Cavity shape reconstruction with a homogeneous Robin condition via a constrained coupled complex boundary method with ADMM

TL;DR

This paper introduces a novel constrained coupled complex boundary method combined with ADMM for reconstructing cavity shapes with Robin boundary conditions, improving stability and accuracy in inverse boundary problems.

Contribution

It applies a new complex boundary formulation with inequality constraints and ADMM to enhance cavity shape reconstruction from boundary data.

Findings

Effective reconstruction of cavity shapes demonstrated in numerical experiments.

The method shows robustness against noisy data and initialization.

Shape derivatives and gradients are explicitly derived for optimization.

Abstract

We revisit the problem of identifying an unknown portion of a boundary subject to a Robin condition based on a pair of Cauchy data on the accessible part of the boundary. It is known that a single measurement may correspond to infinitely many admissible domains. Nonetheless, numerical strategies based on shape optimization have been shown to yield reasonable reconstructions of the unknown boundary. In this study, we propose a new application of the coupled complex boundary method to address this class of inverse boundary identification problems. The overdetermined problem is reformulated as a complex boundary value problem with a complex Robin condition that couples the Cauchy data on the accessible boundary. The reconstruction is achieved by minimizing a cost functional constructed from the imaginary part of the complex-valued solution. To improve stability with respect to noisy data and initialization, we augment the formulation with inequality constraints through prior admissible bounds on the state, leading to a constrained shape optimization problem. The shape derivative of the complex state and the corresponding shape gradient of the cost functional are derived, and the resulting problem is solved using an alternating direction method of multipliers (ADMM) framework. The proposed approach is implemented using the finite element method and validated through various numerical experiments.