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.
Highlights
The idea of the regularization is to introduce conditions for the parameters / inclusions to be reconstructed for the minimization problem, which are based on the monotonicity properties of the Neumann-to-Dirichlet operator and the monotonicity tests
We compare numerical reconstructions of inclusions based on the monotonicity-based regularization with the one-step linearization, which shows the robustness of our method regarding noise in practice
We start with the introduction of the problems of interest, e.g., the direct as well as inverse problem of stationary linear elasticity
Summary
The main motivation is the non-destructive testing of elastic structures, such as is required for material examinations, in exploration geophysics, and for medical diagnostics (elastography). By means of a regularization approach, a stationary elastic inverse problem is solved in Jadamba et al [16] and applied in numerical examples. We want to mention the monotonicity methods for linear elasticity developed by the authors of this paper in Eberle and Harrach [5] as well as its application for the reconstruction of inclusions based on experimental data in Eberle and Moll [7]. The idea of the regularization is to introduce conditions for the parameters / inclusions to be reconstructed for the minimization problem, which are based on the monotonicity properties of the Neumann-to-Dirichlet operator and the monotonicity tests. We compare numerical reconstructions of inclusions based on the monotonicity-based regularization with the one-step linearization, which shows the robustness of our method regarding noise in practice
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