American Institute of Mathematical Sciences

Advanced
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 & 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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The inhomogeneous cavity
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems & Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263
[2]

Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems & Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[3]

Giovanni Bozza, Massimo Brignone, Matteo Pastorino, Andrea Randazzo, Michele Piana. Imaging of unknown targets inside inhomogeneous backgrounds by means of qualitative inverse scattering. Inverse Problems & Imaging, 2009, 3 (2) : 231-241. doi: 10.3934/ipi.2009.3.231
[4]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
[5]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010
[6]

Fang Zeng, Xiaodong Liu, Jiguang Sun, Liwei Xu. The reciprocity gap method for a cavity in an inhomogeneous medium. Inverse Problems & Imaging, 2016, 10 (3) : 855-868. doi: 10.3934/ipi.2016024
[7]

Johannes Elschner, Guanghui Hu, Masahiro Yamamoto. Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type. Inverse Problems & Imaging, 2015, 9 (1) : 127-141. doi: 10.3934/ipi.2015.9.127
[8]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027
[9]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[10]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[11]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343
[12]

Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems & Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537
[13]

Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477
[14]

Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
[15]

Martin Hanke, William Rundell. On rational approximation methods for inverse source problems. Inverse Problems & Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185
[16]

Peter Monk, Jiguang Sun. Inverse scattering using finite elements and gap reciprocity. Inverse Problems & Imaging, 2007, 1 (4) : 643-660. doi: 10.3934/ipi.2007.1.643
[17]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[18]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[19]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291
[20]

Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843

2017 Impact Factor: 1.465

Tools

Metrics

  • PDF downloads (72)
  • HTML views (227)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences