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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...
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.
Building Your Own Wavelets at Home
"... Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what mak ..."
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Cited by 151 (13 self)
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Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what makes them so successful in application areas. The main
Biorthogonal Wavelet Bases for the Boundary Element Method
, 2003
"... As shown by Dahmen, Harbrecht and Schneider [7, 23, 32], the fully discrete wavelet Galerkin scheme for boundary integral equations scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. The supposition is a wavelet basis with a sufficiently ..."
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Cited by 30 (8 self)
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As shown by Dahmen, Harbrecht and Schneider [7, 23, 32], the fully discrete wavelet Galerkin scheme for boundary integral equations scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. The supposition is a wavelet basis with a sufficiently large number of vanishing moments. In this paper we present several constructions of appropriate wavelet bases on manifolds based on the biorthogonal spline wavelets of A. Cohen, I. Daubechies and J.C. Feauveau [4]. By numerical experiments we demonstrate that it is worthwhile to spent effort on their construction to increase the performance of the wavelet Galerkin scheme considerably.
Adaptive Wavelet Schemes for Elliptic Problems  Implementation and Numerical Experiments
 SIAM J. Scient. Comput
, 1999
"... Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution stays proportional to the smallest possible error that can be realized by any linear combination o ..."
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Cited by 27 (13 self)
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Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution stays proportional to the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets. On one hand, the results are purely asymptotic. On the other hand, the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet. It is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results. We briey review rst the main theoretical facts, give a detailed description of the algorithm, highlight the essential data structures and illustrate the results by one and two dimensional numerical examples.
Multiwavelets for Second Kind Integral Equations
, 1997
"... We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a twodimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N² t ..."
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Cited by 25 (14 self)
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We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a twodimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N² to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.
Wavelet Least Square Methods For Boundary Value Problems
 SIAM J. Numer. Anal
, 1999
"... This paper is concerned with least squares methods for the numerical solution of operator equations. Our primary focus is the discussion of the following conceptual issues: the selection of appropriate least squares functionals, their numerical evaluation in the special light of recent developments ..."
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Cited by 21 (13 self)
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This paper is concerned with least squares methods for the numerical solution of operator equations. Our primary focus is the discussion of the following conceptual issues: the selection of appropriate least squares functionals, their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations. We describe first a general format of variational problems that are well posed in a certain natural topology. In order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems, their formulation as a first order system, transmission problems, the system of Stokes equations or more general saddle point problems. Particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions. We propose a unified treatment based on wavelet expansions. In particular, we exploit the fact that weighted s...
Wavelet Methods for PDEs  Some Recent Developments
 J. Comput. Appl. Math
, 1999
"... this article will be on recent developments in the last area. Rather than trying to give an exhaustive account of the state of the art I would like to bring out some mechanisms which are in my opinion important for the application of wavelets to operator equations. To accomplish this I found it nece ..."
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Cited by 16 (4 self)
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this article will be on recent developments in the last area. Rather than trying to give an exhaustive account of the state of the art I would like to bring out some mechanisms which are in my opinion important for the application of wavelets to operator equations. To accomplish this I found it necessary to address some of the pivotal issues in more detail than others. Nevertheless, such a selected `zoom in' supported by an extensive list of references should provide a sound footing for conveying also a good idea about many other related branches that will only be briey touched upon. Of course, the selection of material is biased by my personal experience and therefore is not meant to reect any objective measure of importance. The paper is organized around two essential issues namely adaptivity and the development of concepts for coping with a major obstruction in this context namely practically relevant domain geometries
Multiscale Preconditioning for the Coupling of FEMBEM
, 2000
"... We apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior two dimensional Laplacian. The matrices belonging to the boundary terms of the coupled FEMBEM system are compressed using biorthogonal wavelet bases developed from A. Cohen, I. Daubechies and J ..."
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Cited by 14 (7 self)
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We apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior two dimensional Laplacian. The matrices belonging to the boundary terms of the coupled FEMBEM system are compressed using biorthogonal wavelet bases developed from A. Cohen, I. Daubechies and J.C. Feauveau [5]. We describe different solving and preconditioning techniques. Through numerical experiments we provide results which corroborate the theory of [19] and the present paper.
Wavelets in Computer Graphics
 PROCEEDINGS OF THE IEEE, TO APPEAR
"... One of the perennial goals in computer graphics is realism in realtime. Handling geometrically complex scenes and physically faithful descriptions of their appearance and behavior, clashes with the requirement of multiple frame per second update rates. It is no surprise then that hierarchical modeli ..."
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Cited by 12 (1 self)
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One of the perennial goals in computer graphics is realism in realtime. Handling geometrically complex scenes and physically faithful descriptions of their appearance and behavior, clashes with the requirement of multiple frame per second update rates. It is no surprise then that hierarchical modeling and simulation have already enjoyed a long history in computer graphics. Most recently these ideas have received a significant boost as wavelet based algorithms have entered many areas in computer graphics. We give an overview of some of the areas in which wavelets have already had an impact on the state of the art.