A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data

Heping Dong; Deyue Zhang; Yukun Guo

doi:10.3934/ipi.2019010

Article Contents

2019, Volume 13, Issue 1: 177-195. Doi: 10.3934/ipi.2019010

This issue Previous Article Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line Next Article Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data

1.
School of Mathematics, Jilin University, Changchun 130012, China
2.
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Deyue Zhang
* Corresponding author: Deyue Zhang

Received: March 31, 2018

Revised: July 31, 2018

Published: December 2018

The first author is supported by NSFC grants 11801213 and 11771180. The second author is supported by NSFC grant 11671170. The third author is supported by NSFC grants 11601107, 41474102 and 11671111.

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Abstract

In this paper, we consider the inverse problem of determining the location and the shape of a sound-soft obstacle from the modulus of the far-field data for a single incident plane wave. By adding a reference ball artificially to the inverse scattering system, we propose a system of nonlinear integral equations based iterative scheme to reconstruct both the location and the shape of the obstacle. The reference ball technique causes few extra computational costs, but breaks the translation invariance and brings information about the location of the obstacle. Several validating numerical examples are provided to illustrate the effectiveness and robustness of the proposed inversion algorithm.

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Mathematics Subject Classification: 78A46.

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Figure 1. An illustration of the reference ball technique

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Figure 2. Reconstructions of an apple-shaped domain with $1\%$ noise and $\epsilon = 0.015$

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Figure 3. Reconstructions of an apple-shaped domain with $5\%$ noise and $\epsilon = 0.035$

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Figure 4. Reconstructions of an apple-shaped domain with different reference balls and $1\%$ noise. Here we are using the initial guess $(c_1^{(0)}, c_2^{(0)}) = (-0.4, -0.8), r^{(0)} = 0.1$, and (a) $\epsilon = 0.011$, (b) $\epsilon = 0.02$, (c) $\epsilon = 0.03$, (d) $\epsilon = 0.011$

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Figure 5. Reconstructions of an apple-shaped domain with different incoming wave directions, $1\%$ noise is added. Here, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.7, 0.7), r^{(0)} = 0.1$, the reference ball $(b_1, b_2) = (4, 0), R = 0.4$, and $\epsilon = 0.015$

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Figure 6. Reconstructions of an apple-shaped domain with different locations of the reference ball, $1\%$ noise is added. Here, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.7, 0.7), r^{(0)} = 0.1$ and $\epsilon = 0.015$

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Figure 7. Reconstructions of a peanut-shaped domain with $1\%$ noise and $\epsilon = 0.015$

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Figure 8. {Reconstructions of a peanut-shaped domain with $5\%$ noise and $\epsilon = 0.035$

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Figure 9. Reconstructions of a peanut-shaped domain with different reference balls, $1\%$ noise is added. Here, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.3, -0.6), r^{(0)} = 0.1$ and $\epsilon = 0.015$. (a) $(b_1, b_2) = (3, 0), R = 0.2$. (b) $(b_1, b_2) = (4, 0), R = 0.4$. (c) $(b_1, b_2) = (5.5, 0), R = 0.8$

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Figure 10. Reconstructions of a peanut shaped domain with different initial guesses and $1\%$ noise. Here, we are using the reference ball $(b_1, b_2) = (4, 0), R = 0.4$ and $\epsilon = 0.015$

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Figure 11. Reconstructions of a rectangle shaped domain with $1\%$ and $5\%$ noise data respectively. Here, we are using the initial guess $(c_1^{(0)}, c_2^{(0)}) = (0.4, -0.8), r^{(0)} = 0.1$ and the reference ball $(b_1, b_2) = (4, 0), R = 0.5$

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Figure 12. Reconstructions of a rectangle shaped domain with different initial guesses ($1\%$ noise is added). Here, we are using the incoming wave direction $d = (\cos(2\pi/3), \sin(2\pi/3))$, the reference ball $(b_1, b_2) = (4, 0), R = 0.5$ and $\epsilon = 0.015$

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Figure 13. Reconstructions of a rectangle-shaped domain with different reference balls, $1\%$ noise is added. Here, the incoming wave direction $d = (\cos(\pi/6), \sin(\pi/6))$, the initial guess $(c_1^{(0)}, c_2^{(0)}) = (-0.3, -0.9), r^{(0)} = 0.1$, and $\epsilon = 0.015$

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