A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers

Jun Lai; Ming Li; Peijun Li; Wei Li

doi:10.3934/ipi.2018027

Article Contents

2018, Volume 12, Issue 3: 635-665. Doi: 10.3934/ipi.2018027

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A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers

Jun Lai^1,,
Ming Li^2, ,,
Peijun Li^3, and
Wei Li^4,

1.
School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, China
2.
College of Data Science, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
3.
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
4.
Institute for Mathematics and its Applications, Minneapolis, MN 55455, USA

* Corresponding author: Ming Li
* Corresponding author: Ming Li

Received: June 2017

Revised: September 2017

Published: March 2018

The first author was supported in part by the Funds for Creative Research Groups of NSFC (No. 11621101) and the Major Research Plan of NSFC (No. 91630309). The second author was supported partially by the National Natural Science Foundation of China (Grant no 11771321). The third author was supported in part by the NSF grant DMS-1151308.

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Abstract

Consider the scattering of a time-harmonic plane wave by heterogeneous media consisting of linear or nonlinear point scatterers and extended obstacles. A generalized Foldy–Lax formulation is developed to take fully into account of the multiple scattering by the complex media. A new imaging function is proposed and an FFT-based direct imaging method is developed for the inverse obstacle scattering problem, which is to reconstruct the shape of the extended obstacles. The novel idea is to utilize the nonlinear point scatterers to excite high harmonic generation so that enhanced imaging resolution can be achieved. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

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Mathematics Subject Classification: Primary: 78A46, 78M15; Secondary: 65N21.

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Figure 1. Schematic of the problem geometry

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Figure 2. Schematic of the imaging modality with nonlinear point scatterers

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Figure 3. Imaging of two extended scatterers surrounded by 1000 linear point scatterers. (a) Example 1: $\kappa = 10$; (b) Example 2: $\kappa = 50$

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Figure 4. Example 3: Imaging of one extended scatterer with two fixed quadratically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 2$; (b) Imaging with $\kappa_2 = 4$

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Figure 5. Example 3: Imaging of one extended scatterer with two moving quadratically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 2$; (b) Imaging with $\kappa_2 = 4$

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Figure 6. Example 4: Imaging of two extended scatterers with two quadratically nonlinear point scatterers close by. (a) Imaging with $\kappa_1 = 5$; (b) Imaging with $\kappa_2 = 10$

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Figure 7. Example 4: Imaging of two extended scatterers with two quadratically nonlinear point scatterers far away. (a) Imaging with $\kappa_1 = 5$; (b) Imaging with $\kappa_2 = 10$

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Figure 8. Example 5: Imaging of two extended scatterers with two cubically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 2$; (b) Imaging with $\kappa_3 = 6$

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Figure 9. Example 6: Imaging of two extended scatterers with two cubically nonlinear point scatterers. (a) Imaging with $\kappa_1 = 5$; (b) Imaging with $\kappa_3 = 15$

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Table 1. Parameters used in the numerical experiments

$N_{\rm point}$	number of point scatterers
$N_{\rm boundary}$	number of points to discretize the boundary of extended scatterer(s)
$N_{\rm direction}$	number of incident and observation directions
$N_{\rm sampling}$	number of sampling points along the $x$-and $y$-direction
$T_{\rm invert}$	time to invert (factorize) the scattering matrix
$T_{\rm solver}$	time to solve the linear system for one incidence
$T_{\rm ffp}$	time to evaluate the far-field patterns
$T_{\rm NUFFT}$	time to apply the NUFFT to evaluate the imaging function

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Table 2. Time (in seconds) to solve the linear system (68) on an HP workstation

$N_{\rm point}$	$N_{\rm boundary}$	Method 1	Method 2	Method 3
$1000$	$600$	$0.16$	$1.42$	$0.89$
$10000$	$600$	$8.9$	fail to converge	fail to converge

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Table 3. Results for imaging two extended scatterers surrounded by linear point scatterers

	$\kappa$	$N_{\rm point}$	$N_{\rm boundary}$	$N_{\rm direction}$	$N_{\rm sampling}$
Example 1	10	1000	600	360	500
Example 2	50	1000	4800	1800	500
		$T_{\rm invert}$	$T_{\rm sampling}$	$T_{\rm ffp}$	$T_{\rm NUFFT}$
Example 1		7.95e-2	2.56e-3	2.23e-2	2.46e-1
Example 2		1.61	2.17e-2	3.49e-1	3.70

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Table 4. Results for imaging the extended scatterers surrounded by quadratically nonlinear point scatterers

	$\kappa$	$N_{\rm point}$	$N_{\rm boundary}$	$N_{\rm direction}$	$N_{\rm sampling}$
Example 3	2	2	600	360	500
Example 4	5	2	1200	360	500
		$T_{\rm invert}$	$T_{\rm solver}$	$T_{\rm ffp}$	$T_{\rm NUFFT}$
Example 3		1.22e-3	8.52e-3	1.39e-2	3.39e-1
Example 4		1.90e-1	3.99e-3	1.96e-2	3.84e-1

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Table 5. Results for imaging the extended scatterers surrounded by cubically nonlinear point scatterers

	$\kappa$	$N_{\rm point}$	$N_{\rm boundary}$	$N_{\rm direction}$	$N_{\rm sampling}$
Example 5	2	2	600	360	500
Example 6	5	2	1200	360	500
		$T_{\rm invert}$	$T_{\rm solver}$	$T_{\rm ffp}$	$T_{\rm NUFFT}$
Example 5		1.24e-3	1.00e-2	1.22e-2	4.33e-1
Example 6		4.22e-3	1.91e-2	2.11e-2	4.41e-1

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