284 research outputs found
An algebraic multigrid method for high order time-discretizations of the div-grad and the curl-curl equations
We present an algebraic multigrid algorithm for fully coupled implicit Runge-Kutta and Boundary Value Method time-discretizations of the div-grad and curl-curl equations. The algorithm uses a blocksmoother and a multigrid hierarchy derived from the hierarchy built by any algebraic multigrid algorithm for the stationary version of the problem. By a theoretical analysis and numerical experiments, we show that the convergence is similar to or better than the convergence of the scalar algebraic multigrid algorithm on which it is based. The algorithm benefits from several possibilities for implementation optimization. This results in a computational complexity which, for a modest number of stages, scales almost linearly as a function of the number of variables. © 2008 IMACS
A new level-dependent coarsegrid correction scheme for indefinite Helmholtz problems
In this paper we construct and analyse a level-dependent coarsegrid
correction scheme for indefinite Helmholtz problems. This adapted multigrid
method is capable of solving the Helmholtz equation on the finest grid using a
series of multigrid cycles with a grid-dependent complex shift, leading to a
stable correction scheme on all levels. It is rigourously shown that the
adaptation of the complex shift throughout the multigrid cycle maintains the
functionality of the two-grid correction scheme, as no smooth modes are
amplified in or added to the error. In addition, a sufficiently smoothing
relaxation scheme should be applied to ensure damping of the oscillatory error
components. Numerical experiments on various benchmark problems show the method
to be competitive with or even outperform the current state-of-the-art
multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or
BiCGStab.Comment: 21 page
Asynchronous Stabilisation and Assembly Techniques for Additive Multigrid
Multigrid solvers are among the best solvers in the world, but once
applied in the real world there are issues they must overcome. Many multigrid
phases exhibit low concurrency. Mesh and matrix assembly are challenging to
parallelise and introduce algorithmic latency. Dynamically adaptive codes exacerbate
these issues. Multigrid codes require the computation of a cascade of matrices and
dynamic adaptivity means these matrices are recomputed throughout the solve.
Existing methods to compute the matrices are expensive and delay the solve. Non-
trivial material parameters further increase the cost of accurate equation integration.
We propose to assemble all matrix equations as stencils in a delayed element-wise
fashion. Early multigrid iterations use cheap geometric approximations and more
accurate updated stencil integrations are computed in parallel with the multigrid
cycles. New stencil integrations are evaluated lazily and asynchronously fed to the
solver once they become available. They do not delay multigrid iterations. We
deploy stencil integrations as parallel tasks that are picked up by cores that would
otherwise be idle. Coarse grid solves in multiplicative multigrid also exhibit limited
concurrency. Small coarse mesh sizes correspond to small computational workload
and require costly synchronisation steps. This acts as a bottleneck and delays
solver iterations. Additive multigrid avoids this restriction, but becomes unstable
for non-trivial material parameters as additive coarse grid levels tend to overcorrect.
This leads to oscillations. We propose a new additive variant, adAFAC-x, with a
stabilisation parameter that damps coarse grid corrections to remove oscillations.
Per-level we solve an additional equation that produces an auxiliary correction.
The auxiliary correction can be computed additively to the rest of the solve and
uses ideas similar to smoothed aggregation multigrid to anticipate overcorrections.
Pipelining techniques allow adAFAC-x to be written using single-touch semantics
on a dynamically adaptive mesh
Computing and deflating eigenvalues while solving multiple right hand side linear systems in Quantum Chromodynamics
We present a new algorithm that computes eigenvalues and eigenvectors of a
Hermitian positive definite matrix while solving a linear system of equations
with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could
be saved and recombined through the eigenvectors of the tridiagonal projection
matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm
capitalizes on the iteration vectors produced by CG to update only a small
window of vectors that approximate the eigenvectors. While this window is
restarted in a locally optimal way, the CG algorithm for the linear system is
unaffected. Yet, in all our experiments, this small window converges to the
required eigenvectors at a rate identical to unrestarted Lanczos. After the
solution of the linear system, eigenvectors that have not accurately converged
can be improved in an incremental fashion by solving additional linear systems.
In this case, eigenvectors identified in earlier systems can be used to
deflate, and thus accelerate, the convergence of subsequent systems. We have
used this algorithm with excellent results in lattice QCD applications, where
hundreds of right hand sides may be needed. Specifically, about 70 eigenvectors
are obtained to full accuracy after solving 24 right hand sides. Deflating
these from the large number of subsequent right hand sides removes the dreaded
critical slowdown, where the conditioning of the matrix increases as the quark
mass reaches a critical value. Our experiments show almost a constant number of
iterations for our method, regardless of quark mass, and speedups of 8 over
original CG for light quark masses.Comment: 22 pages, 26 eps figure
Recommended from our members
Schnelle Löser für partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
Compression and Reduced Representation Techniques for Patch-Based Relaxation
Patch-based relaxation refers to a family of methods for solving linear
systems which partitions the matrix into smaller pieces often corresponding to
groups of adjacent degrees of freedom residing within patches of the
computational domain. The two most common families of patch-based methods are
block-Jacobi and Schwarz methods, where the former typically corresponds to
non-overlapping domains and the later implies some overlap. We focus on cases
where each patch consists of the degrees of freedom within a finite element
method mesh cell. Patch methods often capture complex local physics much more
effectively than simpler point-smoothers such as Jacobi; however, forming,
inverting, and applying each patch can be prohibitively expensive in terms of
both storage and computation time. To this end, we propose several approaches
for performing analysis on these patches and constructing a reduced
representation. The compression techniques rely on either matrix norm
comparisons or unsupervised learning via a clustering approach. We illustrate
how it is frequently possible to retain/factor less than 5% of all patches and
still develop a method that converges with the same number of iterations or
slightly more than when all patches are stored/factored.Comment: 16 pages, 5 figure
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