Results 1  10
of
110
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 ..."
Abstract

Cited by 541 (16 self)
 Add to MetaCart
(Show Context)
. 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 ..."
Abstract

Cited by 225 (39 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 100 (22 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 92 (22 self)
 Add to MetaCart
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.
The Wavelet Element Method Part I: Construction and Analysis
, 1997
"... . The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some IR n . The domain of interest is split into subdomains which are mapped to a simple re ..."
Abstract

Cited by 78 (18 self)
 Add to MetaCart
. The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some IR n . The domain of interest is split into subdomains which are mapped to a simple reference domain, here ndimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the reference domain such that mapping them to each subdomain and matching along the interfaces leads to a wavelet system on the domain. In this paper we use adapted biorthogonal wavelet systems on the interval in such a way, that tensor products of these functions can be used for the construction of wavelet bases on the reference domain. We describe the matching procedure in any dimension n in order to impose continuity and prove that it leads to a construction of a biorthogonal wavelet system on the domain. These wavelet systems characterize Sobolev spaces measuring both piecewise and...
Multiresolution and wavelets
 Proc. Edinburgh Math. Soc
, 1994
"... Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general ..."
Abstract

Cited by 69 (32 self)
 Add to MetaCart
Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skewsymmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.
Adaptive Multilevel Methods in Three Space Dimensions
 Int. J. Numer. Methods Eng
, 1993
"... this paper to collect wellknown results on 3D mesh refinement ..."
Abstract

Cited by 56 (13 self)
 Add to MetaCart
(Show Context)
this paper to collect wellknown results on 3D mesh refinement
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 ..."
Abstract

Cited by 55 (17 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 48 (15 self)
 Add to MetaCart
(Show Context)
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.