Monotonicity in inverse scattering for Maxwell's equations

Annalena Albicker; Roland Griesmaier

doi:10.3934/ipi.2022032

Article Contents

2023, Volume 17, Issue 1: 68-105. Doi: 10.3934/ipi.2022032

This issue Previous Article A non-convex PDE-constrained denoising model for impulse and Gaussian noise mixture reduction Next Article Multilevel Markov Chain Monte Carlo for Bayesian inverse problem for Navier-Stokes equation

Monotonicity in inverse scattering for Maxwell's equations

Annalena Albicker and
Roland Griesmaier

Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

Received: October 2021

Revised: April 2022

Early access: June 2022

Published: February 2023

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173

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Abstract

A recent area of research in inverse scattering theory has been the study of monotonicity relations for the eigenvalues of far field operators and their use in shape reconstruction for inverse scattering problems. We develop such monotonicity relations for an electromagnetic inverse scattering problem governed by Maxwell's equations, and we apply them to establish novel rigorous characterizations of the shape of scattering objects in terms of the corresponding far field operators. Along the way we establish the existence of electromagnetic fields that have arbitrarily large energy in some prescribed region, while at the same time having arbitrarily small energy in some other prescribed region. These localized vector wave functions not only play an important role in the proofs of the novel monotonicity based shape characterizations but they are also of independent interest. We conclude with some simple numerical demonstrations of our theoretical results.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 65N21.

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Figure 1. Number of negative eigenvalues (left) and number of positive eigenvalues (right) $ {\rm{Re}}\bigl(\lambda_n(r_D)\bigr) $ (dotted), $ -\alpha \mu_n(r_B) $ (dashed), and $ {\rm{Re}}\bigl(\lambda_n(r_D)\bigr)-\alpha\mu_n(r_B) $ (solid) within the range $ n = 0, \ldots, 1000 $ as function of $ r_B $

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Figure 2. Visualization of exact shape of scattering object in Example 9.1 (left), visualization of isosurface $ I_{20} = 2 $ of indicator function from (65) using simulated far field data without additional noise (center), and visualization of isosurface $ I_{20} = 11 $ using simulated far field data with $ 0.1 \% $ noise (right)

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Figure 3. Visualization of the indicator function $ I_\alpha $ in the $ {\boldsymbol x}_1, {\boldsymbol x}_2 $-plane for $ \alpha \in \{0.01, 0.1, 0.5, 1, 10, 20\} $ using simulated far field data without additional noise. The dashed lines show the exact boundaries of the cross-section of the scatterer

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Figure 4. Visualization of the exact shape of the mixed scattering object in Example 9.2 (left), and of the isosurfaces $ I_{0.5}^+ = 7 $ (center) and $ I_{-1}^- = 1 $ (right)

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