We 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.

我们为逆问题匹配追踪 (IPMP) 算法提出了一种新颖的字典学习插件，用于近似球形逆问题，例如重力势的向下延续。通过该插件，我们的目标是自动选择字典并同时降低计算成本。IPMP 算法迭代地最小化 Tikhonov-Phillips 函数，以便将所谓的字典元素的加权线性组合构造为正则化近似。字典是有意冗余的一组试验函数，例如球谐函数 (SH)、Slepian 函数 (SL) 以及径向基函数 (RBF) 和小波 (RBW)。在以前的工作中，这个字典是手动选择的，这导致了高运行时间和存储需求。而且，也不能排除可能的偏见。我们在这里介绍的附加学习技术允许我们使用无限多的试验函数，同时降低计算成本。这种方法可以量化未来研究中可能存在的偏差。我们解释了一般机制并提供了证明其适用性和效率的数值结果。