431 research outputs found
On the validity of the local Fourier analysis
Local Fourier analysis (LFA) is a useful tool in predicting the convergence
factors of geometric multigrid methods (GMG). As is well known, on rectangular
domains with periodic boundary conditions this analysis gives the exact
convergence factors of such methods. In this work, using the Fourier method, we
extend these results by proving that such analysis yields the exact convergence
factors for a wider class of problems
"Rewiring" Filterbanks for Local Fourier Analysis: Theory and Practice
This article describes a series of new results outlining equivalences between
certain "rewirings" of filterbank system block diagrams, and the corresponding
actions of convolution, modulation, and downsampling operators. This gives rise
to a general framework of reverse-order and convolution subband structures in
filterbank transforms, which we show to be well suited to the analysis of
filterbank coefficients arising from subsampled or multiplexed signals. These
results thus provide a means to understand time-localized aliasing and
modulation properties of such signals and their subband
representations--notions that are notably absent from the global viewpoint
afforded by Fourier analysis. The utility of filterbank rewirings is
demonstrated by the closed-form analysis of signals subject to degradations
such as missing data, spatially or temporally multiplexed data acquisition, or
signal-dependent noise, such as are often encountered in practical signal
processing applications
Using cylindrical algebraic decomposition and local Fourier analysis to study numerical methods: two examples
Local Fourier analysis is a strong and well-established tool for analyzing
the convergence of numerical methods for partial differential equations. The
key idea of local Fourier analysis is to represent the occurring functions in
terms of a Fourier series and to use this representation to study certain
properties of the particular numerical method, like the convergence rate or an
error estimate.
In the process of applying a local Fourier analysis, it is typically
necessary to determine the supremum of a more or less complicated term with
respect to all frequencies and, potentially, other variables. The problem of
computing such a supremum can be rewritten as a quantifier elimination problem,
which can be solved with cylindrical algebraic decomposition, a well-known tool
from symbolic computation.
The combination of local Fourier analysis and cylindrical algebraic
decomposition is a machinery that can be applied to a wide class of problems.
In the present paper, we will discuss two examples. The first example is to
compute the convergence rate of a multigrid method. As second example we will
see that the machinery can also be used to do something rather different: We
will compare approximation error estimates for different kinds of
discretizations.Comment: The research was funded by the Austrian Science Fund (FWF): J3362-N2
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
Extended local fourier analysis for multigrid optimal smoothing, coarse grid correction, and preconditioning
Multigrid methods are fast iterative solvers for partial di erential equations. Especially for elliptic equations they have been proven to be highly e cient. For problems with nonelliptic and nonsymmetric features--as they often occur in typical real life applications--a rigorous mathematical theory is generally not available. For such situations Fourier smoothing and two-grid analysis can be considered as the main analysis tools to obtain quantitative convergence estimates and to optimize different multigrid components like smoothers or inter-grid transfer operators. In general, it is difficult to choose the correct multigrid components for large classes of problems. A popular alternative to construct a robust solver is the use of multigrid as a preconditioner for a Krylov subspace acceleration method like GMRES. Our contributions to the Fourier analysis for multigrid are two-fold. Firstly we extend the range of situations for which the Fourier analysis can be applied. More precisely, the Fourier analysis is generalized to k-grid cycles and to multigrid as a preconditioner. With a k-grid analysis it is possible to investigate real multigrid effects which cannot be captured by the classical two-grid analysis. Moreover, the k-grid analysis allows for a more detailed investigation of possible coarse grid correction difficulties. Additional valuable insight is obtained by evaluating multigrid as a preconditioner for GMRES. Secondly we extend the range of discretizations and multigrid components for which detailed Fourier analysis results exist. We consider four well-known singularly perturbed model problems to demonstrate the usefulness of the above generalizations: The anisotropic Poisson equation, the rotated anisotropic diffusion equation, the convection diffusion equation with dominant convection, and the driven cavity problem governed by the incompressible Navier Stokes equations. Each of these equations represents a larger class of problems with similar features and complications which are of practical relevance. With the help of the newly developed Fourier analysis methods, a comprehensive study of characteristic difficulties for singular perturbation problems can be performed. Based on the insights from this analysis it is possible to identify remedies resulting in an improved multigrid convergence. The theoretical considerations are validated by numerical test calculations
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
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