  We present a fast algorithm for the total variation regularization of the
$3$-D gravity inverse problem. Through imposition of the total variation
regularization, subsurface structures presenting with sharp discontinuities are
preserved better than when using a conventional minimum-structure inversion.
The associated problem formulation for the regularization is non linear but can
be solved using an iteratively reweighted least squares algorithm. For small
scale problems the regularized least squares problem at each iteration can be
solved using the generalized singular value decomposition. This is not feasible
for large scale problems. Instead we introduce the use of a randomized
generalized singular value decomposition in order to reduce the dimensions of
the problem and provide an effective and efficient solution technique. For
further efficiency an alternating direction algorithm is used to implement the
total variation weighting operator within the iteratively reweighted least
squares algorithm. Presented results for synthetic examples demonstrate that
the novel randomized decomposition provides good accuracy for reduced
computational and memory demands as compared to use of classical approaches.
