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978-3-8439-2005-6, Reihe Mathematik

Roger Telschow An Orthogonal Matching Pursuit for the Regularization of Spherical Inverse Problems

179 Seiten, Dissertation Universität Siegen (2014), Softcover, A5

We propose a novel algorithm to solve inverse problems on the sphere which appear in the geosciences. Based on an orthogonal matching pursuit, the signal is expanded in terms of trial functions which are iteratively picked from a large redundant set of spherical functions, the so-called dictionary. The method is capable of combining arbitrary spherical trial functions which is a great advantage to former approximation algorithms. In particular, we combine spherical harmonics of low degrees with localized trial functions such as wavelets of different scales and/or scaling functions for the reconstruction of global trends and regional details of the signal, respectively. In order to stabilize the solution, we use a Tikhonov regularization with a particular spherical Sobolev norm as a penalty term. There is no need to solve any system of equations or any integration problem which provides the ability to handle very large amounts of data or extremely scattered data sets. The outcome of the algorithm is a smooth and sparse approximation of the unknown signal which is locally adapted to the detail structure of the signal as well as to the density of the given data. Moreover, in the case that wavelets are contained in the dictionary, we additionally obtain a multiresolution of the signal. Several numerical experiments are presented, which contain the

approximation of irregularly distributed terrestrial data as well as the regularization of an inverse problem by taking the example of downward continuation of the Earth’s gravitational field.