459 research outputs found
Solving the Poisson equation on small aspect ratio domains using unstructured meshes
We discuss the ill conditioning of the matrix for the discretised Poisson
equation in the small aspect ratio limit, and motivate this problem in the
context of nonhydrostatic ocean modelling. Efficient iterative solvers for the
Poisson equation in small aspect ratio domains are crucial for the successful
development of nonhydrostatic ocean models on unstructured meshes. We introduce
a new multigrid preconditioner for the Poisson problem which can be used with
finite element discretisations on general unstructured meshes; this
preconditioner is motivated by the fact that the Poisson problem has a
condition number which is independent of aspect ratio when Dirichlet boundary
conditions are imposed on the top surface of the domain. This leads to the
first level in an algebraic multigrid solver (which can be extended by further
conventional algebraic multigrid stages), and an additive smoother. We
illustrate the method with numerical tests on unstructured meshes, which show
that the preconditioner makes a dramatic improvement on a more standard
multigrid preconditioner approach, and also show that the additive smoother
produces better results than standard SOR smoothing. This new solver method
makes it feasible to run nonhydrostatic unstructured mesh ocean models in small
aspect ratio domains.Comment: submitted to Ocean Modellin
Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs
We design and implement a parallel algebraic multigrid method for isotropic
graph Laplacian problems on multicore Graphical Processing Units (GPUs). The
proposed AMG method is based on the aggregation framework. The setup phase of
the algorithm uses a parallel maximal independent set algorithm in forming
aggregates and the resulting coarse level hierarchy is then used in a K-cycle
iteration solve phase with a -Jacobi smoother. Numerical tests of a
parallel implementation of the method for graphics processors are presented to
demonstrate its effectiveness.Comment: 18 pages, 3 figure
A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs
A self-learning algebraic multigrid method for dominant and minimal singular
triplets and eigenpairs is described. The method consists of two multilevel
phases. In the first, multiplicative phase (setup phase), tentative singular
triplets are calculated along with a multigrid hierarchy of interpolation
operators that approximately fit the tentative singular vectors in a collective
and self-learning manner, using multiplicative update formulas. In the second,
additive phase (solve phase), the tentative singular triplets are improved up
to the desired accuracy by using an additive correction scheme with fixed
interpolation operators, combined with a Ritz update. A suitable generalization
of the singular value decomposition is formulated that applies to the coarse
levels of the multilevel cycles. The proposed algorithm combines and extends
two existing multigrid approaches for symmetric positive definite eigenvalue
problems to the case of dominant and minimal singular triplets. Numerical tests
on model problems from different areas show that the algorithm converges to
high accuracy in a modest number of iterations, and is flexible enough to deal
with a variety of problems due to its self-learning properties.Comment: 29 page
Aggregation-based aggressive coarsening with polynomial smoothing
This paper develops an algebraic multigrid preconditioner for the graph
Laplacian. The proposed approach uses aggressive coarsening based on the
aggregation framework in the setup phase and a polynomial smoother with
sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration
as a preconditioner to the flexible Conjugate Gradient iteration in the solve
phase. We show that by combining these techniques it is possible to design a
simple and scalable algorithm. Results of the algorithm applied to graph
Laplacian systems arising from the standard linear finite element
discretization of the scalar Poisson problem are reported
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