A robust alternating direction numerical scheme in a shape optimization setting for solving geometric inverse problems

TL;DR

This paper introduces a robust numerical scheme based on the alternating direction method of multipliers for shape optimization, improving the detection of complex geometric features in inverse problems, especially under noisy conditions.

Contribution

The paper develops a novel alternating direction method of multipliers tailored for shape optimization in geometric inverse problems, enhancing accuracy in detecting complex cavities.

Findings

Outperforms traditional methods in shape reconstruction accuracy.

Effective in noisy data scenarios.

Applicable in both 2D and 3D problems.

Abstract

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.