  Gravitational field modelling is an important tool for inferring past and
present dynamic processes of the Earth. Functions on the sphere such as the
gravitational potential are usually expanded in terms of either spherical
harmonics or radial basis functions (RBFs). The (Regularized) Functional
Matching Pursuit ((R)FMP) and its variants use an overcomplete dictionary of
diverse trial functions to build a best basis as a sparse subset of the
dictionary and compute a model, for instance, of the gravity field, in this
best basis. Thus, one advantage is that the dictionary may contain spherical
harmonics and RBFs. Moreover, these methods represent a possibility to obtain
an approximative and stable solution of an ill-posed inverse problem, such as
the downward continuation of gravitational data from the satellite orbit to the
Earth's surface, but also other inverse problems in geomathematics and medical
imaging. A remaining drawback is that in practice, the dictionary has to be
finite and, so far, could only be chosen by rule of thumb or trial-and-error.
In this paper, we develop a strategy for automatically choosing a dictionary by
a novel learning approach. We utilize a non-linear constrained optimization
problem to determine best-fitting RBFs (Abel-Poisson kernels). For this, we use
the Ipopt software package with an HSL subroutine. Details of the algorithm are
explained and first numerical results are shown.
