Uniqueness and numerical resolution via iterated sensitivity equation of an inverse electromagnetic coefficient problem with partial boundary data

Jérémy Heleine

doi:10.3934/ipi.2024042

Article Contents

2025, Volume 19, Issue 3: 498-522. Doi: 10.3934/ipi.2024042

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Uniqueness and numerical resolution via iterated sensitivity equation of an inverse electromagnetic coefficient problem with partial boundary data

Jérémy Heleine^,

Institut de Mathématiques de Toulouse CNRS UMR 5219, Université Paul Sabatier F-31062 Toulouse Cedex 9, France

^*Corresponding author: Jérémy Heleine
^*Corresponding author: Jérémy Heleine

Received: November 2023

Revised: July 2024

Early access: October 10, 2024

Published online: October 10, 2024

Abstract / Introduction Full Text(HTML) Figure(15) / Table(9) Related Papers Cited by

Abstract

In this paper we study an inverse boundary value problem for Maxwell's equations. The goal is to reconstruct perturbations in the refractive index of the medium inside an object from the knowledge of the tangential trace of an electric field on a part of the boundary of the domain. We first provide a uniqueness result for this inverse problem. Then, we propose a new approach to reconstruct numerically the perturbations. This complete procedure is based on the minimization of a cost functional involving an iterated sensitivity equation.

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Mathematics Subject Classification: Primary: 65N21, 35Q61; Secondary: 35R30, 65N20.

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Figure 1. Example of possible configuration for the domain $ \Omega $. The accessible part of the boundary $ \Gamma_0 $ is shown as thick parts on $ \partial\Omega $. The support of the perturbation is here composed of three parts, delimited by dotted lines. This support does not touch $ \mathcal{V} $, the tubular neighborhood where $ \kappa $ is assumed to be known, represented here by the part filled with vertical lines. Finally, $ \Gamma_\text{int} $ is an artificial boundary included in $ \mathcal{V} $, delimiting the subdomain $ U $ represented by the gray part

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Figure 2. Configuration of the domain $ \Omega $ for the first numerical test. The thicker parts on the boundary is $ \Gamma_0 $. The artificial boundary is the dashed line, while the dotted line represents the boundary of the support of the perturbation

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Figure 3. Real part of the reconstructed refractive index in the unit disk. The boundary of the exact support of the perturbation is shown as a white line

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Figure 4. Configuration of the domain $ \Omega $ in 3D. The accessible part $ \Gamma_0 $ is represented by the patches on the sphere

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Figure 5. Real part of the reconstructed refractive index in the 3D unit ball. The boundary of the exact support of the perturbation is shown as a white line

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Figure 6. Configuration of the domain $ \Omega $ for the head profile. The thicker parts on the boundary is $ \Gamma_0 $. The artificial boundary is the dashed line, while the dotted line represents the boundary of the support of the perturbation

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Figure 7. Real part of the reconstructed refractive index in the geometry of a 2D head profile in microwave regime. The boundary of the exact support of the perturbation is shown as a white line

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Figure 8. Real part of the reconstructed refractive index in the unit disk with exact data on the whole boundary. The boundary of the exact support of the perturbation is shown as a white line

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Figure 9. Comparison of the exact tangential trace of the difference field $ (\mathit{\boldsymbol{E}}[\kappa_\text{ex}] - \mathit{\boldsymbol{E}}[\kappa_0]) \times \mathit{\boldsymbol{n}} $ on $ \Gamma_\text{int} $ and its approximation by the quasi-reversibility method. The circle $ \Gamma_\text{int} $ is unfolded: the $ x $-axis shows the angle of a point while $ y $-axis shows the value of the modulus of the trace at this point

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Figure 10. Reconstruction of more complex perturbations in the unit disk in 2D. Top pictures: star-shaped perturbation with constant amplitude. Bottom pictures: disk-shaped perturbation with variable amplitude. Left pictures: real part of the exact refractive index. Right pictures: results of the complete inversion procedures

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Figure 11. Reconstruction of an ellipsoidal perturbation in the unit disk (left picture) and in the unit ball (right picture)

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Figure 12. Reconstruction of a perturbation composed of two connected components in 2D (left picture) and in 3D (right picture)

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Figure 13. Reconstruction of a star-shaped perturbation, using total data on top figures and partial data on bottom figure. Left: approximation by a disk. Right: final result

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Figure 14. Comparison of a tangential trace of the difference field generated by the star-shaped perturbation, after data completion, with the trace generated from the reconstructed perturbation

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Figure 15. Result of the minimization of the cost function without using the previous rough approximation as initial guess

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Table 1. Reconstruction of a spherical perturbation in the unit disk in 2D, with unitary physical parameters

Parameter	Exact value	Approximation	Relative error
Center	($ -0.4 $, 0)	($ -0.37236 $, 0.00482)	$ 7.01355 \cdot 10^{-2} $
Radius	0.2	0.21157	$ 5.78467 \cdot 10^{-2} $
Amplitude	0.1	0.09143	$ 8.57238 \cdot 10^{-2} $

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Table 2. Behavior of the reconstructed amplitude $ a $ with respect to the exact amplitude $ a_\text{ex} $

$ a_\text{ex} $	$ a $	$ a - a_\text{ex} $	$ \frac{\|a - a_\text{ex}\|}{\|a_\text{ex}\|} $
$ -0.30 $	$ -0.26952 $	0.03048	10.15936%
$ -0.25 $	$ -0.22748 $	0.02252	9.00868%
$ -0.20 $	$ -0.18193 $	0.01807	9.03563%
$ -0.15 $	$ -0.13500 $	0.01500	10.00114%
$ -0.10 $	$ -0.09257 $	0.00743	7.43232%
$ -0.05 $	$ -0.04519 $	0.00481	9.61981%
0.05	0.04446	$ -0.00554 $	11.08609%
0.10	0.09143	$ -0.00857 $	8.57175%
0.15	0.13252	$ -0.01748 $	11.65165%
0.20	0.18244	$ -0.01756 $	8.77901%
0.25	0.22534	$ -0.02466 $	9.86586%
0.30	0.27537	$ -0.02463 $	8.21076%

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Table 3. Reconstruction of a spherical perturbation in the unit ball in 3D, with unitary physical parameters

Parameter	Exact value	Approximation	Relative error
Center	($ -0.4 $, 0, 0)	($ -0.35166 $, $ -0.03888 $, $ -0.02515 $)	$ 1.67349 \cdot 10^{-1} $
Radius	0.2	0.19326	$ 3.37106 \cdot 10^{-2} $
Amplitude	0.2	0.21760	$ 8.79817 \cdot 10^{-2} $

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Table 4. Reconstruction of a spherical perturbation in a 2D head profile, with realistic physical parameters (microwave regime)

Parameter	Exact value	Approximation	Relative error
Center	(0.05, 0.4)	(0.05502, 0.38480)	$ 3.97167 \cdot 10^{-2} $
Radius	0.1	0.11796	$ 1.79581 \cdot 10^{-1} $
Amplitude	0.1	0.07054	$ 2.94557 \cdot 10^{-1} $

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Table 5. Reconstruction of a spherical perturbation in the unit disk in 2D, with unitary physical parameters. The transmission step is skipped and exact data are used

Parameter	Exact value	Approximation	Relative error
Center	($ -0.4 $, 0)	($ -0.39754 $, 0.00227)	$ 8.35419 \cdot 10^{-3} $
Radius	0.2	0.20301	$ 1.50750 \cdot 10^{-2} $
Amplitude	0.1	0.09822	$ 1.77511 \cdot 10^{-2} $

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Table 6. Reconstruction of an ellipsoidal perturbation in the unit disk in 2D, with unitary physical parameters

Parameter	Exact value	Approximation	Relative error
Center	($ -0.4 $, 0)	($ -0.39182 $, 0.00187)	$ 2.09871 \cdot 10^{-2} $
$ x $-radius	0.15	0.13962	$ 6.91861 \cdot 10^{-2} $
$ y $-radius	0.25	0.26395	$ 5.58086 \cdot 10^{-2} $
Amplitude	0.1	0.10096	$ 9.63128 \cdot 10^{-3} $

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Table 7. Reconstruction of an ellipsoidal perturbation in the unit ball in 3D, with unitary physical parameters

Parameter	Exact value	Approximation	Relative error
Center	($ -0.4 $, 0, 0)	($ -0.35245 $, $ -0.04526 $, $ -0.02455 $)	$ 1.75205 \cdot 10^{-1} $
$ x $-radius	0.2	0.19363	$ 3.18313 \cdot 10^{-2} $
$ y $-radius	0.4	0.34730	$ 1.31745 \cdot 10^{-1} $
$ z $-radius	0.3	0.28110	$ 6.30023 \cdot 10^{-2} $
Amplitude	0.2	0.24738	$ 2.36877 \cdot 10^{-1} $

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Table 8. Reconstruction of a perturbation having two connected components in the unit disk in 2D, with unitary physical parameters

Parameter	Exact value	Approximation	Relative error
Center 1	($ -0.55 $, $ -0.45 $)	($ -0.53157 $, $ -0.43996 $)	$ 2.95271 \cdot 10^{-2} $
Radius 1	0.1	0.12189	$ 2.18877 \cdot 10^{-1} $
Center 2	(0.4, 0.6)	(0.38488, 0.58559)	$ 2.89726 \cdot 10^{-2} $
Radius 2	0.07	0.08498	$ 2.14044 \cdot 10^{-1} $
Amplitude	0.1	0.06953	$ 3.04699 \cdot 10^{-1} $

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Table 9. Reconstruction of a perturbation having two connected components in the unit ball in 3D, with unitary physical parameters

Parameter	Exact value	Approximation	Relative error
Center 1	($ -0.5 $, 0, 0)	($ -0.42968 $, $ -0.03325 $, $ -0.02998 $)	$ 1.66729 \cdot 10^{-1} $
Radius 1	0.2	0.18773	$ 6.13715 \cdot 10^{-2} $
Center 2	(0, 0, 0.6)	($ -0.03367 $, 0.01041, 0.50099)	$ 1.75165 \cdot 10^{-1} $
Radius 2	0.1	0.09801	$ 1.98898 \cdot 10^{-2} $
Amplitude	0.2	0.24617	$ 2.30842 \cdot 10^{-1} $

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