American Institute of Mathematical Sciences

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April  2018, 12(2): 281-291. doi: 10.3934/ipi.2018012

On recovery of an inhomogeneous cavity in inverse acoustic scattering

Fenglong Qu 1, and  Jiaqing Yang 2,,

1. 

School of Mathematics and Informational Science, Yantai University, Yantai, Shandong 264005, China
2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China

* The corresponding author

Received  August 2016 Revised  December 2017 Published  February 2018

Fund Project: Fenglong Qu is supported by the NNSF of China under grant No. 11401513 and NSF of Shandong Province of China grant No. ZR2017MA044. Jiaqing Yang is supported by the NNSF of China under grant No. 11401568 and No. 11771349, by the China Postdoctoral Science Foundation under grant No. 2015M580827 and No. 2016T90900, and by Postdoctoral research project of Shaanxi Province of China under grant No. 2016BSHYDZZ52.

Consider the time-harmonic acoustic scattering of an incident point source inside an inhomogeneous cavity. By constructing an equivalent integral equation, the well-posedness of the direct problem is proved in $L^p$ with using the classical Fredholm theory. Motivated by the previous work [10], a novel uniqueness result is then established for the inverse problem of recovering the refractive index of piecewise constant function from the wave fields measured on a closed surface inside the cavity.

Keywords: Uniqueness, inverse scattering, refractive index, inhomogeneous cavity, point source.
Mathematics Subject Classification: Primary: 35R30, 35P25; Secondary: 78A46.
Citation: Fenglong Qu, Jiaqing Yang. On recovery of an inhomogeneous cavity in inverse acoustic scattering. Inverse Problems and Imaging, 2018, 12 (2) : 281-291. doi: 10.3934/ipi.2018012
References:
[1]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.

[2]

F. Cakoni and D. Colton, Qualitative Method in Inverse Scattering Theory, Springer. Berlin, 2006.

[3]

F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemp. Math., 615 (2014), 71-88. 

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977.

[5]

P. Jakubik and R. Potthast, Testing the integrity of some cavity-the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914.  doi: 10.1016/j.apnum.2007.04.007.

[6]

X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18pp.

[7]

S. Meng, H. Haddar and F. Cakoni, The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20pp.

[8]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17pp.

[9]

H. Qin and D. Colton, The inverse scattering problem for cavities, J. Appl. Numer. Math., 62 (2015), 699-708.  doi: 10.1016/j.apnum.2010.10.011.

[10]

J. Yang, H. Zhang and B. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles, arXiv: 1305.0917.

[11]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17pp.

show all references

References:
[1]

F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.  doi: 10.1137/090769338.

[2]

F. Cakoni and D. Colton, Qualitative Method in Inverse Scattering Theory, Springer. Berlin, 2006.

[3]

F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemp. Math., 615 (2014), 71-88. 

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977.

[5]

P. Jakubik and R. Potthast, Testing the integrity of some cavity-the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914.  doi: 10.1016/j.apnum.2007.04.007.

[6]

X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18pp.

[7]

S. Meng, H. Haddar and F. Cakoni, The factorization method for a cavity in an inhomogeneous medium, Inverse Problems, 30 (2014), 045008, 20pp.

[8]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17pp.

[9]

H. Qin and D. Colton, The inverse scattering problem for cavities, J. Appl. Numer. Math., 62 (2015), 699-708.  doi: 10.1016/j.apnum.2010.10.011.

[10]

J. Yang, H. Zhang and B. Zhang, Uniqueness in inverse acoustic and electromagnetic scattering by penetrable obstacles, arXiv: 1305.0917.

[11]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17pp.

Figure 1.  The inhomogeneous cavity
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Figure 2.  The inhomogeneous cavity
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