A dual approach to Kohn–Vogelius regularization applied to data completion problem

Abstract This paper focuses on the data completion problem, which is well known to be an ill-posed inverse problem. We propose a dual regularization strategy without regularization parameter, based on the minimization of a functional which, instead of acting on the space of solutions, acts on the space of data. We prove the well-posedness of the minimization problem and the convergence of our regularized solution to the exact solution when the amount of noise on the data goes to 0. Moreover we prove that the regularized solution satisfies the well-known Morozov discrepancy principle . Our regularization strategy is closely related to the usual Kohn–Vogelius minimization strategy . In particular, we show that it allows not only to stably obtain a good reconstruction of the missing data of the Cauchy problem but also to determine the unique parameter of regularization for the Kohn–Vogelius strategy that satisfies the Morozov discrepancy principle. Finally, we present numerical results, in two and three dimensions, to highlight the efficiency of the proposed method.