Shape Reconstruction in Linear Elasticity: One-step Linearization and Monotonicity-based Regularization

We deal with the shape reconstruction of inclusions in elastic bodies. Therefore, we review the monotonicity methods introduced in a former work of the authors. These monotonicity methods build the basis for the improvement of the standard one-step linearization method. The one-step linearization method consists of solving a minimization problem with standard regularization techniques, which is commonly used in practice but builds only a heuristical approach since there is no proven theory of its convergence. In contrary, the monotonicity-based regularization, where we introduce constraints for the minimization problem via the monotonicity methods, has a rigorously proven theory, i.e., we prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. Finally, we present numerical experiments.