156 research outputs found
h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
In this work we exploit agglomeration based -multigrid preconditioners to
speed-up the iterative solution of discontinuous Galerkin discretizations of
the Stokes and Navier-Stokes equations. As a distinctive feature -coarsened
mesh sequences are generated by recursive agglomeration of a fine grid,
admitting arbitrarily unstructured grids of complex domains, and agglomeration
based discontinuous Galerkin discretizations are employed to deal with
agglomerated elements of coarse levels. Both the expense of building coarse
grid operators and the performance of the resulting multigrid iteration are
investigated. For the sake of efficiency coarse grid operators are inherited
through element-by-element projections, avoiding the cost of numerical
integration over agglomerated elements. Specific care is devoted to the
projection of viscous terms discretized by means of the BR2 dG method. We
demonstrate that enforcing the correct amount of stabilization on coarse grids
levels is mandatory for achieving uniform convergence with respect to the
number of levels. The numerical solution of steady and unsteady, linear and
non-linear problems is considered tackling challenging 2D test cases and 3D
real life computations on parallel architectures. Significant execution time
gains are documented.Comment: 78 pages, 7 figure
A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems
Classical iterative methods for tomographic reconstruction include the class
of Algebraic Reconstruction Techniques (ART). Convergence of these stationary
linear iterative methods is however notably slow. In this paper we propose the
use of Krylov solvers for tomographic linear inversion problems. These advanced
iterative methods feature fast convergence at the expense of a higher
computational cost per iteration, causing them to be generally uncompetitive
without the inclusion of a suitable preconditioner. Combining elements from
standard multigrid (MG) solvers and the theory of wavelets, a novel
wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to
significantly speed-up Krylov convergence. The performance of the
WMG-preconditioned Krylov method is analyzed through a spectral analysis, and
the approach is compared to existing methods like the classical Simultaneous
Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods
on a 2D tomographic benchmark problem. Numerical experiments are promising,
showing the method to be competitive with the classical Algebraic
Reconstruction Techniques in terms of convergence speed and overall performance
(CPU time) as well as precision of the reconstruction.Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13
figures, 3 table
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
An overlapped grid method for multigrid, finite volume/difference flow solvers: MaGGiE
The objective is to develop a domain decomposition method via overlapping/embedding the component grids, which is to be used by upwind, multi-grid, finite volume solution algorithms. A computer code, given the name MaGGiE (Multi-Geometry Grid Embedder) is developed to meet this objective. MaGGiE takes independently generated component grids as input, and automatically constructs the composite mesh and interpolation data, which can be used by the finite volume solution methods with or without multigrid convergence acceleration. Six demonstrative examples showing various aspects of the overlap technique are presented and discussed. These cases are used for developing the procedure for overlapping grids of different topologies, and to evaluate the grid connection and interpolation data for finite volume calculations on a composite mesh. Time fluxes are transferred between mesh interfaces using a trilinear interpolation procedure. Conservation losses are minimal at the interfaces using this method. The multi-grid solution algorithm, using the coaser grid connections, improves the convergence time history as compared to the solution on composite mesh without multi-gridding
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature
Bridging the gap between geometric and algebraic multi-grid methods
In this paper, a multi-grid solver for the discretisation of partial differential equations on complicated domains is developed. The algorithm requires as input the given discretisation only instead of a hierarchy of discretisations on coarser grids. Such auxiliary grids and discretisations are generated in a black-box fashion and are employed to define purely algebraic intergrid transfer operators. The geometric interpretation of the algorithm allows one to use the framework of geometric multigrid methods to prove its convergence. The focus of this paper is on the formulation of the algorithm and the demonstration of its efficiency by numerical experiments, while the analysis is carried out for some model problems
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