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

TL;DR

The main finding is that the proposed alternating direction method of multipliers within a shape optimization framework significantly outperforms conventional shape optimization methods in reconstructing unknown cavity shapes.

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.

Figures (17)

• Figure 1: Reconstructions with exact data at varying values of $\beta$
• Figure 2: Histories of costs, gradient norms, and Hausdorff distances with respect to the final computed shape ${\partial}{\omega}^{N}$ (third row), $N=300$, and the exact cavity shape ${\partial}{\omega}^{\ast}$ (last row) corresponding to the case of the L-shape cavity shown in Figure \ref{['fig:figure1a']} with $\beta = 0.001$
• Figure 3: Reconstructions with noisy data at varying noise levels ($\delta = 3\%, 5\%, 9\%$) with $\beta=0.001$ and without regularization (i.e., $\gamma = 0$)
• Figure 4: Histories of costs, gradient norms, and Hausdorff distances with respect to the final computed shape ${\partial}{\omega}^{N}$ (third row), $N=300$, and the exact cavity shape ${\partial}{\omega}^{\ast}$ (last row) corresponding to the case of the L-shape cavity shown in Figure \ref{['fig:figure1c']} when $\delta = 9\%$
• Figure 5: Reconstructions with noisy data at noise level $\delta = 9\%$, with $\beta=0.001$ at varying levels of regularization ($\gamma = 0.001, 0.003, 0.005$)

Theorems & Definitions (7)

• Theorem 1.1: BourgeoisDarde2010
• Definition 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5