We deal with the shape reconstruction of inclusions in elastic bodies. For
solving this inverse problem in practice, data fitting functionals are used.
Those work better than the rigorous monotonicity methods from [5], but have no
rigorously proven convergence theory. Therefore we show how the monotonicity
methods can be converted into a regularization method for a data-fitting
functional without losing the convergence properties of the monotonicity
methods. This is a great advantage and a significant improvement over standard
regularization techniques. In more detail, we introduce constraints on the
minimization problem of the residual based on the monotonicity methods and
prove the existence and uniqueness of a minimizer as well as the convergence of
the method for noisy data. In addition, we compare numerical reconstructions of
inclusions based on the monotonicity-based regularization with a standard
approach (one-step linearization with Tikhonov-like regularization), which also
shows the robustness of our method regarding noise in practice.Comment: 26 pages, 15 figure
