THE REGULARIZED ORTHOGONAL FUNCTIONAL MATCHING PURSUIT FOR ILL-POSED INVERSE PROBLEMS

We propose a novel algorithm to solve a general class of linear ill-posed inverse problems. For our numerical tests, we consider ill-posed problems on the sphere as they appear in the geosciences. Based on an iterative greedy algorithm, called the orthogonal matching pursuit, the signal is expanded in terms of trial functions which are picked from a large redundant set of functions, the so-called dictionary. The method is able to combine arbitrary trial functions which is a great advantage to former approximation algorithms. In particular, we combine orthogonal polynomials (such as spherical harmonics in the case of the sphere) of low degrees with localized trial functions such as wavelets and/or scaling functions for the reconstruction of global trends and regional details of the signal, respectively. Since we deal with ill-posed problems, we use a Tikhonov-type regularization with a penalty term based on a (spherical) Sobolev norm. 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.