices to
problems in higher dimensions. Namely, we construct a class of preconditioners
which can be computed and applied at a linear computational cost. Their use
with appropriate Krylov methods leads to algorithms of nearly linear
complexity
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