  Inverse problems defined on the sphere arise in many fields, including
seismology and cosmology where problems are defined on the globe and the cosmic
sphere. These are generally high-dimensional and computationally very complex
and, as a result, sampling the posterior of spherical inverse problems is a
challenging task. In this work, we describe a framework that leverages a
proximal Markov chain Monte Carlo (MCMC) algorithm to efficiently sample the
high-dimensional space of spherical inverse problems with a sparsity-promoting
wavelet prior. We detail the modifications needed for the algorithm to be
applied to spherical problems, and give special consideration to the crucial
forward modelling step which contains spherical harmonic transforms that are
computationally expensive. By sampling the posterior, our framework allows for
full and flexible uncertainty quantification, something which is not possible
with other methods based on, for example, convex optimisation. We demonstrate
our framework in practice on full-sky cosmological mass-mapping and on a common
problem in global seismic tomography. We find that our approach is potentially
useful at moderate resolutions, such as those of interest in seismology. Our
framework is generally limited by resolution requirements, such as those
required for astrophysical applications, due to the poor scaling of the
complexity of spherical harmonic transforms with resolution. A new Python
package, pxmcmc, containing the proximal MCMC sampler, measurement operators,
wavelet transforms and sparse priors is made publicly available.
