  In this paper, we are concerned with efficiently solving the sequences of
regularized linear least squares problems associated with employing
Tikhonov-type regularization with regularization operators designed to enforce
edge recovery. An optimal regularization parameter, which balances the fidelity
to the data with the edge-enforcing constraint term, is typically not known a
priori. This adds to the total number of regularized linear least squares
problems that must be solved before the final image can be recovered.
Therefore, in this paper, we determine effective multigrid preconditioners for
these sequences of systems. We focus our approach on the sequences that arise
as a result of the edge-preserving method introduced in [6], where we can
exploit an interpretation of the regularization term as a diffusion operator;
however, our methods are also applicable in other edge-preserving settings,
such as iteratively reweighted least squares problems. Particular attention is
paid to the selection of components of the multigrid preconditioner in order to
achieve robustness for different ranges of the regularization parameter value.
In addition, we present a parameter culling approach that, when used with the
L-curve heuristic, reduces the total number of solves required. We demonstrate
our preconditioning and parameter culling routines on examples in computed
tomography and image deblurring.
