Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A direct imaging method for inverse scattering problem of biharmonic wave with phased and phaseless data

Tielei Zhu, Zhihao Ge

TL;DR

The paper addresses inverse biharmonic obstacle scattering in two dimensions with phased and phaseless data. It proposes a direct imaging method based on reverse-time migration (RTM) that builds imaging functions from a single measurement type and leverages the Helmholtz fundamental solution to reduce computational cost, enabling efficient obstacle localization without solving forward problems iteratively. A rigorous resolution analysis shows that the imaging functions exhibit contrast when sampling points are near the obstacle boundary and decay away from it, and numerical experiments confirm accurate reconstructions of complex geometries under moderate noise. By extending RTM to fourth-order biharmonic problems and accommodating multiple data modalities and boundary conditions, the work provides a flexible, robust framework for fast inverse scattering with phaseless and phased measurements.

Abstract

This paper investigates the inverse biharmonic scattering problems of identifying the shape and location of the obstacle with phased and phaseless measurement data. A direct imaging method based on reverse time migration is proposed for reconstructing the extended obstacle with one of four types of boundary conditions on the obstacle. The newly developed imaging functions are constructed by utilizing merely one of various measurement data, including the scattered field, its normal derivative, the bending moment, the transverse force, its far-field and the phaseless total field. Our resolution analysis demonstrates that these imaging functions have a contrast when sampling points are near or far from the boundary of the obstacle. Numerical experiments are further presented to show the algorithm's efficiency to accurately reconstruct complex scatter geometries and its robustness to noise.

A direct imaging method for inverse scattering problem of biharmonic wave with phased and phaseless data

TL;DR

The paper addresses inverse biharmonic obstacle scattering in two dimensions with phased and phaseless data. It proposes a direct imaging method based on reverse-time migration (RTM) that builds imaging functions from a single measurement type and leverages the Helmholtz fundamental solution to reduce computational cost, enabling efficient obstacle localization without solving forward problems iteratively. A rigorous resolution analysis shows that the imaging functions exhibit contrast when sampling points are near the obstacle boundary and decay away from it, and numerical experiments confirm accurate reconstructions of complex geometries under moderate noise. By extending RTM to fourth-order biharmonic problems and accommodating multiple data modalities and boundary conditions, the work provides a flexible, robust framework for fast inverse scattering with phaseless and phased measurements.

Abstract

This paper investigates the inverse biharmonic scattering problems of identifying the shape and location of the obstacle with phased and phaseless measurement data. A direct imaging method based on reverse time migration is proposed for reconstructing the extended obstacle with one of four types of boundary conditions on the obstacle. The newly developed imaging functions are constructed by utilizing merely one of various measurement data, including the scattered field, its normal derivative, the bending moment, the transverse force, its far-field and the phaseless total field. Our resolution analysis demonstrates that these imaging functions have a contrast when sampling points are near or far from the boundary of the obstacle. Numerical experiments are further presented to show the algorithm's efficiency to accurately reconstruct complex scatter geometries and its robustness to noise.
Paper Structure (7 sections, 11 theorems, 83 equations, 7 figures)

This paper contains 7 sections, 11 theorems, 83 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Formulation of the biharmonic wave scattering problem
  3. The direct scattering problem
  4. The inverse scattering problems for extended obstacles
  5. The direct imaging method with phased data
  6. The direct imaging method with phaseless data
  7. Numerical experiments

Key Result

Lemma 2.1

The boundary value problem eq:bvp admits a unique solution $w$ in $H^{2}_{\mathrm{ loc } }(D^c)$

Figures (7)

  • Figure 1: Example 1: The reconstruction results with no noisy data
  • Figure 2: Example 2: The $j-$th row presents the reconstruction results of the indicator function $I_j$$(j=1,\cdots,11)$.
  • Figure 3: Example 3: The $j-$th row presents the reconstruction results of the indicator function $I_j$$(j=1,\cdots,11)$.
  • Figure :
  • Figure :
  • ...and 2 more figures

Theorems & Definitions (18)

  • Lemma 2.1: see bourgeois_well_2020
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1: see chen_reverse_2013
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Theorem 3.4
  • ...and 8 more