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Factorization method for a simply supported obstacle from point source measurements via far--field transformation

Isaac Harris, Andreas Kleefeld

Abstract

We consider an inverse shape problem for recovering an unknown simply supported obstacle in two dimensions from near--field point--source measurements for the biharmonic Helmholtz equation. The measured data consist of the scattered field and its Laplacian on a closed measurement curve surrounding the obstacle. By exploiting an operator splitting of the biharmonic operator, we decouple the scattered field into propagating and evanescent components. This decoupling allows us to reformulate the measured data in terms of an acoustic near--field operator for a sound--soft scatterer. Since the acoustic near--field operator does not directly admit the symmetric factorization required by the factorization method, we introduce a far--field transformation (defined independently of the obstacle) that augments the near--field operator into a far--field operator with a symmetric factorization. This yields a rigorous factorization method characterization of the obstacle and leads to a practical reconstruction algorithm based on spectral data of the transformed operator. Finally, we present numerical experiments with synthetic data that demonstrate stable reconstructions under noise and illustrate the role of regularization, including a variant that uses only the scattered field data.

Factorization method for a simply supported obstacle from point source measurements via far--field transformation

Abstract

We consider an inverse shape problem for recovering an unknown simply supported obstacle in two dimensions from near--field point--source measurements for the biharmonic Helmholtz equation. The measured data consist of the scattered field and its Laplacian on a closed measurement curve surrounding the obstacle. By exploiting an operator splitting of the biharmonic operator, we decouple the scattered field into propagating and evanescent components. This decoupling allows us to reformulate the measured data in terms of an acoustic near--field operator for a sound--soft scatterer. Since the acoustic near--field operator does not directly admit the symmetric factorization required by the factorization method, we introduce a far--field transformation (defined independently of the obstacle) that augments the near--field operator into a far--field operator with a symmetric factorization. This yields a rigorous factorization method characterization of the obstacle and leads to a practical reconstruction algorithm based on spectral data of the transformed operator. Finally, we present numerical experiments with synthetic data that demonstrate stable reconstructions under noise and illustrate the role of regularization, including a variant that uses only the scattered field data.

Paper Structure

This paper contains 12 sections, 4 theorems, 112 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Direct Scattering Problem
  3. Solution to the Inverse Shape Problem
  4. Uniqueness of the Inverse Problem
  5. Factorization Method via Far--Field Transform
  6. Numerical Experiments
  7. Boundary Integral Equations for the Scattering Problem
  8. Numerical Reconstructions
  9. Reconstructions with and without regularization
  10. Using different regularization parameters
  11. Reconstructions without Laplacian data
  12. Conclusion

Key Result

Theorem 2.1

The biharmonic scattering problem for a simply supported obstacle by a point source incident wave biharmonic--SRC has a unique solution $u^{\mathrm{scat}}( \cdot \, ,y) \in H^2_{loc}(\mathbb{R}^2 \setminus \overline{D})$ for all $y \in \Gamma$ that satisfies the estimate for some constant $C>0$.

Figures (11)

  • Figure 1: Visual representations of the three scatterers defined in Table \ref{['scatterers']}.
  • Figure 2: Reconstruction of the kite--shaped scatterer with and without regularization where we add $5\%$ random noise to the data. Here we use the Tikhonov filter function given in \ref{['filters']} with $\alpha = 0.0001$. Left: reconstruction without regularization and Right: reconstruction with regularization.
  • Figure 3: Reconstruction of the star--shaped scatterer with and without regularization where we add $2\%$ random noise to the data. Here we use the Tikhonov filter function given in \ref{['filters']} with $\alpha = 0.0001$. Left: reconstruction without regularization and Right: reconstruction with regularization.
  • Figure 4: Reconstruction of the star--shaped scatterer with and without regularization where we add $0\%$ random noise to the data. Here we use the Tikhonov filter function given in \ref{['filters']} with $\alpha = 0.0001$. Left: reconstruction without regularization and Right: reconstruction with regularization.
  • Figure 5: Reconstruction of the kite--shaped scatterer with $k= 2.209855$ i.e. the first Dirichlet eigenvalue of the region. Here we use the GLSM filter function given in \ref{['filters']} with $\alpha = 0.0001$. Left: reconstruction without added noise and Right: reconstruction with 5$\%$ added noise.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3