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Inverse wave-number-dependent source problems for the Helmholtz equation with partial information on radiating period

Hongxia Guo, Guanghui Hu, Guanqiu Ma

TL;DR

This work develops a frequency-domain factorization method to image the support of a wave-number-dependent source in the Helmholtz equation when the radiating period is only partially known. By introducing new test functions and leveraging data from two opposite observation directions, the authors characterize a $Θ$-convex hull of the source and, with full angular data, recover the convex hull of the support; they also extend the framework to near-field data and to sources with two disconnected components. Key contributions include a novel parameter in the test function design, a range-identity-based imaging criterion, and rigorous uniqueness results for strips and convex-hull reconstruction, validated by extensive 2D and 3D numerical experiments. The results provide a frequency-domain perspective on inverse time-domain source problems and demonstrate practical feasibility with limited directional data. The methodology has potential impact for imaging transient sources in acoustics and related wave problems using sparse measurement configurations.

Abstract

This paper addresses a factorization method for imaging the support of a wave-number-dependent source function from multi-frequency data measured at a finite pair of symmetric receivers in opposite directions. The source function is given by the inverse Fourier transform of a compactly supported time-dependent source whose initial moment or terminal moment for radiating is unknown. Using the multi-frequency far-field data at two opposite observation directions, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the directions. A new parameter is incorporated into the design of test functions for indicating the unknown moment. The data from a finite pair of opposite directions can be used to recover the $Θ$-convex polygon of the support. Uniqueness in recovering the convex hull of the support is obtained as a by-product of our analysis using all observation directions. Similar results are also discussed with the multi-frequency near-field data from a finite pair of observation positions in three dimensions. We further comment on possible extensions to source functions with two disconnected supports. Extensive numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. The theoretical framework explored here should be seen as the frequency-domain analysis for inverse source problems in the time domain.

Inverse wave-number-dependent source problems for the Helmholtz equation with partial information on radiating period

TL;DR

This work develops a frequency-domain factorization method to image the support of a wave-number-dependent source in the Helmholtz equation when the radiating period is only partially known. By introducing new test functions and leveraging data from two opposite observation directions, the authors characterize a -convex hull of the source and, with full angular data, recover the convex hull of the support; they also extend the framework to near-field data and to sources with two disconnected components. Key contributions include a novel parameter in the test function design, a range-identity-based imaging criterion, and rigorous uniqueness results for strips and convex-hull reconstruction, validated by extensive 2D and 3D numerical experiments. The results provide a frequency-domain perspective on inverse time-domain source problems and demonstrate practical feasibility with limited directional data. The methodology has potential impact for imaging transient sources in acoustics and related wave problems using sparse measurement configurations.

Abstract

This paper addresses a factorization method for imaging the support of a wave-number-dependent source function from multi-frequency data measured at a finite pair of symmetric receivers in opposite directions. The source function is given by the inverse Fourier transform of a compactly supported time-dependent source whose initial moment or terminal moment for radiating is unknown. Using the multi-frequency far-field data at two opposite observation directions, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the directions. A new parameter is incorporated into the design of test functions for indicating the unknown moment. The data from a finite pair of opposite directions can be used to recover the -convex polygon of the support. Uniqueness in recovering the convex hull of the support is obtained as a by-product of our analysis using all observation directions. Similar results are also discussed with the multi-frequency near-field data from a finite pair of observation positions in three dimensions. We further comment on possible extensions to source functions with two disconnected supports. Extensive numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. The theoretical framework explored here should be seen as the frequency-domain analysis for inverse source problems in the time domain.
Paper Structure (19 sections, 15 theorems, 103 equations, 24 figures)

This paper contains 19 sections, 15 theorems, 103 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Literature review and comments
  4. Review of the factorization of far-field and near-field operators
  5. Test and indicator functions with multi-frequency far-field data
  6. The terminal moment $t_{\max}$ is unknown
  7. The initial moment $t_{\min}$ is unknown
  8. Indicator functions and inversion algorithms
  9. Uniqueness and algorithm at two opposite observation directions
  10. Algorithm and uniqueness at a finite pair of opposite directions
  11. Inversion algorithm with multi-frequency near-field data
  12. Discussions on source functions with two disconnected supports
  13. Numerical examples
  14. Numerical examples for the far-field case in ${\mathbb R}^2$ and ${\mathbb R}^3$
  15. Recovery of $K_D^{(\hat{x})}$ and $\Theta_D$ with far-field measurements in ${\mathbb R}^2$.
  16. ...and 4 more sections

Key Result

Lemma 2.1

(GGH2022) We have $F=L\mathcal{T}L^*$, where $L=L_D^{(\hat{x})}: X_D\rightarrow L^2(0, K )$ is defined by for all $u\in X_D$, and $\mathcal{T}: X_D\rightarrow X_D$ is a multiplication operator defined by

Figures (24)

  • Figure 1: Illustration of the strips $K_D^{(\hat{x})}$ (green area) and $K _{D,\eta}^{(\hat{x})}$ (union of green and blue area) with $\hat{x} = (1, 0)$ when $\eta<t_{\max}$.
  • Figure 2: Illustration of the annuluses $A_D^{({x})}$ (green area) and $A_{D,\eta}^{({x})}$ (union of green and blue area) defined in \ref{['annulus-a']} and \ref{['annulus-b']}.
  • Figure 3: Reconstructions using multi-frequency far-field data from a single observation direction for a peanut-shaped source support with the auxiliary indicator function $1/I^{(\hat{x})}_{\eta}$ defined in (\ref{['indicator5']}).
  • Figure 4: Reconstructions using multi-frequency far-field data from a single observation direction for a peanut-shaped support. We employ an auxiliary indicator function as defined in (\ref{['indicator5']}) and indicator function as defined in (\ref{['indicator7']}) with $M=1$.
  • Figure 5: Reconstructions using multi-frequency far-field data from $M$ pairs of opposite observation directions for a peanut-shaped source support.
  • ...and 19 more figures

Theorems & Definitions (28)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Theorem 3.5: Determination of the strip $K_D^{(\hat{x})}$
  • proof
  • ...and 18 more