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The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
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Cited by 541 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
Wavelets on Closed Subsets of the Real Line
 in: Topics in the Theory and Applications of Wavelets, L.L. Schumaker and
"... . We construct orthogonal and biorthogonal wavelets on a given closed subset of the real line. We also study wavelets satisfying certain types of boundary conditions. We introduce the concept of "wavelet probing ", which is closely related to our construction of wavelets. This technique al ..."
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Cited by 74 (5 self)
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. We construct orthogonal and biorthogonal wavelets on a given closed subset of the real line. We also study wavelets satisfying certain types of boundary conditions. We introduce the concept of "wavelet probing ", which is closely related to our construction of wavelets. This technique allows us to very quickly perform a number of different numerical tasks associated with wavelets. x1. Introduction Wavelets and multiscale analysis have emerged in a number of different fields, from harmonic analysis and partial differential equations in pure mathematics to signal and image processing in computer science and electrical engineering. Typically a general function, signal, or image is broken up into linear combinations of translated and scaled versions of some simple, basic building blocks. Multiscale analysis comes with a natural hierarchical structure obtained by only considering the linear combinations of building blocks up to a certain scale. This hierarchical structure is particularly...
Smooth Wavelet Decompositions with Blocky Coefficient Kernels
, 1993
"... We describe bases of smooth wavelets where the coefficients are obtained by integration against (finite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were first developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approac ..."
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Cited by 60 (12 self)
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We describe bases of smooth wavelets where the coefficients are obtained by integration against (finite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were first developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approach emphasizes the idea of averageinterpolation  synthesizing a smooth function on the line having prescribed boxcar averages  and the link between averageinterpolation and DubucDeslauriers interpolation. We also emphasize characterizations of smooth functions via their coefficients. We describe boundarycorrected expansions for the interval, which have a simple and revealing form. We use these results to reinterpret the empirical wavelet transform  i.e. finite, discrete wavelet transforms of data arising from boxcar integrators (e.g. CCD devices).
Tensor Product Type Subspace Splittings And Multilevel Iterative Methods For Anisotropic Problems
, 1994
"... We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in R d . Our analysis is based on the theory of additive subspace correction methods and applies to finiteelement and prewaveletschemes. We present multilevel and ..."
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Cited by 55 (17 self)
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We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in R d . Our analysis is based on the theory of additive subspace correction methods and applies to finiteelement and prewaveletschemes. We present multilevel and prewaveletbased methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.
Optimized general sparse grid approximation spaces for operator equations
 MATHEMATICS OF COMPUTATIONS
, 2008
"... This paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, we construct operatoradapted subspaces with a ..."
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Cited by 30 (5 self)
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This paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, we construct operatoradapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces, provided that certain additional regularity assumptions on the solution are fulfilled. Specifically for operators of positive order, their dimension is O(2^J) independent of the dimension n of the problem, compared to O(2^Jn) for the full grid space. Also, for operators of negative order the overall cost is significantly in favor of the new approximation spaces. We give cost estimates for the case of continuous linear information. We show these results in a constructive manner by proposing a Galerkin method together with optimal preconditioning. The theory covers elliptic boundary value problems as well as boundary integral equations.
Waveletbased Algorithms for Fast PDE Solvers
, 1995
"... Wavelet techniques are used to construct fast solvers for elliptic and parabolic equations. The space and frequency localization properties of wavelet bases make them attractive for handling problems with strongly varying coefficients. Three major themes are treated: building solution operators, pre ..."
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Cited by 11 (2 self)
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Wavelet techniques are used to construct fast solvers for elliptic and parabolic equations. The space and frequency localization properties of wavelet bases make them attractive for handling problems with strongly varying coefficients. Three major themes are treated: building solution operators, preconditioners and homogenized equations. In classical coordinates, the discrete solution operators of elliptic and parabolic equations are dense matrices. It is well known that in wavelet bases they are compressed to accommodate sparse representations. A fast timemarching algorithm for building a discrete solution operator of parabolic equations (in wavelet coordinates) is given. The algorithm can even handle highly oscillatory source terms. Sparcity is achieved by ignoring small entries at each time iteration. The error of this procedure is shown to be wellcontrolled. Experiments indicate that explicit discretizations yield more efficient algorithms for loworder methods. The structure of ...
Optimized Approximation Spaces for Operator Equations
"... This paper is concerned with the construction of optimized grids and approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, we construct operator adapted finite element sub ..."
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Cited by 8 (5 self)
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This paper is concerned with the construction of optimized grids and approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, we construct operator adapted finite element subspaces with less dimension than the standard full grid spaces that keep the approximation order of the standard full grid spaces provided that certain additional regularity assumptions on the solution are fulfilled. Specifically for operators of positive order, the dimension is O(2^J) independent of the dimension n of the problem compared to O(2^Jn) for the full grid space. Also, for operators of negative order the overall complexity is signi cantly in favor of the new approximation spaces. We give complexity estimates for the case of continuous linear information. We show these results in a constructive manner by proposing a finite element method together with optimal preconditioning. The theory
Wavelets in econometrics: An application to outlier testing
, 1994
"... Abstract. In recent years, wavelets have become widely used in physics, engineering, and mathematics. They have been used for signal processing, image processing, numerical computation, and data compression. Wavelets have not, however, been used very much in the elds of Economics, Econometrics, and ..."
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Cited by 8 (0 self)
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Abstract. In recent years, wavelets have become widely used in physics, engineering, and mathematics. They have been used for signal processing, image processing, numerical computation, and data compression. Wavelets have not, however, been used very much in the elds of Economics, Econometrics, and Finance. In this study, We will look at the wavelet transform in the context of multiresolution analysis, discuss its uses in other elds, and present an Econometric application of wavelets to outlier detection. Postal address for correspondence: Dr. Seth A. Greenblatt