14 research outputs found
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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'
Optimal Chebyshev Smoothers and One-sided V-cycles
The solution to the Poisson equation arising from the spectral element
discretization of the incompressible Navier-Stokes equation requires robust
preconditioning strategies. One such strategy is multigrid. To realize the
potential of multigrid methods, effective smoothing strategies are needed.
Chebyshev polynomial smoothing proves to be an effective smoother. However,
there are several improvements to be made, especially at the cost of symmetry.
For the same cost per iteration, a full V-cycle with order Chebyshev
polynomial smoothing may be substituted with a half V-cycle with order
Chebyshev polynomial smoothing, wherein the smoother is omitted on the up-leg
of the V-cycle. The choice of omitting the post-smoother in favor of higher
order Chebyshev pre-smoothing is shown to be advantageous in cases where the
multigrid approximation property constant, , is large. Results utilizing
Lottes's fourth-kind Chebyshev polynomial smoother are shown. These methods
demonstrate substantial improvement over the standard Chebyshev polynomial
smoother. The authors demonstrate the effectiveness of this scheme in
-geometric multigrid, as well as a 2D model problem with finite differences.Comment: 35 pages, 27 figures, 5 tables (including supplementary materials
Algorithmic Monotone Multiscale Finite Volume Methods for Porous Media Flow
Multiscale finite volume methods are known to produce reduced systems with
multipoint stencils which, in turn, could give non-monotone and out-of-bound
solutions. We propose a novel solution to the monotonicity issue of multiscale
methods. The proposed algorithmic monotone (AM- MsFV/MsRSB) framework is based
on an algebraic modification to the original MsFV/MsRSB coarse-scale stencil.
The AM-MsFV/MsRSB guarantees monotonic and within bound solutions without
compromising accuracy for various coarsening ratios; hence, it effectively
addresses the challenge of multiscale methods' sensitivity to coarse grid
partitioning choices. Moreover, by preserving the near null space of the
original operator, the AM-MsRSB showed promising performance when integrated in
iterative formulations using both the control volume and the Galerkin-type
restriction operators. We also propose a new approach to enhance the
performance of MsRSB for MPFA discretized systems, particularly targeting the
construction of the prolongation operator. Results show the potential of our
approach in terms of accuracy of the computed basis functions and the overall
convergence behavior of the multiscale solver while ensuring a monotone
solution at all times.Comment: 29 pages, 20 figure
A robust adaptive algebraic multigrid linear solver for structural mechanics
The numerical simulation of structural mechanics applications via finite
elements usually requires the solution of large-size and ill-conditioned linear
systems, especially when accurate results are sought for derived variables
interpolated with lower order functions, like stress or deformation fields.
Such task represents the most time-consuming kernel in commercial simulators;
thus, it is of significant interest the development of robust and efficient
linear solvers for such applications. In this context, direct solvers, which
are based on LU factorization techniques, are often used due to their
robustness and easy setup; however, they can reach only superlinear complexity,
in the best case, thus, have limited applicability depending on the problem
size. On the other hand, iterative solvers based on algebraic multigrid (AMG)
preconditioners can reach up to linear complexity for sufficiently regular
problems but do not always converge and require more knowledge from the user
for an efficient setup. In this work, we present an adaptive AMG method
specifically designed to improve its usability and efficiency in the solution
of structural problems. We show numerical results for several practical
applications with millions of unknowns and compare our method with two
state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM
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Schnelle Löser für Partielle Differentialgleichungen
This workshop was well attended by 52 participants with broad geographic representation from 11 countries and 3 continents. It was a nice blend of researchers with various backgrounds