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A quantitative sampling method for elastic and electromagnetic sources

Xiaodong Liu, Qingxiang Shi

TL;DR

This work tackles inverse source problems for elastic and electromagnetic waves by introducing a quantitative sampling framework that leverages full or partial far-field patterns to reconstruct source functions. It defines elastic indicators $\mathcal{I}_f$, $\mathcal{I}_p$, $\mathcal{I}_s$ and electromagnetic indicators $\mathcal{I}_E$, $\mathcal{I}_H$ that yield exact source recoveries under appropriate conditions (e.g., $\mathcal{I}_f=S$, $\mathcal{I}_E=J$ when divergence-free, $\mathcal{I}_H={\rm curl}\,J$), thereby enabling stable, quantitative reconstructions from finite data. The paper provides explicit stability estimates for the discrete indicators obtained from finitely many directions and frequencies, showing how accuracy improves with larger data sets and lower noise, and demonstrates the approach with comprehensive numerical simulations in ${\mathbb R}^2$ and ${\mathbb R}^3$. The results establish a practical, low-cost alternative to qualitative sampling methods, offering constructive uniqueness and robust recovery of both source magnitudes and derivative information in elastic and EM settings, with clear pathways for experimental data integration.

Abstract

This work is dedicated to a novel sampling method for accurately reconstructing elastic and electromagnetic sources from the far field patterns. We show that the proposed indicators in the form of integrals with full far field patterns are exactly the source functions. These facts not only give constructive uniqueness proofs of the inverse source problems, but also establish the theoretical basis of the proposed sampling methods. Furthermore, we derive the stability estimates for the corresponding discrete indicators using the far field patterns with finitely many observations and frequencies. We have also proposed the indicators with partial far field patterns and proved their validity for providing the derivative information of the unknown sources. Numerical examples are presented to verify the accuracy and stability of the proposed quantitative sampling method.

A quantitative sampling method for elastic and electromagnetic sources

TL;DR

This work tackles inverse source problems for elastic and electromagnetic waves by introducing a quantitative sampling framework that leverages full or partial far-field patterns to reconstruct source functions. It defines elastic indicators , , and electromagnetic indicators , that yield exact source recoveries under appropriate conditions (e.g., , when divergence-free, ), thereby enabling stable, quantitative reconstructions from finite data. The paper provides explicit stability estimates for the discrete indicators obtained from finitely many directions and frequencies, showing how accuracy improves with larger data sets and lower noise, and demonstrates the approach with comprehensive numerical simulations in and . The results establish a practical, low-cost alternative to qualitative sampling methods, offering constructive uniqueness and robust recovery of both source magnitudes and derivative information in elastic and EM settings, with clear pathways for experimental data integration.

Abstract

This work is dedicated to a novel sampling method for accurately reconstructing elastic and electromagnetic sources from the far field patterns. We show that the proposed indicators in the form of integrals with full far field patterns are exactly the source functions. These facts not only give constructive uniqueness proofs of the inverse source problems, but also establish the theoretical basis of the proposed sampling methods. Furthermore, we derive the stability estimates for the corresponding discrete indicators using the far field patterns with finitely many observations and frequencies. We have also proposed the indicators with partial far field patterns and proved their validity for providing the derivative information of the unknown sources. Numerical examples are presented to verify the accuracy and stability of the proposed quantitative sampling method.

Paper Structure

This paper contains 11 sections, 6 theorems, 79 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A quantitative sampling method for elastic sources
  3. Direct elastic source scattering problems
  4. Indicators with full or partial far field patterns for elastic sources
  5. A quantitative sampling method for electromagnetic sources
  6. Direct electromagnetic source scattering problems
  7. Indicators for electromagnetic sources
  8. Stability estimates for the discrete indicators
  9. Numerical simulations
  10. Reconstruction for elastic wave source in $\mathbb R^2$.
  11. Reconstruction for electromagnetic sources in $\mathbb R^3$.

Key Result

Theorem 2.1

Let $S\in\left(L^2(\mathbb R^n)\right)^n$, then

Figures (10)

  • Figure 1: True source function $S$ and its support $\Omega$ in Example one.
  • Figure 2: Reconstructions with $\omega_{\Lambda}=40$ and different $L$. Top row: reconstructions by plotting $(I_f)_{(1)}$. Bottom row: reconstructions by plotting $(I_f^{\epsilon})_{(1)}$.
  • Figure 3: Reconstructions with $L=51$ and different $\omega_{\Lambda}$. Top row: reconstructions by plotting $(I_f)_{(2)}$. Bottom row: reconstructions by plotting $(I_f^{\epsilon})_{(2)}$.
  • Figure 4: Reconstructions of $I_p$ and $I_s$ for Example one with $L=51$, $\omega_{\Lambda}=40$.
  • Figure 5: Reconstructions of $S_1$ and $S_2$ by $I_f$. We take $L=51$ and $\omega_{\Lambda}=40$.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Theorem 4.2