## 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.

Author: | Willi Freeden, F. Schneider |
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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 |

Tag: | compact operator equation ; mutiresolution; regularization wavelets |

Source: | Inverse Problems (1998) |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 510 Mathematik |

MSC-Classification (mathematics): | 41A58 Series expansions (e.g. Taylor, Lidstone series, but not Fourier series) |

45B05 Fredholm integral equations | |

45L05 Theoretical approximation of solutions (For numerical analysis, see 65Rxx) | |

47B07 Operators defined by compactness properties | |

47B38 Operators on function spaces (general) | |

65R30 Improperly posed problems | |

86A22 Inverse problems [See also 35R30] |