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Linearized Boundary Control Method for Damping Reconstruction in an Acoustic Inverse Boundary Value Problem

Tianyu Yang, Yang Yang

Abstract

We develop a linearized boundary control method for the inverse boundary value problem of determining the damping coefficient in the damped wave equation. The objective is to reconstruct an unknown perturbation in a known background damping from the linearized Neumann-to-Dirichlet map. When the linearization is at a constant background damping, we derive a reconstructive algorithm with stability estimates based on the boundary control method in dimension $n\geq 1$. The reconstruction algorithm is implemented in one dimension to validate its numerical feasibility. When the linearization is at a non-constant background damping, we establish an increasing stability estimate in the time domain in dimension $n\geq 3$.

Linearized Boundary Control Method for Damping Reconstruction in an Acoustic Inverse Boundary Value Problem

Abstract

We develop a linearized boundary control method for the inverse boundary value problem of determining the damping coefficient in the damped wave equation. The objective is to reconstruct an unknown perturbation in a known background damping from the linearized Neumann-to-Dirichlet map. When the linearization is at a constant background damping, we derive a reconstructive algorithm with stability estimates based on the boundary control method in dimension . The reconstruction algorithm is implemented in one dimension to validate its numerical feasibility. When the linearization is at a non-constant background damping, we establish an increasing stability estimate in the time domain in dimension .
Paper Structure (11 sections, 11 theorems, 122 equations, 6 figures, 1 algorithm)

This paper contains 11 sections, 11 theorems, 122 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Linearized Blagoves̆c̆enskiı̆ Identity with a Free Complex-Valued Parameter
  4. A Boundary Control Result
  5. Reconstruction and Stability
  6. Constant $\sigma_0$
  7. Non-constant $\sigma_0$
  8. Numerical Implementation
  9. Computing Boundary Controls using Time Reversal
  10. Numerical experiments
  11. Appendix

Key Result

Proposition 1

For any real number $s\geq \frac{1}{2}$, $\dot\Lambda:H_{00}^s((0,2T)\times\partial\Omega) \rightarrow H^s((0,2T)\times\partial\Omega)$ is a bounded linear operator.

Figures (6)

  • Figure 1: Continuous ground truth $\dot{\sigma} = \cos(\pi x) + \cos(2\pi x) + \cos(3\pi x) + \sin(4\pi x)+4$.
  • Figure 2: Left: Reconstructed $\dot\sigma$ with $0\%,1\%,5\%$ Gaussian noise and the ground truth. Right: The corresponding error between the reconstruction result and the ground truth. The relative $L^2$-errors are $0.2\%,3.5\%$ and $16.2\%$, respectively.
  • Figure 3: Piecewise-constant ground truth $\dot\sigma$ and its Fourier truncation $\dot\sigma_N$ with $N=10$.
  • Figure 4: Left: The reconstructions with $0\%,1\%,5\%$ Gaussian noise and the orthogonal projection $\dot\sigma_N$. Right: Errors between the reconstructions and the orthogonal projection $\dot\sigma_N$. The relative $L^2$-errors are $0.2\%,3.0\%$ and $22.5\%$, respectively.
  • Figure 5: Ground truth $\dot\sigma$
  • ...and 1 more figures

Theorems & Definitions (26)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 4
  • Proposition 5
  • proof
  • Remark 6
  • ...and 16 more