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Shape reconstruction of inclusions based on noisy data via monotonicity methods for the time harmonic elastic wave equation

Sarah Eberle-Blick

TL;DR

This work addresses shape reconstruction of inclusions in an elastic medium from noisy measurements for the time-harmonic Navier equation. It adapts the standard and linearized monotonicity tests by perturbing the Neumann-to-Dirichlet map to a noisy operator $\Lambda^\delta$ and proves the existence of a maximal noise level $\delta_0$ ensuring consistent reconstruction. The paper provides rigorous eigenvalue-based criteria and demonstrates through numerical simulations that inclusions can be detected and their shapes recovered under realistic noise, with standard monotonicity offering greater noise robustness at the cost of computation, and linearized monotonicity offering faster results albeit with reduced resilience to noise. The results advance practical applicability to laboratory and real-world data and point toward monotonicity-based regularization and multi-frequency extensions as future directions.

Abstract

In this paper, we extend our research concerning the standard and linearized monotonicity methods for the inverse problem of the time harmonic elastic wave equation and introduce the modification of these methods for noisy data. In more detail, the methods must provide consistent results when using noisy data in order to be able to perform simulations with real world data, e.g., laboratory data. We therefore consider the disturbed Neumann-to-Dirichlet operator and modify the bound of the eigenvalues in the monotonicity tests for reconstructing unknown inclusions with noisy data. In doing so, we show that there exists a noise level $δ_0$ so that the inclusions are detected and their shape is reconstructed for all noise levels $δ< δ_0$. Finally, we present some numerical simulations based on noisy data.

Shape reconstruction of inclusions based on noisy data via monotonicity methods for the time harmonic elastic wave equation

TL;DR

This work addresses shape reconstruction of inclusions in an elastic medium from noisy measurements for the time-harmonic Navier equation. It adapts the standard and linearized monotonicity tests by perturbing the Neumann-to-Dirichlet map to a noisy operator and proves the existence of a maximal noise level ensuring consistent reconstruction. The paper provides rigorous eigenvalue-based criteria and demonstrates through numerical simulations that inclusions can be detected and their shapes recovered under realistic noise, with standard monotonicity offering greater noise robustness at the cost of computation, and linearized monotonicity offering faster results albeit with reduced resilience to noise. The results advance practical applicability to laboratory and real-world data and point toward monotonicity-based regularization and multi-frequency extensions as future directions.

Abstract

In this paper, we extend our research concerning the standard and linearized monotonicity methods for the inverse problem of the time harmonic elastic wave equation and introduce the modification of these methods for noisy data. In more detail, the methods must provide consistent results when using noisy data in order to be able to perform simulations with real world data, e.g., laboratory data. We therefore consider the disturbed Neumann-to-Dirichlet operator and modify the bound of the eigenvalues in the monotonicity tests for reconstructing unknown inclusions with noisy data. In doing so, we show that there exists a noise level so that the inclusions are detected and their shape is reconstructed for all noise levels . Finally, we present some numerical simulations based on noisy data.

Paper Structure

This paper contains 12 sections, 5 theorems, 44 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and other preliminaries
  3. Summary of the monotonicty methods
  4. Standard monotonicty test
  5. Linearized monotonicity test
  6. Monotonicity tests for noisy data
  7. Standard monotonicity test
  8. Linearized monotonicity test
  9. Numerical simulations
  10. Standard monotonicity test
  11. Linearized monotonicity test
  12. Conclusion and Outlook

Key Result

Theorem 3.1

Let $D := D_1 \cup D_2 \cup D_3$, where the sets are as in eq_lambdaMuRho and $B \subset \Omega$ and $\alpha_j > 0$ be as in eq_testCoeff, and set $\alpha:=(\alpha_1,\alpha_2,\alpha_3)$. Let $M_s\in\mathbb{R}$ be defined as where $d(\lambda_0,\mu_0,\rho_0)$ is the number of positive eigenvalues of $L_{\lambda_0,\mu_0,\rho_0}$ as defined in (eq_def_d). The following holds: where $\Lambda$ is the

Figures (9)

  • Figure 1: Cube with two inclusions (blue) (similar as the model in EH21).
  • Figure 2: Plot of the number of negative eigenvalues (left) and the reconstruction (right) for $\omega=50\,rad/s$ ($l_p=1.79\,m$, $l_s=0.18\,m$ for the homogenous background material) for noise level $\eta=0$ and $\tilde{M}_s=107$.
  • Figure 3: Plot of the number of negative eigenvalues (left) and the reconstruction (right) for $\omega=50\,rad/s$ ($l_p=1.79\,m$, $l_s=0.18\,m$ for the homogenous background material) for noise level $\eta=0.005$, $\delta=1\cdot 10^{-6}$ and $\tilde{M}_s=107$.
  • Figure 4: Plot of the number of negative eigenvalues (left) and the reconstruction (right) for $\omega=50\,rad/s$ ($l_p=1.79\,m$, $l_s=0.18\,m$ for the homogenous background material) for noise level $\eta=0.025$, $\delta=4.54\cdot 10^{-6}$ and $\tilde{M}_s=107$.
  • Figure 5: Plot of the relation of the bounds $M_{\min}$ and $M_{\max}$ and $\delta$ for different noise levels $\eta$.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Theorem 3.1: see EP24
  • Theorem 3.2: see EP24a
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof