Regularization Wavelets and Multiresolution

  • Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind \(A F = G\). Due to the ill-posedness of the equation a variety of regularization methods are in discussion for an approximate solution, where particular emphasize must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular value decomposition as a wavelet analysis. This enables us to perform, e.g., Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another one by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz. satellite gravity gradiometry.

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Author:Willi Freeden, F. Schneider
URN (permanent link):urn:nbn:de:hbz:386-kluedo-5783
Serie (Series number):Berichte der Arbeitsgruppe Technomathematik (AGTM Report) (174)
Document Type:Preprint
Language of publication:English
Year of Completion:1997
Year of Publication:1997
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Tag:compact operator equation ; mutiresolution; regularization wavelets
Source:Inverse Problems (1998)
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification (mathematics):41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
45-XX INTEGRAL EQUATIONS / 45Bxx Fredholm integral equations / 45B05 Fredholm integral equations
45-XX INTEGRAL EQUATIONS / 45Lxx Theoretical approximation of solutions (For numerical analysis, see 65Rxx) / 45L05 Theoretical approximation of solutions (For numerical analysis, see 65Rxx)
47-XX OPERATOR THEORY / 47Bxx Special classes of linear operators / 47B07 Operators defined by compactness properties
47-XX OPERATOR THEORY / 47Bxx Special classes of linear operators / 47B38 Operators on function spaces (general)
65-XX NUMERICAL ANALYSIS / 65Rxx Integral equations, integral transforms / 65R30 Improperly posed problems
86-XX GEOPHYSICS [See also 76U05, 76V05] / 86Axx Geophysics [See also 76U05, 76V05] / 86A22 Inverse problems [See also 35R30]
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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