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Local Fourier Analysis
"... Classic convergence analysis for geometric multigrid The constant coefficient case The classic convergence analysis for multigrid methods assumes: • d-dimensional PDE with constant coefficients (−1)q d∑ i=1 d 2q dx2qi u(x) = g(x), x ∈ Ω = (0, 1)d, q ≥ 1. • Periodic boundary conditions on ∂Ω or an i ..."
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Classic convergence analysis for geometric multigrid The constant coefficient case The classic convergence analysis for multigrid methods assumes: • d-dimensional PDE with constant coefficients (−1)q d∑ i=1 d 2q dx2qi u(x) = g(x), x ∈ Ω = (0, 1)d, q ≥ 1. • Periodic boundary conditions on ∂Ω
Local Fourier Analysis for Tensor-Product Multigrid
"... Abstract. We present a new formulation of multigrid, the so-called tensor-product multigrid method, which can be used to solve Lyapunov equations. These matrix equations are of considerable importance in control theory and model reduction. Since they are formulated on a tensor product space, they ar ..."
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, they are of possibly very large dimension and one needs an efficient solver like multigrid with optimal chosen components. We show that this can be done by computing the convergence factors with Local Fourier Analysis adapted for this tensor-product multigrid method.
COARSE GRID APPROXIMATION GOVERNED BY LOCAL FOURIER ANALYSIS
"... Abstract. Solving discrete boundary value problems with the help of an appro-priate multigrid method [1, 4, 5, 6] necessitates the construction of a sequence of coarse grids with corresponding coarse grid approximations for the given fine grid discretization. Popular choices in this context are the ..."
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frequencies) of the Fourier symbol of the fine grid operator. This strategy is abbreviated by FCA since the design of coarse grid approximations is based on local Fourier analysis [5, 7]. The entries of the coarse grid stencils are simply given by linear combinations of the fine grid entries. As a consequence
Phase space decompositions: Local Fourier analysis on spheres
- SIGNAL PROC. 43
, 1993
"... Continuous wavelet analysis and Gabor analysis have proven to be very useful tools for the analysis of signals in which local frequencies can be extracted. Both techniques can be described in the same footing using the theory of square-integrable group representations or derived theories. It is show ..."
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Cited by 9 (0 self)
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Continuous wavelet analysis and Gabor analysis have proven to be very useful tools for the analysis of signals in which local frequencies can be extracted. Both techniques can be described in the same footing using the theory of square-integrable group representations or derived theories
A NONPARAMETRIC TEST FOR STATIONARITY BASED ON LOCAL FOURIER ANALYSIS
"... In this paper we propose a nonparametric hypothesis test for stationarity based on local Fourier analysis. We employ a test statistic that measures the variation of time-localized estimates of the power spectral density of an observed random process. For the case of a white Gaussian noise process, w ..."
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Cited by 7 (2 self)
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In this paper we propose a nonparametric hypothesis test for stationarity based on local Fourier analysis. We employ a test statistic that measures the variation of time-localized estimates of the power spectral density of an observed random process. For the case of a white Gaussian noise process
LOCAL FOURIER ANALYSIS OF SPACE-TIME RELAXATION AND MULTIGRID SCHEMES ∗
"... Abstract. We consider numerical methods for generalized diffusion equations that are motivated by the transport problems arising in electron beam radiation therapy planning. While Monte Carlo methods are typically used for simulations of the forward-peaked scattering behavior of electron beams, roug ..."
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Cited by 1 (0 self)
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] shows the necessary optimal scaling with some dependence on the choice of scattering kernel. In order to understand this behavior, local Fourier analysis can be applied to the two-grid cycle. Using this approach, expressions for the error-propagation operators of the coarsegrid correction and relaxation
A LOCAL FOURIER ANALYSIS FRAMEWORK FOR FINITE-ELEMENT DISCRETIZATIONS OF SYSTEMS OF PDES ∗
"... Abstract. Since their popularization in the late 1970s and early 1980s, multigrid methods have been a central tool in the numerical solution of the linear and nonlinear systems that arise from the discretization of many PDEs. In this paper, we present a local Fourier analysis (LFA, or local mode ana ..."
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Abstract. Since their popularization in the late 1970s and early 1980s, multigrid methods have been a central tool in the numerical solution of the linear and nonlinear systems that arise from the discretization of many PDEs. In this paper, we present a local Fourier analysis (LFA, or local mode
Local Fourier analysis of multigrid methods with polynomial smoothers and aggressive coarsening
"... Abstract. We focus on the study of multigrid methods with aggressive coarsening and polynomial smoothers for the solution of the linear systems corresponding to finite dif-ference/element discretizations of the Laplace equation. Using local Fourier analysis we determine automatically the optimal val ..."
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Abstract. We focus on the study of multigrid methods with aggressive coarsening and polynomial smoothers for the solution of the linear systems corresponding to finite dif-ference/element discretizations of the Laplace equation. Using local Fourier analysis we determine automatically the optimal
(F.W.O.-Vlaanderen). LOCAL FOURIER ANALYSIS OF MULTIGRID FOR THE
, 2006
"... We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell’s equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk and Winther are considered. The key to our approach is the identifi ..."
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We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell’s equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk and Winther are considered. The key to our approach
SUBMITTED MANUSCRIPT 1 “Rewiring ” Filterbanks for Local Fourier Analysis: Theory and Practice
, 909
"... Abstract — 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 convolu ..."
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-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
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