  Discrete inverse problems correspond to solving a system of equations in a
stable way with respect to noise in the data. A typical approach to enforce
uniqueness and select a meaningful solution is to introduce a regularizer.
While for most applications the regularizer is convex, in many cases it is not
smooth nor strongly convex. In this paper, we propose and study two new
iterative regularization methods, based on a primal-dual algorithm, to solve
inverse problems efficiently. Our analysis, in the noise free case, provides
convergence rates for the Lagrangian and the feasibility gap. In the noisy
case, it provides stability bounds and early-stopping rules with theoretical
guarantees. The main novelty of our work is the exploitation of some a priori
knowledge about the solution set, i.e. redundant information. More precisely we
show that the linear systems can be used more than once along the iteration.
Despite the simplicity of the idea, we show that this procedure brings
surprising advantages in the numerical applications. We discuss various
approaches to take advantage of redundant information, that are at the same
time consistent with our assumptions and flexible in the implementation.
Finally, we illustrate our theoretical findings with numerical simulations for
robust sparse recovery and image reconstruction through total variation. We
confirm the efficiency of the proposed procedures, comparing the results with
state-of-the-art methods.
