A dictionary learning add-on for spherical downward continuation

TL;DR

This paper introduces a novel dictionary learning extension for spherical inverse problem algorithms, enabling the use of infinitely many trial functions and improving approximation accuracy in gravitational potential downward continuation.

Contribution

It presents a new learning technique that allows IPMP algorithms to utilize an infinitely large, learned dictionary, enhancing their flexibility and reducing bias in spherical inverse problems.

Findings

The learned dictionary improves approximation accuracy.

The method is applicable to various trial functions like spherical harmonics and wavelets.

Numerical results demonstrate the efficiency of the approach.

Abstract

We propose a novel dictionary learning add-on for existing approximation algorithms for spherical inverse problems such as the downward continuation of the gravitational potential. The Inverse Problem Matching Pursuit (IPMP) algorithms iteratively minimize the Tikhonov functional in order to construct a weighted linear combination of so-called dictionary elements as a regularized approximation. A dictionary is a set that contains trial functions such as spherical harmonics (SHs), Slepian functions (SLs) as well as radial basis functions (RBFs) and wavelets (RBWs). Previously, the IPMP algorithms worked with finite dictionaries which are vulnerable regarding a possible biasing of the outcome. Here, we propose an additional learning technique that allows us to work with infinitely many trial functions and provides us with a learnt dictionary for future use in the IPMP algorithms. We explain the general mechanism and provide numerical results that prove its applicability and efficiency.