Deterministic-statistical approach for an inverse acoustic source problem using multiple frequency limited aperture data

Yanfang Liu; Zhizhang Wu; Jiguang Sun; Zhiwen Zhang

doi:10.3934/ipi.2023018

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2023. Doi: 10.3934/ipi.2023018

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Deterministic-statistical approach for an inverse acoustic source problem using multiple frequency limited aperture data

1.
Department of Mathematics, The George Washington University, Washington, DC 20052, U.S.A
2.
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China
3.
Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A

^*Corresponding author: Zhizhang Wu
^*Corresponding author: Zhizhang Wu

Received: September 2022

Revised: March 2023

Early access: April 2023

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Abstract

We propose a deterministic-statistical method for an inverse source problem using multiple frequency limited aperture far field data. The direct sampling method is used to obtain a disc such that it contains the compact support of the source. The Dirichlet eigenfunctions of the disc are used to expand the source function. Then the inverse problem is recast as a statistical inference problem and the Bayesian inversion is employed to reconstruct the coefficients of the eigen-expansion for the source function. The stability of the statistical inverse problem with respect to the measured data is justified in the sense of Hellinger distance. A preconditioned Crank-Nicolson (pCN) Metropolis-Hastings (MH) algorithm is implemented to explore the posterior density function. Numerical examples show that the proposed method is effective for both smooth and non-smooth sources.

Keywords:

Mathematics Subject Classification: Primary: 35R30; Secondary: 62F15, 65N21.

Citation:

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Figure 1. Exact source function $ f $ for Examples $ 1-5 $

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Figure 2. Example 1 (exact support known). Top row: the histograms of the coefficients for $ f_{BE} $. Second row: $ f_{BE} $. Third row: the reconstruction error $ f-f_{BE} $. Left column: $ \Gamma_1 $. Middle column: $ \Gamma_2 $. Right column: $ \Gamma_3 $

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Figure 3. Example 1 (reconstructed support). First row: plots of the indicators for the DSM. Second row: $ f_{BE} $'s. Third row: the reconstruction error $ f-f_{BE} $. Left column: $ \Gamma_1 $. Middle column: $ \Gamma_2 $. Right column: $ \Gamma_3 $

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Figure 4. Example 2. First row: plots of the indicators for the DSM. Second row: the reconstructed $ f_{BE} $. Third row: the reconstruction error $ f-f_{BE} $. Left column: $ \Gamma_1 $. Middle column: $ \Gamma_2 $. Right column: $ \Gamma_3 $

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Figure 5. Example 3. First row: plots of the indicators for the DSM. Second row: the reconstructed $ f_{BE} $. Third row: the reconstruction error $ f-f_{BE} $. Left column: $ \Gamma_1 $. Middle column: $ \Gamma_2 $. Right column: $ \Gamma_3 $

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Figure 6. Example 4. First row: plots of the indicators for the DSM. Second row: the reconstructed $ f_{BE} $. Third row: the reconstruction error $ f-f_{BE} $. Left column: $ \Gamma_1 $. Middle column: $ \Gamma_2 $. Right column: $ \Gamma_3 $

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Figure 7. Example 5. First row: plots of the indicators for the DSM. Second row: the reconstructed $ f_{BE} $. Third row: the reconstruction error $ f-f_{BE} $. Left column: $ \Gamma_1 $. Middle column: $ \Gamma_2 $. Right column: $ \Gamma_3 $

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Table 1. Exact support of $ f(x) $ and the radii of the discs $ \hat{B} $'s by the DSM

Exact support	Example 1	Example 2	Example 3	Example 4	Example 5
Exact support	B(0, 0.9)	B(0, 0.9)	$ B $(0, 0.7471)$ ^* $	$ a,b=0.9,1.08 $	B(0, 0.9)
$ \Gamma_1 $	1.3601	0.9055	0.8246	1.7205	0.9849
$ \Gamma_2 $	1.4422	1.1402	1.0198	1.5000	1.2369
$ \Gamma_3 $	1.1314	1.0817	1.0817	1.2806	1.1705

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Table 2. Absolute error (AE) $ \|f-f_{BE}\|_2 $ and the relative error (RE) $ \frac{\|f_{BE}-f\|_2}{\|f\|_2} $

	Example 1				Example 2		Example 3		Example 4		Example 5
	AE$^e$	RE$^e$	AE	RE	AE	RE	AE	RE	AE	RE	AE	RE
$\Gamma_1$	0.0336	1.59%	0.0865	4.10%	0.0386	2.59%	0.0653	6.37%	0.3513	26.72%	0.2758	13.74%
$\Gamma_2$	0.0463	2.19%	0.1107	5.25%	0.0604	4.05%	0.1707	16.78%	0.3282	24.87%	0.2745	17.40%
$\Gamma_3$	0.1200	5.67%	0.1140	5.39%	0.0583	3.91%	0.2641	26.01%	0.4367	33.03%	0.2956	18.72%

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