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The Radon Transform-Based Sampling Methods for Biharmonic Sources from the Scattered Fields

Xiaodong Liu, Qingxiang Shi, Jing Wang

TL;DR

The paper tackles inverse biharmonic source problems using multi-frequency scattered field data, introducing three quantitative sampling methods that leverage a Radon-transform relationship between the source and the measured fields. It provides constructive uniqueness results: an explicit S reconstruction from full-aperture data, and a singularity-based indicator approach for recovering the source support under sparse sensing, including annular, polygonal, and mixed geometries. Two practical indicators for reconstructing the source function are proposed: I_S^(1) based on Im(u^s) and a more efficient I_S^(2) that uses both Im(u^s) and ∆u^s. Numerical experiments demonstrate high-resolution boundary and source imaging with sparse, noisy data, highlighting the near-field advantages for resolving polygonal boundaries and providing explicit sensor-count criteria for unique recovery.

Abstract

This paper presents three quantitative sampling methods for reconstructing extended sources of the biharmonic wave equation using scattered field data. The first method employs an indicator function that solely relies on scattered fields $ u^s$ measured on a single circle, eliminating the need for Laplacian or derivative data. Its theoretical foundation lies in an explicit formula for the source function, which also serves as a constructive proof of uniqueness. To improve computational efficiency, we introduce a simplified double integral formula for the source function, at the cost of requiring additional measurements $Δu^s$. This advancement motivates the second indicator function, which outperforms the first method in both computational speed and reconstruction accuracy. The third indicator function is proposed to reconstruct the support boundary of extended sources from the scattered fields $ u^s$ at a finite number of sensors. By analyzing singularities induced by the source boundary, we establish the uniqueness of annulus and polygon-shaped sources. A key characteristic of the first and third indicator functions is their link between scattered fields and the Radon transform of the source function. Numerical experiments demonstrate that the proposed sampling methods achieve high-resolution imaging of the source support or the source function itself.

The Radon Transform-Based Sampling Methods for Biharmonic Sources from the Scattered Fields

TL;DR

The paper tackles inverse biharmonic source problems using multi-frequency scattered field data, introducing three quantitative sampling methods that leverage a Radon-transform relationship between the source and the measured fields. It provides constructive uniqueness results: an explicit S reconstruction from full-aperture data, and a singularity-based indicator approach for recovering the source support under sparse sensing, including annular, polygonal, and mixed geometries. Two practical indicators for reconstructing the source function are proposed: I_S^(1) based on Im(u^s) and a more efficient I_S^(2) that uses both Im(u^s) and ∆u^s. Numerical experiments demonstrate high-resolution boundary and source imaging with sparse, noisy data, highlighting the near-field advantages for resolving polygonal boundaries and providing explicit sensor-count criteria for unique recovery.

Abstract

This paper presents three quantitative sampling methods for reconstructing extended sources of the biharmonic wave equation using scattered field data. The first method employs an indicator function that solely relies on scattered fields measured on a single circle, eliminating the need for Laplacian or derivative data. Its theoretical foundation lies in an explicit formula for the source function, which also serves as a constructive proof of uniqueness. To improve computational efficiency, we introduce a simplified double integral formula for the source function, at the cost of requiring additional measurements . This advancement motivates the second indicator function, which outperforms the first method in both computational speed and reconstruction accuracy. The third indicator function is proposed to reconstruct the support boundary of extended sources from the scattered fields at a finite number of sensors. By analyzing singularities induced by the source boundary, we establish the uniqueness of annulus and polygon-shaped sources. A key characteristic of the first and third indicator functions is their link between scattered fields and the Radon transform of the source function. Numerical experiments demonstrate that the proposed sampling methods achieve high-resolution imaging of the source support or the source function itself.

Paper Structure

This paper contains 13 sections, 12 theorems, 100 equations, 14 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Indicators and their applications for uniqueness
  3. Identification of the source function with multi-frequency scattered fields
  4. Reconstruction of the support Ω using the multi-frequency sparse scattered fields
  5. Annular sources
  6. Polygonal sources
  7. A combination of annular and polygonal sources
  8. Numerical algorithms
  9. The indicator function I_ Ω
  10. The indicator function
  11. The indicator function
  12. Numerical examples and discussions
  13. Concluding remarks

Key Result

Lemma 2.1

For a function $S(z)\in L^2_{comp}({\mathbb R}^2)$ with compact support in $B_R(0)$. We have the following formula about $S(z)$: where $\nu(x)$ is the unit outward normal vector at $x\in\partial B_R(0)$ and

Figures (14)

  • Figure 1: Illustrations for the Case 1.
  • Figure 2: Illustrations for the Case 2.
  • Figure 3: Illustrations for the Case 3.
  • Figure 4: Illustrations for the Case 4.
  • Figure 5: The true sources.
  • ...and 9 more figures

Theorems & Definitions (23)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Lemma 2.7
  • ...and 13 more