Experimenting with automatized numerical methods

The approximation of the gravitational potential is still of interest in geodesy as it is utilized, e.g., for the mass transport of the Earth. The Inverse Problem Matching Pursuits (IPMPs) were proposed as alternative solvers for these kind of problems. They were successfully tested on diverse applications, including the downward continuation of the gravitational potential.

It is well-known that, for such linear inverse problems on the sphere, there exist a variety of global as well as local basis systems, e.g. spherical harmonics, Slepian functions as well as radial basis functions and wavelets. Each type has its specific pros and cons. Nonetheless, approximations are often represented in only one of them. On the contrary, the IPMPs enable an approximation as a mixture of diverse trial functions. They are chosen iteratively from an intentionally overcomplete dictionary such that the Tikhonov functional is reduced. However, an a-priori defined, finite dictionary has its own drawbacks, in particular with respect to efficiency.

Thus, we developed a learning add-on which uses an infinite dictionary instead while simultaneously reducing the computational cost. The add-on is implemented as constrained non-linear optimization problems with respect to the characteristic parameters of the different basis systems. In this talk, we give details on the matching pursuits and, in particular, the learning add-on and show recent numerical results with respect to the downward continuation of the gravitational potential.