A direct reconstruction method for radiating sources in Maxwell's equations with single-frequency data

TL;DR

This work tackles the inverse source problem for time-harmonic Maxwell equations with boundary Cauchy data at a fixed frequency, aiming to recover the number, locations, and moment vectors of point-like sources. It introduces a direct imaging approach based on the base function $I(\mathbf{z},\mathbf{q})$ and the enhanced imaging functional $\widetilde{I}_s(\mathbf{z})$, which yields peaks at source locations and allows robust moment estimation, even when magnitudes vary. The methodology extends to small-volume sources via an asymptotic expansion of the radiated field, enabling simultaneous localization and direction recovery for the source moments. Comprehensive 3D numerical experiments demonstrate the method’s accuracy, stability to noise, and flexibility across wavenumbers and imaging distances from the data boundary, making it well-suited for antenna design, medical imaging, and environmental tracing.

Abstract

This paper presents a fast and robust numerical method for reconstructing point-like sources in the time-harmonic Maxwell's equations given Cauchy data at a fixed frequency. This is an electromagnetic inverse source problem with broad applications, such as antenna synthesis and design, medical imaging, and pollution source tracing. We introduce new imaging functions and a computational algorithm to determine the number of point sources, their locations, and associated moment vectors, even when these vectors have notably different magnitudes. The number of sources and locations are estimated using significant peaks of the imaging functions, and the moment vectors are computed via explicitly simple formulas. The theoretical analysis and stability of the imaging functions are investigated, where the main challenge lies in analyzing the behavior of the dot products between the columns of the imaginary part of the Green's tensor and the unknown moment vectors. Additionally, we extend our method to reconstruct small-volume sources using an asymptotic expansion of their radiated electric field. We provide numerical examples in three dimensions to demonstrate the performance of our method.

Figures (10)

• Figure 1: Reconstruction results for a point source on $\{z=0.5\}$. True location at $(-0.5,0,0.5)^\top$ is marked by green crosses. (a) For $\mathbf{p}=(-17,-7,-8)^\top$, $|I(\mathbf{z},\mathbf{q})|$ peaks at $(-0.495, 0, 0.495)^\top$, relative error $1\%$. (b) For $\mathbf{p}=(17,-7-8)^\top$, $|I(\mathbf{z},\mathbf{q})|$ peaks at $(-0.462, -0.10, 0.375)^\top$, relative error $24.267\%$. (c) For $\mathbf{p}=(17,-7-8)^\top$, $\widetilde{I}_1(\mathbf{z})$ peaks at $(-0.495, 0, 0.495)^\top$, relative error $1\%$. Here, $\mathbf{z} \in [-2,2]^3$, $k=20,\mathbf{q} =(1,1,1)^\top$. Computed locations are rounded to three decimal digits.
• Figure 2: For $\mathbf{p}=(17,-7,-8)^\top$, all three terms $|\mathbf{p} \cdot \mathrm{Im}\, \mathbb{G}(\mathbf{x},\mathbf{z}){\bf{e}}_1|^2$, $|\mathbf{p} \cdot \mathrm{Im}\, \mathbb{G}(\mathbf{x},\mathbf{z}){\bf{e}}_2|^2$, $|\mathbf{p} \cdot \mathrm{Im}\, \mathbb{G}(\mathbf{x},\mathbf{z}){\bf{e}}_3|^2$ and their sum $\widetilde{I}_2(\mathbf{z})$ attain their maximum at $\mathbf{x}=(-0.5,0,0.5)^\top$ in a small neighborhood of $\mathbf{x}$.
• Figure 3: For $\mathbf{p}=(2,9,-1)^\top$, only one term $|\mathbf{p} \cdot \mathrm{Im}\, \mathbb{G}(\mathbf{x},\mathbf{z}){\bf{e}}_2|^2$ attains its maximum at $\mathbf{x}=(0,0,0)^\top$ in a small neighborhood of $\mathbf{x}$, but dominates the values of the other terms that do not peak at $\mathbf{x}$. As a result, their sum $\widetilde{I}_2(\mathbf{z})$ still attains a maximum at $\mathbf{x}$ in this neighborhood.
• Figure 4: Reconstruction results for the three point sources in Table \ref{['Ta:3points']} for $k=20, s=4$. Isosurface visualizations for the true locations in (a), for $\widetilde{I}^\text{re}_4(\mathbf{z})$ in (b), and for $\widetilde{I}^\text{im}_4(\mathbf{z})$ in (c). Cross-sectional views restricted to 2D domains of $\widetilde{I}^\text{re}_4(\mathbf{z})$ in (d)--(e) and of $\widetilde{I}^\text{im}_4(\mathbf{z})$ in (f), with the true locations marked with green crosses.
• Figure 5: Reconstruction results for the six point sources in Table \ref{['Ta:6points']} for $k=20, s=4$. Isosurface visualizations for the true locations in (a), for $\widetilde{I}^\text{re}_4(\mathbf{z})$ and their updates in (b), and for $\widetilde{I}^\text{im}_4(\mathbf{z})$ and their updates in (c).

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Remark 3
• Lemma 4
• proof
• Theorem 5
• proof
• Remark 6