496 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'
Hierarchical LU preconditioning for the time-harmonic Maxwell equations
The time-harmonic Maxwell equations are used to study the effect of electric
and magnetic fields on each other. Although the linear systems resulting from
solving this system using FEMs are sparse, direct solvers cannot reach the
linear complexity. In fact, due to the indefinite system matrix, iterative
solvers suffer from slow convergence. In this work, we study the effect of
using the inverse of -matrix approximations of the Galerkin
matrices arising from N\'ed\'elec's edge FEM discretization to solve the linear
system directly. We also investigate the impact of applying an
factorization as a preconditioner and we study the number of iterations to
solve the linear system using iterative solvers
Frequency-robust preconditioning of boundary integral equations for acoustic transmission
The scattering and transmission of harmonic acoustic waves at a penetrable
material are commonly modelled by a set of Helmholtz equations. This system of
partial differential equations can be rewritten into boundary integral
equations defined at the surface of the objects and solved with the boundary
element method (BEM). High frequencies or geometrical details require a fine
surface mesh, which increases the number of degrees of freedom in the weak
formulation. Then, matrix compression techniques need to be combined with
iterative linear solvers to limit the computational footprint. Moreover, the
convergence of the iterative linear solvers often depends on the frequency of
the wave field and the objects' characteristic size. Here, the robust PMCHWT
formulation is used to solve the acoustic transmission problem. An operator
preconditioner based on on-surface radiation conditions (OSRC) is designed that
yields frequency-robust convergence characteristics. Computational benchmarks
compare the performance of this novel preconditioned formulation with other
preconditioners and boundary integral formulations. The OSRC preconditioned
PMCHWT formulation effectively simulates large-scale problems of engineering
interest, such as focused ultrasound treatment of osteoid osteoma
Schnelle Löser für Partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
Computational Electromagnetism and Acoustics
It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems
Accelerated Calder\'on preconditioning for Maxwell transmission problems
We investigate a range of techniques for the acceleration of Calder\'on
(operator) preconditioning in the context of boundary integral equation methods
for electromagnetic transmission problems. Our objective is to mitigate as far
as possible the high computational cost of the barycentrically-refined meshes
necessary for the stable discretisation of operator products. Our focus is on
the well-known PMCHWT formulation, but the techniques we introduce can be
applied generically. By using barycentric meshes only for the preconditioner
and not for the original boundary integral operator, we achieve significant
reductions in computational cost by (i) using "reduced" Calder\'on
preconditioners obtained by discarding constituent boundary integral operators
that are not essential for regularisation, and (ii) adopting a
``bi-parametric'' approach in which we use a lower quality (cheaper)
-matrix assembly routine for the preconditioner than for the
original operator, including a novel approach of discarding far-field
interactions in the preconditioner. Using the boundary element software Bempp
(www.bempp.com), we compare the performance of different combinations of these
techniques in the context of scattering by multiple dielectric particles.
Applying our accelerated implementation to 3D electromagnetic scattering by an
aggregate consisting of 8 monomer ice crystals of overall diameter 1cm at
664GHz leads to a 99% reduction in memory cost and at least a 75% reduction in
total computation time compared to a non-accelerated implementation
Fast multipole method applied to 3D frequency domain elastodynamics
This article is concerned with the formulation and implementation of a fast multipole-accelerated BEM for 3-D elastodynamics in the frequency domain, based on the so-called diagonal form for the expansion of the elastodynamic fundamental solution, a multi-level strategy. As usual with the FM-BEM, the linear system of BEM equations is solved by GMRES, and the matrix is never explicitly formed. The truncation parameter in the multipole expansion is adjusted to the level, a feature known from recent published studies for the Maxwell equations. A preconditioning strategy based on the concept of sparse approximate inverse (SPAI) is presented and implemented. The proposed formulation is assessed on numerical examples involving BEM unknowns, which show in particular that, as expected, the proposed FM-BEM is much faster than the traditional BEM, and that the GMRES iteration count is significantly reduced when the SPAI preconditioner is used
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