  We develop variational regularization methods which leverage
sparsity-promoting priors to solve severely ill posed inverse problems defined
on the 3D ball (i.e. the solid sphere). Our method solves the problem natively
on the ball and thus does not suffer from discontinuities that plague alternate
approaches where each spherical shell is considered independently.
Additionally, we leverage advances in probability density theory to produce
Bayesian variational methods which benefit from the computational efficiency of
advanced convex optimization algorithms, whilst supporting principled
uncertainty quantification. We showcase these variational regularization and
uncertainty quantification techniques on an illustrative example. The C++ code
discussed throughout is provided under a GNU general public license.
