20 research outputs found
A rapidly converging domain decomposition method for the Helmholtz equation
A new domain decomposition method is introduced for the heterogeneous 2-D and
3-D Helmholtz equations. Transmission conditions based on the perfectly matched
layer (PML) are derived that avoid artificial reflections and match incoming
and outgoing waves at the subdomain interfaces. We focus on a subdivision of
the rectangular domain into many thin subdomains along one of the axes, in
combination with a certain ordering for solving the subdomain problems and a
GMRES outer iteration. When combined with multifrontal methods, the solver has
near-linear cost in examples, due to very small iteration numbers that are
essentially independent of problem size and number of subdomains. It is to our
knowledge only the second method with this property next to the moving PML
sweeping method.Comment: 16 pages, 3 figures, 6 tables - v2 accepted for publication in the
Journal of Computational Physic
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
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
Parallel Controllability Methods For the Helmholtz Equation
The Helmholtz equation is notoriously difficult to solve with standard
numerical methods, increasingly so, in fact, at higher frequencies.
Controllability methods instead transform the problem back to the time-domain,
where they seek the time-harmonic solution of the corresponding time-dependent
wave equation. Two different approaches are considered here based either on the
first or second-order formulation of the wave equation. Both are extended to
general boundary-value problems governed by the Helmholtz equation and lead to
robust and inherently parallel algorithms. Numerical results illustrate the
accuracy, convergence and strong scalability of controllability methods for the
solution of high frequency Helmholtz equations with up to a billion unknowns on
massively parallel architectures
A multigrid method for the Helmholtz equation with optimized coarse grid corrections
We study the convergence of multigrid schemes for the Helmholtz equation,
focusing in particular on the choice of the coarse scale operators. Let G_c
denote the number of points per wavelength at the coarse level. If the coarse
scale solutions are to approximate the true solutions, then the oscillatory
nature of the solutions implies the requirement G_c > 2. However, in examples
the requirement is more like G_c >= 10, in a trade-off involving also the
amount of damping present and the number of multigrid iterations. We conjecture
that this is caused by the difference in phase speeds between the coarse and
fine scale operators. Standard 5-point finite differences in 2-D are our first
example. A new coarse scale 9-point operator is constructed to match the fine
scale phase speeds. We then compare phase speeds and multigrid performance of
standard schemes with a scheme using the new operator. The required G_c is
reduced from about 10 to about 3.5, with less damping present so that waves
propagate over > 100 wavelengths in the new scheme. Next we consider extensions
of the method to more general cases. In 3-D comparable results are obtained
with standard 7-point differences and optimized 27-point coarse grid operators,
leading to an order of magnitude reduction in the number of unknowns for the
coarsest scale linear system. Finally we show how to include PML boundary
layers, using a regular grid finite element method. Matching coarse scale
operators can easily be constructed for other discretizations. The method is
therefore potentially useful for a large class of discretized high-frequency
Helmholtz equations.Comment: Coarse scale operators are simplified and only standard smoothers
used in v3; 5 figures, 12 table
A fast and robust computational method for the ionization cross sections of the driven Schroedinger equation using an O(N) multigrid-based scheme
This paper improves the convergence and robustness of a multigrid-based
solver for the cross sections of the driven Schroedinger equation. Adding an
Coupled Channel Correction Step (CCCS) after each multigrid (MG) V-cycle
efficiently removes the errors that remain after the V-cycle sweep. The
combined iterative solution scheme (MG-CCCS) is shown to feature significantly
improved convergence rates over the classical MG method at energies where bound
states dominate the solution, resulting in a fast and scalable solution method
for the complex-valued Schroedinger break-up problem for any energy regime. The
proposed solver displays optimal scaling; a solution is found in a time that is
linear in the number of unknowns. The method is validated on a 2D Temkin-Poet
model problem, and convergence results both as a solver and preconditioner are
provided to support the O(N) scalability of the method. This paper extends the
applicability of the complex contour approach for far field map computation [S.
Cools, B. Reps, W. Vanroose, An Efficient Multigrid Calculation of the Far
Field Map for Helmholtz and Schroedinger Equations, SIAM J. Sci. Comp. 36(3)
B367--B395, 2014].Comment: 24 pages, 10 figures, 1 tabl