Regularization of ill-posed inverse problems with tolerances and sparsity in the parameter space

We consider the solution of ill-posed inverse problems using regularization with tolerances. In particular, we are interested in the reconstruction of solutions that lie within or close to an area outlined by a tolerance measure. To approximate the true solution of the problem in a stable way, we propose a Tikhonov functional with a tolerance function in the regularization term. The tolerances allow us to neglect errors in the penalty term up to a certain threshold. Our theoretical analysis proves that the proposed method complies with all the requirements of variational regularization methods. In addition, we establish convergence rates for the convergence of minimizers to the true solution.

Moreover, we are interested in obtaining sparse solutions. For this purpose, we extend the proposed approach with the idea of elastic net regularization by introducing an additional penalty term that promotes the sparsity of the solution. We establish theoretical results for this elastic net approach and give a convergence rate analysis for the minimizers. To confirm our analytical findings, we illustrate the effect of tolerances in the computed regularized solutions on some numerical examples.