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Inverse problems defined naturally on the sphere are becoming increasingly ofinterest. In this article we provide a general framework for evaluation ofinverse problems on the sphere, with a strong emphasis on flexibility andscalability. We consider flexibility with respect to the prior selection(regularization), the problem definition - specifically the problem formulation(constrained/unconstrained) and problem setting (analysis/synthesis) - andoptimization adopted to solve the problem. We discuss and quantify thetrade-offs between problem formulation and setting. Crucially, we consider theBayesian interpretation of the unconstrained problem which, combined withrecent developments in probability density theory, permits rapid, statisticallyprincipled uncertainty quantification (UQ) in the spherical setting. Linearityis exploited to significantly increase the computational efficiency of such UQtechniques, which in some cases are shown to permit analytic solutions. Weshowcase this reconstruction framework and UQ techniques on a variety ofspherical inverse problems. The code discussed throughout is provided under aGNU general public license, in both C++ and Python.

Sparse image reconstruction on the sphere: a general approach with uncertainty quantification