236 research outputs found
Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer
This paper analyzes the Krylov convergence rate of a Helmholtz problem
preconditioned with Multigrid. The multigrid method is applied to the Helmholtz
problem formulated on a complex contour and uses GMRES as a smoother substitute
at each level. A one-dimensional model is analyzed both in a continuous and
discrete way. It is shown that the Krylov convergence rate of the continuous
problem is independent of the wave number. The discrete problem, however, can
deviate significantly from this bound due to a pitchfork in the spectrum. It is
further shown in numerical experiments that the convergence rate of the Krylov
method approaches the continuous bound as the grid distance gets small
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
On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning
This paper studies and analyzes a preconditioned Krylov solver for Helmholtz
problems that are formulated with absorbing boundary layers based on complex
coordinate stretching. The preconditioner problem is a Helmholtz problem where
not only the coordinates in the absorbing layer have an imaginary part, but
also the coordinates in the interior region. This results into a preconditioner
problem that is invertible with a multigrid cycle. We give a numerical analysis
based on the eigenvalues and evaluate the performance with several numerical
experiments. The method is an alternative to the complex shifted Laplacian and
it gives a comparable performance for the studied model problems
Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems
In this paper we solve the Helmholtz equation with multigrid preconditioned
Krylov subspace methods. The class of Shifted Laplacian preconditioners are
known to significantly speed-up Krylov convergence. However, these
preconditioners have a parameter beta, a measure of the complex shift. Due to
contradictory requirements for the multigrid and Krylov convergence, the choice
of this shift parameter can be a bottleneck in applying the method. In this
paper, we propose a wavenumber-dependent minimal complex shift parameter which
is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid
scheme. We claim that, given any (regionally constant) wavenumber, this minimal
complex shift parameter provides the reader with a parameter choice that leads
to efficient Krylov convergence. Numerical experiments in one and two spatial
dimensions validate the theoretical results. It appears that the proposed
complex shift is both the minimal requirement for a multigrid V-cycle to
converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page
Absolute value preconditioning for symmetric indefinite linear systems
We introduce a novel strategy for constructing symmetric positive definite
(SPD) preconditioners for linear systems with symmetric indefinite matrices.
The strategy, called absolute value preconditioning, is motivated by the
observation that the preconditioned minimal residual method with the inverse of
the absolute value of the matrix as a preconditioner converges to the exact
solution of the system in at most two steps. Neither the exact absolute value
of the matrix nor its exact inverse are computationally feasible to construct
in general. However, we provide a practical example of an SPD preconditioner
that is based on the suggested approach. In this example we consider a model
problem with a shifted discrete negative Laplacian, and suggest a geometric
multigrid (MG) preconditioner, where the inverse of the matrix absolute value
appears only on the coarse grid, while operations on finer grids are based on
the Laplacian. Our numerical tests demonstrate practical effectiveness of the
new MG preconditioner, which leads to a robust iterative scheme with minimalist
memory requirements
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