Learning dictionaries for inverse problems on the sphere

The downward continuation of the gravitational potential from satellite data is a spherical ill-posed inverse problem in the geosciences. The GRACE mission (NASA/DLR) supplied us, e. g., with monthly deviations from this potential. This enables us to consider the mass transport on the Earth and, thus, visualize the climate change.
Classically, an approximative solution of an inverse problem is expanded in a suitable basis like spherical harmonics. Alternatively, the Inverse Problem Matching Pursuit (IPMP) algorithms iteratively build a linear combination of dictionary elements. A dictionary usually consists of global and local trial functions, such as spherical harmonics, Slepian functions and radial basis functions. Consequently, an a-priori choice of a finite dictionary is of major importance for the IPMP algorithms.
Therefore, we develop the Learning Inverse Problem Matching Pursuit (LIPMP) algorithms in this thesis. Here, the choice of a dictionary element in each iteration is extended by several non-linear optimization problems. In this way, the LIPMP algorithms automatically learn a finite number of optimized trial functions from infinitely many possible ones.
First, we introduce the basic results necessary for the understanding of the novel learning methods. Then we develop the LIPMP algorithms. There, we have a closer look on the downward continuation of satellite data of the gravitational potential. We also consider some of the theoretical aspects of the methods. Further, we present numerical experiments which underline the applicability of the strategies for the downward continuation problem.