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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.
Multiscale Methods For Boundary Integral Equations And Their Application To Boundary Value Problems In Scattering Theory And Geodesy
, 1996
"... ..."
Towards Object Oriented Software Tools for Numerical Multiscale Methods for P.D.E.s using Wavelets
 RWTH Aachen, Preprint IGPM No. 127
, 1996
"... . The enormous increase of computer power in the past years has lead to the possibility of solving more and more complex partial differential and integral equations numerically. All the more one is faced with the problem of developing efficient software routines that can be handled and overlooked by ..."
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Cited by 6 (5 self)
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. The enormous increase of computer power in the past years has lead to the possibility of solving more and more complex partial differential and integral equations numerically. All the more one is faced with the problem of developing efficient software routines that can be handled and overlooked by the users. A corresponding list of basic requirements including a unified disciplined and modular structure lead to object oriented programming that, in particular, can handle operator equations in higher spatial dimensions. One of the characteristics of multiscale basisoriented schemes for solving operator equations numerically featuring wavelets is that they do not rely on geometric decompositions of the underlying domain. During the past few years, we have developed such a software package for a class of p.d.e.s. We describe some of our software tools and provide examples for the use of the package. Keywords: Multiscale methods, wavelets, object oriented software tools. AMS subject c...
Surface Wavelets: A Multiresolution Signal Processing Tool For 3D Computational Modeling
, 2000
"... In this paper, we provide an introduction to wavelet representations for complex surfaces (surface wavelets), with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a mu ..."
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Cited by 4 (2 self)
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In this paper, we provide an introduction to wavelet representations for complex surfaces (surface wavelets), with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those encountered in finite element modeling. This motivates the study of surface wavelets as an efficient representation for the modeling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, stated either in integral form or in differential form. We analyze and implement the wavelet approach for a model 3D potential problem using a surface wavelet basis with linear interpolating properties. We show both theoretically and experimentally that an ) ( 2 n h O convergence rate, n h being the mesh size, can be obtained by retaining only () ( ) N N O 2 7 log entries in the discrete operator matrix, where N is the number of unknowns. The principles described here may also be extended to volumetric discretizations.
Wavelet Techniques for the Fictitious Domain  Lagrange Multiplier Approach
, 2000
"... We consider second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. This is combined with a ctitious domain approach into which the physical domain is embedded. The resulting saddle point problem will be discretized in te ..."
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We consider second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. This is combined with a ctitious domain approach into which the physical domain is embedded. The resulting saddle point problem will be discretized in terms of wavelets, resulting in an operator equation in ` 2 . Stability of the discretization and consequently the uniform boundedness of the condition number of the nite{dimensional operator independent of the discretization is guaranteed by an appropriate LBB condition. For the iterative solution of the saddle point system, an incomplete Uzawa algorithm is employed. It can be shown that the iterative scheme combined with a nested iteration strategy is asymptotically optimal in the sense that it provides the solution up to discretization error on discretization level J in an overall amount of iterations of order O(N J ) where N J is the number of unknowns on level J . Finally, numeri...
OPTIMAL DISCRETIZATION OF INVERSE PROBLEMS IN HILBERT SCALES. REGULARIZATION AND SELFREGULARIZATION OF PROJECTION METHODS∗
"... Abstract. We study the efficiency of the approximate solution of illposed problems, based on discretized noisy observations, which we assume to be given beforehand. A basic purpose of the paper is the consideration of stochastic noise, but deterministic noise is also briefly discussed. We restrict ..."
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Abstract. We study the efficiency of the approximate solution of illposed problems, based on discretized noisy observations, which we assume to be given beforehand. A basic purpose of the paper is the consideration of stochastic noise, but deterministic noise is also briefly discussed. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framework we shall quantify the degree of illposedness, provide general conditions on projection schemes to achieve the best possible order of accuracy. We pay particular attention on the problem of selfregularization vs. Tikhonov regularization. Moreover, we study the information complexity. Asymptotically, any method which achieves the best possible order of accuracy must use at least such amount of noisy observations.
An adaptive discretization for TikhonovPhillips regularization with a posteriori parameter selection
"... The aim of this paper is to describe an efficient adaptive strategy for discretizing illposed linear operator equations of the first kind: we consider TikhonovPhillips regularization x ffi ff = (A A + ffI) \Gamma1 A y ffi with a finite dimensional approximation A n instead of A. We prop ..."
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The aim of this paper is to describe an efficient adaptive strategy for discretizing illposed linear operator equations of the first kind: we consider TikhonovPhillips regularization x ffi ff = (A A + ffI) \Gamma1 A y ffi with a finite dimensional approximation A n instead of A. We propose a sparse matrix structure which still leads to optimal convergences rates but requires substantially less scalar products for computing A n compared with standard methods.