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 .
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, which do not have a rigorous theory of convergence. 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 .
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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
For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from
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, 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
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one-step linearization method. The one-step linearization method consists of solving a minimization problem with standard
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is commonly used in practice but builds only a heuristical approach since there is no proven theory of its
do not have a rigorous theory of
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contrary, the monotonicity-based regularization, where
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via the monotonicity methods, has a rigorously proven theory, i.e., we
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Finally, we present numerical experiments
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