Automatic balancing parameter selection for Tikhonov-TV regularization

This paper considers large-scale linear ill-posed inverse problems whose solutions can be represented as sums of smooth and piecewise constant components. To solve such problems we consider regularizers consisting of two terms that must be balanced. Namely, a Tikhonov term guarantees the smoothness of the smooth solution component, while a total-variation (TV) regularizer promotes blockiness of the non-smooth solution component. A scalar parameter allows to balance between these two terms and, hence, to appropriately separate and regularize the smooth and non-smooth components of the solution. This paper proposes an efficient algorithm to solve this regularization problem by the alternating direction method of multipliers (ADMM). Furthermore, a novel algorithm for automatic choice of the balancing parameter is introduced, using robust statistics. The proposed approach is supported by some theoretical analysis, and numerical experiments concerned with different inverse problems are presented to validate the choice of the balancing parameter.