  The alternating direction method of multipliers within a shape optimization
framework is developed for solving geometric inverse problems, focusing on a
cavity identification problem from the perspective of non-destructive testing
and evaluation techniques. The rationale behind this method is to achieve more
accurate detection of unknown inclusions with pronounced concavities,
emphasizing the aspect of shape optimization. Several numerical results to
illustrate the applicability and efficiency of the method are presented for
various shape detection problems. These numerical experiments are conducted in
both two- and three-dimensional settings, with a focus on cases involving
noise-contaminated data. The main finding of the study is that the proposed
method significantly outperforms conventional shape optimization methods in
reconstructing unknown cavity shapes.
