1,207 research outputs found
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
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
Least Squares Ranking on Graphs
Given a set of alternatives to be ranked, and some pairwise comparison data,
ranking is a least squares computation on a graph. The vertices are the
alternatives, and the edge values comprise the comparison data. The basic idea
is very simple and old: come up with values on vertices such that their
differences match the given edge data. Since an exact match will usually be
impossible, one settles for matching in a least squares sense. This formulation
was first described by Leake in 1976 for rankingfootball teams and appears as
an example in Professor Gilbert Strang's classic linear algebra textbook. If
one is willing to look into the residual a little further, then the problem
really comes alive, as shown effectively by the remarkable recent paper of
Jiang et al. With or without this twist, the humble least squares problem on
graphs has far-reaching connections with many current areas ofresearch. These
connections are to theoretical computer science (spectral graph theory, and
multilevel methods for graph Laplacian systems); numerical analysis (algebraic
multigrid, and finite element exterior calculus); other mathematics (Hodge
decomposition, and random clique complexes); and applications (arbitrage, and
ranking of sports teams). Not all of these connections are explored in this
paper, but many are. The underlying ideas are easy to explain, requiring only
the four fundamental subspaces from elementary linear algebra. One of our aims
is to explain these basic ideas and connections, to get researchers in many
fields interested in this topic. Another aim is to use our numerical
experiments for guidance on selecting methods and exposing the need for further
development.Comment: Added missing references, comparison of linear solvers overhauled,
conclusion section added, some new figures adde
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
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
This work develops a nonlinear multigrid method for diffusion problems
discretized by cell-centered finite volume methods on general unstructured
grids. The multigrid hierarchy is constructed algebraically using aggregation
of degrees of freedom and spectral decomposition of reference linear operators
associated with the aggregates. For rapid convergence, it is important that the
resulting coarse spaces have good approximation properties. In our approach,
the approximation quality can be directly improved by including more spectral
degrees of freedom in the coarsening process. Further, by exploiting local
coarsening and a piecewise-constant approximation when evaluating the nonlinear
component, the coarse level problems are assembled and solved without ever
re-visiting the fine level, an essential element for multigrid algorithms to
achieve optimal scalability. Numerical examples comparing relative performance
of the proposed nonlinear multigrid solvers with standard single-level
approaches -- Picard's and Newton's methods -- are presented. Results show that
the proposed solver consistently outperforms the single-level methods, both in
efficiency and robustness
- …