In the last 10 years, the Inverse Problem Matching Pursuits (IPMPs) were proposed as alternative solvers for linear inverse problems on the sphere and the ball, e.g. from the geosciences. They were constantly further developed and tested on diverse applications, e.g. on the downward continuation of the gravitational potential. This task remains a priority in geodesy due to significant contemporary challenges like the climate change.
It is well-known that, for 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. All of these system have their specific pros and cons. Nonetheless, approximations are often represented in only one of the systems.
On the contrary, as matching pursuits, the IPMPs realize the following line of thought: an approximation is built in a so-called best basis, i.e. a mixture of diverse trial functions. Such a basis is chosen iteratively from an intentionally overcomplete dictionary which contains several types of the mentioned global and local functions. The choice of the next best basis element aims to reduce the Tikhonov functional.
In practice, an a-priori, finite set of trial functions was usually used which was highly inefficient. We developed a learning add-on which enables us to work with an infinite dictionary instead while simultaneously reducing the computational cost. Moreover, it automatized the dictionary choice as well. 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 explain the learning add-on and show recent numerical results with respect to the downward continuation of the gravitational potential.