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Simultaneously recover two constant coefficients and a polygon with a single pair of Cauchy data for the Helmholtz equation

Xiaoxu Xu, Guanghui Hu

TL;DR

This work addresses the inverse boundary value problem for the Helmholtz equation with constant coefficients by seeking simultaneous recovery of $\sigma$, $q$ and a Dirichlet polygonal obstacle $D$ from a single Cauchy data pair on $\partial B$. It advances a one-wave factorization approach built on the Dirichlet-to-Neumann map, augmented with modified factorizations and auxiliary Green functions to handle spectral issues. The key contributions include (i) uniqueness results for the coefficient pair and the polygon under a priori assumptions, (ii) explicit sampling indicators that link data to the unknowns, and (iii) numerical demonstrations confirming accurate, non-iterative reconstruction of coefficients and geometry from one data set. The method is robust to eigenvalue obstructions through artificial impedance or refractive-index constructs and remains applicable to near-field and limited-aperture data, offering a practically impactful tool for noninvasive imaging and nondestructive testing.

Abstract

This paper is concerned with an inverse boundary value problem for the Helmholtz equation over a bounded domain. The aim is to reconstruct two constant coefficients together with the location and shape of a Dirichlet polygonal obstacle from a single pair of Cauchy data. Uniqueness results are verified under some a priori assumptions and the one-wave factorization method has been adapted to recover the polygonal obstacle as well as the two coefficients. A modified factorization using the Dirichlet-to-Neumann operator is employed to overcome difficulties arising from possible eigenvalues. Intensive numerical examples indicate that our method is efficient.

Simultaneously recover two constant coefficients and a polygon with a single pair of Cauchy data for the Helmholtz equation

TL;DR

This work addresses the inverse boundary value problem for the Helmholtz equation with constant coefficients by seeking simultaneous recovery of , and a Dirichlet polygonal obstacle from a single Cauchy data pair on . It advances a one-wave factorization approach built on the Dirichlet-to-Neumann map, augmented with modified factorizations and auxiliary Green functions to handle spectral issues. The key contributions include (i) uniqueness results for the coefficient pair and the polygon under a priori assumptions, (ii) explicit sampling indicators that link data to the unknowns, and (iii) numerical demonstrations confirming accurate, non-iterative reconstruction of coefficients and geometry from one data set. The method is robust to eigenvalue obstructions through artificial impedance or refractive-index constructs and remains applicable to near-field and limited-aperture data, offering a practically impactful tool for noninvasive imaging and nondestructive testing.

Abstract

This paper is concerned with an inverse boundary value problem for the Helmholtz equation over a bounded domain. The aim is to reconstruct two constant coefficients together with the location and shape of a Dirichlet polygonal obstacle from a single pair of Cauchy data. Uniqueness results are verified under some a priori assumptions and the one-wave factorization method has been adapted to recover the polygonal obstacle as well as the two coefficients. A modified factorization using the Dirichlet-to-Neumann operator is employed to overcome difficulties arising from possible eigenvalues. Intensive numerical examples indicate that our method is efficient.

Paper Structure

This paper contains 11 sections, 21 theorems, 82 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Factorization of Dirichlet-to-Neumann operators
  3. Dirichlet-to-Neumann operators and the range identity
  4. Revisited factorizations
  5. Determination of the two coefficients
  6. Reconstruction of the polygon
  7. Numerical examples
  8. Numerical examples for the factorization method
  9. Numerical examples for recovering the coefficients
  10. Numerical examples for recovering the polygonal obstacle
  11. Conclusion

Key Result

Theorem 2.2

For any integer $m\geq1$ the following hold: (a) If $k^2$ is not an eigenvalue of 1'--3', then $G(k^2,\Omega):H^{-1/2}(\partial\Omega)\rightarrow H^{m-3/2}(\partial B)$ is bounded and $G(k^2,\Omega):H^{-1/2}(\partial\Omega)\rightarrow L^2(\partial B)$ is compact. (b) If $k^2$ is not an eigenvalue of

Figures (11)

  • Figure 1: Numerical examples for Example \ref{['Ex0819']}, where the red solid line, black solid line and black dashed line representing the disk $B$, the true shape of the unknown object $\Omega$ (or $\mho$) and $\widetilde{\Omega}$ inside obstacle $\Omega$ (or $\widetilde{\mho}$ inside the medium with support $\overline{\mho}$), respectively.
  • Figure 2: The geometry of Example \ref{['ex1complex']}. The black line represents $\partial D$, the red line represents $\partial\Omega$ (or $\partial\mho$), the circle with knots represents $\partial B$, and the blue line represents $\partial\widetilde{\Omega}$ (or $\partial\widetilde{\mho}$). The boundary value $f$ takes value $1$ at black knots and takes value $0$ at gray knots on $\partial B$.
  • Figure 3: Numerical results for Example \ref{['ex1complex']} to simultaneously reconstruct two coefficients $\sigma=1$ and $k=2$ ($q=4$). $(\lambda_n,f_n)$ in \ref{['ind+']} is the eigensystem of $[A(\kappa^2,\Omega)-A_0(\kappa^2)]_\#$ in (a), (b) and of $[A(\kappa^2,\mho)-A_0(\kappa^2)]_\#$ in (c), (d). $(\lambda_n,f_n)$ in \ref{['ind+tilde']} is the eigensystem of $[A(\kappa^2,\Omega)-A(\kappa^2,\widetilde{\Omega})]_\#$ in (e), (f) and of $[A(\kappa^2,\mho)-A(\kappa^2,\widetilde{\mho})]_\#$ in (g), (h). $\tau_j=j/20+0.5$ and $\kappa_\ell=\ell/20+1.5$, $j,\ell=0,1\cdots,20$. The black dot represents the true value of $(\tau,k)$.
  • Figure 4: Numerical results for Example \ref{['Ex2']}. The black line represents $\partial D$, the circle with knots represents $\partial B$. The boundary value $f$ takes value $0$, $1$, and $2$ at gray, black, and blue knots on $\partial B$, respectively. The colored circles represent $\partial\Omega_P^{(\ell)}$ for different $P$ and $\ell$ with its color indicating the value of $I_2(\Omega_P^{(\ell)})$ in the sense of (\ref{['1029-3']}).
  • Figure 5: Numerical results for Example \ref{['Ex2']}. The black line represents $\partial D$, the circle with knots represents $\partial B$. The boundary value $f$ takes value $0$, $1$, and $2$ at gray, black, and blue knots on $\partial B$, respectively. The colored circles represent $\partial\Omega_P^{(\ell)}$ for different $P$ and $\ell$ with its color indicating the value of $\widetilde{I}_2(\Omega_P^{(\ell)})$ in the sense of (\ref{['1029-3']}).
  • ...and 6 more figures

Theorems & Definitions (42)

  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 32 more