A dictionary learning add-on for spherical downward continuation

Abstract

AbstractWe propose a novel dictionary learning add-on for the Inverse Problem Matching Pursuit (IPMP) algorithms for approximating spherical inverse problems such as the downward continuation of the gravitational potential. With the add-on, we aim to automatize the choice of dictionary and simultaneously reduce the computational costs. The IPMP algorithms iteratively minimize the Tikhonov–Phillips functional in order to construct a weighted linear combination of so-called dictionary elements as a regularized approximation. A dictionary is an intentionally redundant set of trial functions such as spherical harmonics (SHs), Slepian functions (SLs) as well as radial basis functions (RBFs) and wavelets (RBWs). In previous works, this dictionary was chosen manually which resulted in high runtimes and storage demand. Moreover, a possible bias could also not be ruled out. The additional learning technique we present here allows us to work with infinitely many trial functions while reducing the computational costs. This approach may enable a quantification of a possible bias in future research. We explain the general mechanism and provide numerical results that prove its applicability and efficiency.