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.