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95
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...
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 225 (39 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
Adaptive wavelet methods for elliptic operator equations— convergence rates
 Math. Comput
, 2001
"... Abstract. This paper is concerned with the construction and analysis of waveletbased adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance ..."
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Cited by 174 (33 self)
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Abstract. This paper is concerned with the construction and analysis of waveletbased adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called Nterm approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N −s)in the energy norm, whenever such a rate is possible by Nterm approximation. The range of s>0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization. 1.
Wavelets on Manifolds I: Construction and Domain Decomposition
 SIAM J. Math. Anal
, 1998
"... The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features ar ..."
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Cited by 100 (22 self)
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The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...
Biorthogonal SplineWavelets on the Interval  Stability and Moment Conditions
 Appl. Comp. Harm. Anal
, 1997
"... This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and co ..."
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Cited by 99 (48 self)
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This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly...
Composite Wavelet Bases for Operator Equations
 MATH. COMP
, 1996
"... This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary ..."
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Cited by 92 (22 self)
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This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudodifferential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions.
Stable Multiscale Bases and Local Error Estimation for Elliptic Problems
 Appl. Numer. Math
, 1996
"... This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient aposteriori erro ..."
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Cited by 71 (36 self)
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This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient aposteriori error estimators for Galerkin schemes which are based on stable multiscale bases. It is shown that the locality of corresponding multiresolution processes combined with certain norm equivalences involving weighted sequence norms leads to adaptive space refinement strategies which are guaranteed to converge in a wide range of cases, again including operators of negative order. Key words: Stable multiscale bases, norm equivalences, elliptic operator equations, Galerkin schemes, aposteriori error estimators, convergence of adaptive schemes AMS subject classification: 65N55, 65N30, 65N38, 65N12 1 Introduction The increasing importance of adaptive techniques in large scale computation is reflected...
Compression Techniques for Boundary Integral Equations  Optimal Complexity Estimates
, 2002
"... In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offe ..."
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Cited by 48 (15 self)
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In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional aposteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.
Fully Discrete Multiscale Galerkin BEM
 MULTISCALE WAVELET METHODS FOR PDES
, 1997
"... We analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in R³. Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy n ..."
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Cited by 44 (4 self)
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We analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in R³. Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy norm and in certain negative norms can be achieved with "compressed" stiffness matrices containing O(N (log N)²) nonvanishing entries where N denotes the number of degrees of freedom on the boundary manifold. We analyze a quadrature scheme giving rise to fully discrete methods. We show that the fully discrete scheme preserves the asymptotic accuracy of the Galerkin scheme with exact integration and without compression. The overall computational complexity of our algorithm is O(N (log N) 4 ) kernel evaluations. The implications of the results for the numerical solution of elliptic boundary value problems in or exterior to bounded, threedimensional domains are discussed.
ElementByElement Construction Of Wavelets Satisfying Stability And Moment Conditions
 SIAM J. NUMER. ANAL
, 1998
"... In this paper, we construct a class of locally supported wavelet bases for C 0 Lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H s for jsj ! 3 2 (jsj 1 on Lipschitz' manifolds), and ..."
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Cited by 43 (13 self)
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In this paper, we construct a class of locally supported wavelet bases for C 0 Lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H s for jsj ! 3 2 (jsj 1 on Lipschitz' manifolds), and the wavelets can, in principal, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used e.g. for constructing an optimal solver of discretized H s elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet ba...