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We consider the variational regularization for inverse problems in a general form. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level and is therefore purely data driven. Under variational source conditions, we obtain a posteriori error estimates. According to the Bakushinskii veto, convergence in the worst case scenario cannot be expected in general. However, by imposing certain conditions on the noisy data, we establish a convergence result for the heuristic rule. Applications of the results are addressed and numerical simulations are reported. [ABSTRACT FROM AUTHOR]

We investigate an inverse problem of identifying a Robin coefficient with a sparse structure in the Laplace equation from noisy boundary measurements. The sparse structure of the Robin coefficient γ is understood as a small perturbation of a reference profile γ in the sense that their difference γ− γ has a small support. This problem is formulated as an optimal control problem with an L-regularization term. An iteratively reweighted least-squares algorithm with an inner semismooth Newton iteration is employed to solve the resulting optimization problem, and the convergence of the iteratively weighted least-squares algorithm is established. Numerical results for two-dimensional problems are presented to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]