The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

Juan A. Barceló; Carlos Castro; Fabricio Macià; Cristóbal J. Meroño

doi:10.3934/ipi.2023029

Article Contents

2024, Volume 18, Issue 1: 183-207. Doi: 10.3934/ipi.2023029

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The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

1.
M$^2$ASAI. Universidad Politécnica de Madrid, ETSI Caminos, C. Profesor Aranguren s/n, 28040, Madrid, Spain
2.
M$^2$ASAI. Universidad Politécnica de Madrid, ETSI Navales, Avda. de la Memoria, 4, 28040, Madrid, Spain

^*Corresponding author: Cristóbal J. Meroño
^*Corresponding author: Cristóbal J. Meroño

Received: September 1, 2022

Revised: April 1, 2023

Early access: June 15, 2023

Published online: February 1, 2024

This research was supported by Grant MTM2017-85934-C3-3-P of Agencia Estatal de Investigación (Spain).

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Abstract

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $ B_R \subset \mathbb{R}^3 $ of radius $ R $, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $ L^{\infty} $ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $ \partial B_R $. We then investigate how increasing the radius $ R $ affects the quality of the Born approximation, and the existence of a scattering limit as $ R\to \infty $. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $ -\Delta +q $, and strong recovery of singularity results are observed in this case.

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Mathematics Subject Classification: Primary: 35R30, 65N21; Secondary: 35P25.

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Figure 1. Experiment 1: A step conductivity and its Born approximation (left), and a comparison of their Fourier transforms (right). Lower simulations correspond to a conductivity closer to the reference conductivity $ \gamma = 1 $

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Figure 2. Experiment 2: Three different conductivities that coincide in the interval $ (1/3, 1) $ (left) and their Born approximations (right). We observe that they also coincide in this same interval $ (1/3, 1) $

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Figure 3. Experiment 3: Scattering limit. A conductivity $ \gamma $, its Born approximation $ \gamma_{\mathrm{exp}} $ given by (5) with $ R = 1 $, and the scattering limit $ \gamma_{\mathrm{exp}}(\centerdot, \infty) $ given by formula (8) (left)

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Figure 4. Experiment 4: Born approximations (left) and their Fourier transform (right) for smooth conductivities with different sizes

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Figure 5. Experiment 5: Born approximations (left) and their Fourier transform (right) for a step conductivity with noisy data

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Figure 6. Experiment 6: Born approximation of a step potentials (left) and its Fourier transform (right), that illustrate the recovery of singularities property

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Figure 7. Experiment 7: average error distribution of the Born approximation, $ e_\alpha(x) $, computed with 100 samples, where Fourier coefficients are chosen randomly in different intervals

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Figure 8. Experiment 8: Born approximation of a potential which is zero in a neighborhood of $ r = 0 $ (left) and its Fourier transform (right)

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Figure 9. Experiment 9: Born approximation of a potential (left) and its Fourier transform (right) when we consider domains with $ R = 1 $ (upper figure), $ R = 5 $ (medium figure) $ R = \infty $ (bottom one)

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Figure 10. Experiment 10: Born approximation of a negative potential (left) and its Fourier transform (right)

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Figure 11. Experiment 11: Approximation of a Lipschitz conductivity (left simulation) and a smooth one (right simulation) by the iterative algorithm (9)

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Figure 12. Experiment 11: $ L^2 $-error (left) and $ L^\infty $-error (right) in $ log_{10} $-scale for the iterative algorithm when considering the piecewise constant function in the upper simulation of Figure 1, together with the Lipschitz and smooth conductivities of Figure 11

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