Monotonicity-based regularization for shape reconstruction in linear elasticity

Abstract

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 Eberle and Harrach (Inverse Probl 37(4):045006, 2021), 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.

Resolution guarantees for the reconstruction of inclusions in linear elasticity based on monotonicity methods

“…This is in accordance with the results in other papers, e.g. in [6]. testcubes and Neumann patches are given in the caption of the figure.…”

“…[11-13, 15, 16]) and then on other problems such as elasticity (see, e.g. [5][6][7]). In short, for this method the monotonicity properties of the Neumann-to-Dirichlet operator play an essential role.…”

Resolution Guarantees for the Reconstruction of Inclusions in Linear Elasticity Based on Monotonicity Methods

“…All in all, our results are of special importance, when considering simulations based on real data, e.g., in [8] or in the framework of monotonicity-based regularization (see, e.g. [6]).…”

“…We focus on the monotonicity methods which are built on the examinations in [36,37]. These methods were first used for EIT (see, e.g., [11,12,13,14,15]) and then on other problems such as elasticity (see, e.g., [5,6,7]). In short, for this method the monotonicity properties of Neumannto-Dirichlet operator plays an essential role.…”

The monotonicity method for inclusion detection and the time harmonic elastic wave equation

“…Thus, this will build the basis for further examinations and the development of the 'linearised monotonicity methods' (see [EH21]) for the stationary case) which will allow us a faster implementation and as such, a testing with more test inclusions. A further extension could be the consideration of a monotonicity-based regularisation as well as the examination of the resolution guarantee (similar as considered in for the stationary elastic inverse problem in [EH22,EBH23], respectively).…”