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A direct sampling method for inverse time-dependent electromagnetic source problems: reconstruction of the radiating time and spatial support

Fenglin Sun, Hongxia Guo

Abstract

This paper investigates inverse source problems for time-dependent electromagnetic waves governed by Maxwell's equations. After applying the Fourier transform with respect to time, the problem leads to a frequency-domain electromagnetic system with a frequency-dependent source term. We propose a novel direct sampling method for reconstructing such radiating time and spatial space of sources from multi-frequency far-field measurements. By using a pair of multi-frequency data from opposite observation directions, we can obtain the radiating time of the signal. Based on this, the smallest region between two hyperplanes containing the support of source can be reconstructed using multi frequency data from one observation direction. The Theta convex hull of the source support can be reconstructed from multi-frequency data from sparse observation directions. Compared with existing sampling methods that mainly focus on reconstructing the spatial support, the proposed approach allows for the simultaneous reconstruction of both spatial and temporal features of the source.Three-dimensional numerical examples are conducted to validate the effectiveness of the algorithm.

A direct sampling method for inverse time-dependent electromagnetic source problems: reconstruction of the radiating time and spatial support

Abstract

This paper investigates inverse source problems for time-dependent electromagnetic waves governed by Maxwell's equations. After applying the Fourier transform with respect to time, the problem leads to a frequency-domain electromagnetic system with a frequency-dependent source term. We propose a novel direct sampling method for reconstructing such radiating time and spatial space of sources from multi-frequency far-field measurements. By using a pair of multi-frequency data from opposite observation directions, we can obtain the radiating time of the signal. Based on this, the smallest region between two hyperplanes containing the support of source can be reconstructed using multi frequency data from one observation direction. The Theta convex hull of the source support can be reconstructed from multi-frequency data from sparse observation directions. Compared with existing sampling methods that mainly focus on reconstructing the spatial support, the proposed approach allows for the simultaneous reconstruction of both spatial and temporal features of the source.Three-dimensional numerical examples are conducted to validate the effectiveness of the algorithm.
Paper Structure (18 sections, 12 theorems, 90 equations, 8 figures)

This paper contains 18 sections, 12 theorems, 90 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulations
  3. Literature review
  4. Inverse source problem 1
  5. Inverse Fourier transform of far-field pattern
  6. Indicator and test functions
  7. Inverse source problem 2
  8. Inverse Fourier transform of far-field pattern
  9. Indicator and test functions
  10. Numerical examples
  11. Frequency discretization and implementation details
  12. Recovery of the shifted slabs $K_{D,\eta}^{(\hat{x})}$ and $K_{D,\eta}^{(-\hat{x})}$ in ${\mathbb R}^3$
  13. Determination of the excitation time $t_0$ and reconstruction of source supports
  14. Example 1: Cubic source.
  15. Example 2: Spherical source.
  16. ...and 3 more sections

Key Result

Lemma 2.1

For any fixed $\hat{\boldsymbol{x}}\in\mathbb{S}^2$. The supporting interval of $\mathcal{F}^{-1}(\mathbb{E}^{\infty}(\hat{\boldsymbol{x}},\omega))(t)$ is $H:=(c^{-1}\inf(\hat{\boldsymbol{x}}\cdot D)-t_0,c^{-1}\sup(\hat{\boldsymbol{x}}\cdot D)-t_0)$.

Figures (8)

  • Figure 1: Reconstruction of $K_{D,\eta}^{(\hat{x})}$ using multi-frequency data from a single observation direction $\hat{x}=(0,0,1)$, the source is supported in a cube-shaped.
  • Figure 2: Reconstruction of $K_{D,\eta}^{(-\hat{x})}$ using multi-frequency data from a single observation direction $-\hat{x}=(0,0,-1)$, the source is supported in a cube-shaped.
  • Figure 3: Reconstruction of $K_{D,\eta}^{(\hat{x})}\bigcap K_{D,\eta}^{(-\hat{x})}$using multi-frequency data from a single observation direction $\hat{x}=(0,0,1),\mu=4,\epsilon=1$, the source is supported in a cube-shaped.
  • Figure 4: Reconstruction of a cubic support. $\mu=1, \epsilon=1,t_0=3$.
  • Figure 5: Reconstruction of $K_{D,\eta}^{(\hat{x})} \cap K_{D,\eta}^{(-\hat{x})}$. $\mu=1, \epsilon=1,t_0=3$.
  • ...and 3 more figures

Theorems & Definitions (30)

  • Definition 1.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Definition 2.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.2
  • proof
  • ...and 20 more