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Boundary shape reconstruction with Robin condition: existence result, stability analysis, and inversion via multiple measurements

Lekbir Afraites, Julius Fergy Tiongson Rabago

TL;DR

This work addresses the inverse problem of reconstructing an unknown interior Robin boundary $\Gamma$ from exterior Cauchy data for a harmonic field, formulating two boundary-data-tracking shape optimizations. It proves existence of minimizers under suitable regularity and convergence notions, and shows ill-posedness via the compactness of the shape Hessian, indicating limited stability in the single-measurement setting. Numerically, the study adopts Sobolev-gradient-based descent and demonstrates that using multiple Cauchy data markedly improves reconstruction of concavities in 2D and, with more modest gains, in 3D. The results highlight the practical value of multi-measurement data for boundary identification in ill-posed shape inverse problems and set the stage for regularization and higher-order methods in future work.

Abstract

This study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals. Firstly, it rigorously addresses the existence of optimal shape solutions, thus filling a gap in the literature. The argumentation utilized in the proof strategy is contingent upon the specific formulation under consideration. Secondly, it demonstrates the ill-posed nature of the two shape optimization formulations by establishing the compactness of the Riesz operator associated with the quadratic shape Hessian corresponding to each cost functional. Lastly, the study employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary. Numerical experiments in two and three dimensions illustrate the numerical procedure relying on Sobolev gradients proposed herein.

Boundary shape reconstruction with Robin condition: existence result, stability analysis, and inversion via multiple measurements

TL;DR

This work addresses the inverse problem of reconstructing an unknown interior Robin boundary from exterior Cauchy data for a harmonic field, formulating two boundary-data-tracking shape optimizations. It proves existence of minimizers under suitable regularity and convergence notions, and shows ill-posedness via the compactness of the shape Hessian, indicating limited stability in the single-measurement setting. Numerically, the study adopts Sobolev-gradient-based descent and demonstrates that using multiple Cauchy data markedly improves reconstruction of concavities in 2D and, with more modest gains, in 3D. The results highlight the practical value of multi-measurement data for boundary identification in ill-posed shape inverse problems and set the stage for regularization and higher-order methods in future work.

Abstract

This study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals. Firstly, it rigorously addresses the existence of optimal shape solutions, thus filling a gap in the literature. The argumentation utilized in the proof strategy is contingent upon the specific formulation under consideration. Secondly, it demonstrates the ill-posed nature of the two shape optimization formulations by establishing the compactness of the Riesz operator associated with the quadratic shape Hessian corresponding to each cost functional. Lastly, the study employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary. Numerical experiments in two and three dimensions illustrate the numerical procedure relying on Sobolev gradients proposed herein.
Paper Structure (24 sections, 20 theorems, 84 equations, 12 figures, 1 table)

This paper contains 24 sections, 20 theorems, 84 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Shape optimization formulations
  3. Tracking the Neumann data in least-squares approach
  4. Existence of a shape solution
  5. Tracking the Dirichlet data in least-squares approach
  6. Shape optimization formulation
  7. Existence of a shape solution
  8. Second-order analyses and stability issue
  9. Some elements of shape calculus
  10. Shape derivative of $u_{{D}}$
  11. Shape gradient of $J_{N}$
  12. Shape Hessian of $J_{N}$ at a critical shape
  13. Instability analysis of the critical shape of $J_{N}$
  14. Shape Hessian of $J_{D}$ and stability issue concerning \ref{['equa:shop']}
  15. Numerical implementation and experiments
  16. ...and 9 more sections

Key Result

Lemma 2.1.1

Let $\{\boldsymbol{\mathbf{\phi}}^{(k)}\} \subset \mathcal{U}_{\text{ad}}$ and $\boldsymbol{\mathbf{\phi}} \in \mathcal{U}_{\text{ad}}$. For any sequence of domains $\{\Omega(\boldsymbol{\mathbf{\phi}}^{(n)})\}$, $\Omega(\boldsymbol{\mathbf{\phi}}^{(n)}) \in \mathcal{O}_\text{ad}$, there is a subseq

Figures (12)

  • Figure 1: Results of the detections with a single boundary measurement
  • Figure 2: Results of the detections with multiple boundary measurements
  • Figure 3: Histories of (normalized) cost values, Sobolev gradient norms, and Hausdorff distances $d_{H}(\Gamma^{(k)},\Gamma^{\ast})$ for Test($\Gamma_{\text{K}}$)
  • Figure 4: Results of the detections with a single boundary measurement
  • Figure 5: Results of the detections with multiple boundary measurements ($\Gamma^{(0)} = C(\boldsymbol{\mathbf{0}},0.3)$)
  • ...and 7 more figures

Theorems & Definitions (43)

  • Remark 1.1
  • Remark 2.1: Well-posedness of \ref{['eq:state_ud']}
  • Definition 2.1.1
  • Remark 2.2
  • Lemma 2.1.1
  • Remark 2.3
  • Theorem 2.1.1
  • Proposition 2.1.1
  • Lemma 2.1.2
  • proof : Proof of Lemma \ref{['lem:estimates']}
  • ...and 33 more