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Direct sampling for recovering a clamped cavity from biharmonic far field data

Isaac Harris, Heejin Lee, Peijun Li

TL;DR

This work addresses the inverse-shape problem for a clamped cavity embedded in a thin infinite plate governed by the 2D biharmonic equation $\Delta^2 u-k^4u=0$. It extends the direct sampling method to biharmonic waves by deriving a factorization of the far-field operator $F$ using boundary-integral representations and Herglotz wave functions, and by exploiting the Funk–Hecke identity to define two robust imaging functionals based on the far-field data. The authors prove quantitative decay rates for the imaging functionals, show their equivalence, and validate the approach numerically with synthetic data for star- and peanut-shaped cavities, including partial-aperture cases and moderate noise levels. The results establish a simple, non-iterative framework for reconstructing biharmonic clamped cavities and suggest avenues for extending the method to penetrable cavities and to transmission eigenvalue analyses.

Abstract

This paper concerns the inverse shape problem of recovering an unknown clamped cavity embedded in a thin infinite plate. The model problem is assumed to be governed by the two-dimensional biharmonic wave equation in the frequency domain. Based on the far-field data, a resolution analysis is conducted for cavity recovery via the direct sampling method. The Funk--Hecke integral identity is employed to analyze the performance of two imaging functions. Our analysis demonstrates that the same imaging functions commonly used for acoustic inverse shape problems are applicable to the biharmonic wave context. This work presents the first extension of direct sampling methods to biharmonic waves using far-field data. Numerical examples are provided to illustrate the effectiveness of these imaging functions in recovering a clamped cavity.

Direct sampling for recovering a clamped cavity from biharmonic far field data

TL;DR

This work addresses the inverse-shape problem for a clamped cavity embedded in a thin infinite plate governed by the 2D biharmonic equation . It extends the direct sampling method to biharmonic waves by deriving a factorization of the far-field operator using boundary-integral representations and Herglotz wave functions, and by exploiting the Funk–Hecke identity to define two robust imaging functionals based on the far-field data. The authors prove quantitative decay rates for the imaging functionals, show their equivalence, and validate the approach numerically with synthetic data for star- and peanut-shaped cavities, including partial-aperture cases and moderate noise levels. The results establish a simple, non-iterative framework for reconstructing biharmonic clamped cavities and suggest avenues for extending the method to penetrable cavities and to transmission eigenvalue analyses.

Abstract

This paper concerns the inverse shape problem of recovering an unknown clamped cavity embedded in a thin infinite plate. The model problem is assumed to be governed by the two-dimensional biharmonic wave equation in the frequency domain. Based on the far-field data, a resolution analysis is conducted for cavity recovery via the direct sampling method. The Funk--Hecke integral identity is employed to analyze the performance of two imaging functions. Our analysis demonstrates that the same imaging functions commonly used for acoustic inverse shape problems are applicable to the biharmonic wave context. This work presents the first extension of direct sampling methods to biharmonic waves using far-field data. Numerical examples are provided to illustrate the effectiveness of these imaging functions in recovering a clamped cavity.
Paper Structure (8 sections, 7 theorems, 88 equations, 8 figures)

This paper contains 8 sections, 7 theorems, 88 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Formulation of the Problem
  3. Direct Sampling Method
  4. Numerical Validation
  5. Example 1. A star-shaped cavity
  6. Example 2. A peanut-shaped cavity
  7. Example 3. Partial apertures
  8. Conclusion

Key Result

Theorem 3.1

The far-field operator $F: L^2(\mathbb{S}^1) \to L^2(\mathbb{S}^1)$, as defined by ffop, corresponding to the biharmonic clamped scattering problem biharmonic--SRC, is compact.

Figures (8)

  • Figure 1: The reconstruction of the star-shaped cavity by the imaging function $W_{\text{ip}}(z)$. (Left) the reconstruction with no error added to the far-field data; (Right) the reconstruction with $\delta=0.02$, which corresponds to a 2$\%$ noise level.
  • Figure 2: The reconstruction of the star-shaped cavity by the imaging function $W_{\text{norm}}(z)$. (Left) the reconstruction with no error added to the far-field data; (Right) the reconstruction with $\delta=0.02$, which corresponds to a 2$\%$ noise level.
  • Figure 3: The reconstruction of the star-shaped cavity. (Left) the reconstruction via $W_{\text{ip}}(z)$ with $\rho=4$; (Right) the reconstruction via $W_{\text{norm}}(z)$ with $\rho=8$.
  • Figure 4: The reconstruction of the peanut-shaped cavity with $\delta=0.3$, which corresponds to a 30$\%$ noise level. (Left) the reconstruction via $W_{\text{ip}}(z)$; (Right) the reconstruction via $W_{\text{norm}}(z)$.
  • Figure 5: The reconstruction of the peanut-shaped cavity with $\delta=0.1$, which corresponds to a 10$\%$ noise level via the imaging function $W_{\text{ip}}(z)$. (Left) the reconstruction with $\rho=1$; (Right) the reconstruction with $\rho=1/2$.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 1 more