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Inverse parameter and shape problem for an isotropic scatterer with two conductivity coefficients

Rafael Ceja Ayala, Isaac Harris, Andreas Kleefeld

Abstract

In this paper, we consider the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. First, we show the uniqueness for recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we address the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. Furthermore, we examine the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable with respect to noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance by the direct sampling method.

Inverse parameter and shape problem for an isotropic scatterer with two conductivity coefficients

Abstract

In this paper, we consider the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. First, we show the uniqueness for recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we address the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. Furthermore, we examine the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable with respect to noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance by the direct sampling method.
Paper Structure (12 sections, 7 theorems, 81 equations, 9 figures, 1 table)

This paper contains 12 sections, 7 theorems, 81 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The direct scattering problem
  3. Uniqueness result for the coefficients
  4. Discreteness of Transmission Eigenvalue Problem
  5. Uniqueness for two of the material parameters
  6. Direct Sampling Method
  7. Factorization of the Scattered Field with Full Aperture
  8. Remarks on Limited Aperture Data in $\mathbb{R}^2$
  9. Numerical Validation
  10. Boundary Integral Equations
  11. Numerical Examples
  12. Conclusion

Key Result

Theorem 3.1

Assume that $|n_1-n_2|^{-1}\in L^{\infty}(D)$ and $|\eta_1-\eta_2|^{-1}\in L^{\infty}(\partial D)$, then the set of transmission eigenvalues in traceeigproblem1--traceeigproblem2 is at most discrete.

Figures (9)

  • Figure 1: Left to right: plot of \ref{['expansionofsum']} up to $s=15$ and plot of the partial derivative of \ref{['expansionofsum']} with respect to $r=|x|$ for $s=15$.
  • Figure 2: Reconstruction star region by the DSM without noise and with 10% noise.
  • Figure 3: Reconstruction of peanut region by the DSM without noise and with 5% noise.
  • Figure 4: Reconstruction of kite region by the DSM without noise and with 15% noise.
  • Figure 5: Reconstruction of kite region using limited receivers by the DSM without noise and with 15% noise.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 1 more