Sampling type methods for an inverse waveguide problem

Previous Article
A TV Bregman iterative model of Retinex theory
IPI Home
This Issue
Next Article
Inverse acoustic obstacle scattering problems using multifrequency measurements

November 2012, 6(4): 709-747. doi: 10.3934/ipi.2012.6.709

Sampling type methods for an inverse waveguide problem

1.	Department of Mathematics, University of Delaware, Newark, DE 19716
2.	Departamento de Matemáticas, Escuela Politécnica de Ingeniería de Gijón, Universidad de Oviedo, 33203 Gijón, Spain

Received November 2011 Revised August 2012 Published November 2012

Abstract
Related Papers

We consider the problem of locating a penetrable obstacle in an acoustic waveguide from measurements of pressure waves due to point sources inside the waveguide. More precisely, we assume that we are given the scattered field and its normal derivative for any source point and receiver placed on a pair of surfaces known as the source and the measurement surfaces, respectively. A novel feature of this work is that the obstacle is allowed to touch the boundary of the pipe.
We first analyze the associated interior transmission problem. Then, we adapt and analyze the Reciprocity Gap Method (RGM) and the Linear Sampling Method (LSM) to deal with the inverse problem. We also study the relationship between these two methods and provide numerical results.

Keywords: inverse scattering, waveguide., Linear sampling method, reciprocity gap method.

Mathematics Subject Classification: 65N21, 74F1.

Citation: Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709

References:

[1]	T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a 3D homogeneous planar waveguide,, SIAM J. Appl. Math., 71 (2011), 753. doi: 10.1137/100806333.
[2]	T. Arens, D. Gintides and A. Lechleiter, Variational formulations for scattering in a three-dimensional acoustic waveguide,, Math. Meth. Appl. Sci., 31 (2007), 821. doi: 10.1002/mma.947.
[3]	L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation,, Inv. Prob., 24 (2008).
[4]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006).
[5]	F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects,, Inv. Prob., 22 (2006), 845. doi: 10.1088/0266-5611/22/3/007.
[6]	F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in, (2003), 613.
[7]	D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory,, Inv. Prob., 21 (2005), 383. doi: 10.1088/0266-5611/21/1/023.
[8]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inv. Prob., 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003.
[9]	D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.
[10]	D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems,, Inv. Prob., 13 (1997), 1477. doi: 10.1088/0266-5611/13/6/005.
[11]	K. Horoshenkov, R. Ashley and J. Blanksby, Determination of sewer roughness and sediment properties using acoustic techniques,, Water Science and Technology, 47 (2003), 87.
[12]	P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 5 (2011), 465.
[13]	B. Pincon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors,, Inv. Prob., 23 (2007), 1.
[14]	F. Podd, M. Ali, K. Horoshenkov, A. Wood, S. Tait, J. Boot, R. Long and A. Saul, Rapid sonic characterisation of sewer change and obstructions,, Water Science and Technology, 56 (2007), 131.
[15]	P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing,, J. Acoust. Soc. Am., 107 (2000), 2418. doi: 10.1121/1.428628.
[16]	P. Roux, B. Roman and M. Fink, Time-reversal in an ultrasonic waveguide,, Appl. Phys. Lett., 70 (1997), 1811.
[17]	A. Tolstoy, K. Horoshenkov and M. Bin Ali, Detecting pipe changes via acoustic matched field processing,, Applied Acoustics, 70 (2009), 695. doi: 10.1016/j.apacoust.2008.08.007.
[18]	Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging,, Inv. Prob., 16 (2000), 1761. doi: 10.1088/0266-5611/16/6/311.

show all references

References:

[1]	T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a 3D homogeneous planar waveguide,, SIAM J. Appl. Math., 71 (2011), 753. doi: 10.1137/100806333.
[2]	T. Arens, D. Gintides and A. Lechleiter, Variational formulations for scattering in a three-dimensional acoustic waveguide,, Math. Meth. Appl. Sci., 31 (2007), 821. doi: 10.1002/mma.947.
[3]	L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation,, Inv. Prob., 24 (2008).
[4]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory. An Introduction,", Interaction of Mechanics and Mathematics, (2006).
[5]	F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects,, Inv. Prob., 22 (2006), 845. doi: 10.1088/0266-5611/22/3/007.
[6]	F. Cakoni and H. Haddar, Interior transmission problem for anisotropic media,, in, (2003), 613.
[7]	D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory,, Inv. Prob., 21 (2005), 383. doi: 10.1088/0266-5611/21/1/023.
[8]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region,, Inv. Prob., 12 (1996), 383. doi: 10.1088/0266-5611/12/4/003.
[9]	D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem,, Inverse Problems and Imaging, 1 (2007), 13.
[10]	D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems,, Inv. Prob., 13 (1997), 1477. doi: 10.1088/0266-5611/13/6/005.
[11]	K. Horoshenkov, R. Ashley and J. Blanksby, Determination of sewer roughness and sediment properties using acoustic techniques,, Water Science and Technology, 47 (2003), 87.
[12]	P. Monk and V. Selgas, Near field sampling type methods for the inverse fluid-solid interaction problem,, Inverse Problems and Imaging, 5 (2011), 465.
[13]	B. Pincon and K. Ramdani, Selective focusing on small scatterers in acoustic waveguides using time reversal mirrors,, Inv. Prob., 23 (2007), 1.
[14]	F. Podd, M. Ali, K. Horoshenkov, A. Wood, S. Tait, J. Boot, R. Long and A. Saul, Rapid sonic characterisation of sewer change and obstructions,, Water Science and Technology, 56 (2007), 131.
[15]	P. Roux and M. Fink, Time reversal in a waveguide: Study of the temporal and spatial focusing,, J. Acoust. Soc. Am., 107 (2000), 2418. doi: 10.1121/1.428628.
[16]	P. Roux, B. Roman and M. Fink, Time-reversal in an ultrasonic waveguide,, Appl. Phys. Lett., 70 (1997), 1811.
[17]	A. Tolstoy, K. Horoshenkov and M. Bin Ali, Detecting pipe changes via acoustic matched field processing,, Applied Acoustics, 70 (2009), 695. doi: 10.1016/j.apacoust.2008.08.007.
[18]	Y. Xu, C. Matawa and W. Lin, Generalized dual space indicator method for underwater imaging,, Inv. Prob., 16 (2000), 1761. doi: 10.1088/0266-5611/16/6/311.

[1]	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
[2]	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
[3]	Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
[4]	Roland Griesmaier. Reciprocity gap music imaging for an inverse scattering problem in two-layered media. Inverse Problems & Imaging, 2009, 3 (3) : 389-403. doi: 10.3934/ipi.2009.3.389
[5]	Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems & Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577
[6]	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
[7]	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
[8]	Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems & Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681
[9]	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
[10]	Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131
[11]	Pedro Serranho. A hybrid method for inverse scattering for Sound-soft obstacles in R3. Inverse Problems & Imaging, 2007, 1 (4) : 691-712. doi: 10.3934/ipi.2007.1.691
[12]	Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042
[13]	Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems & Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016
[14]	Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547
[15]	Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[16]	Teemu Tyni, Valery Serov. Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line. Inverse Problems & Imaging, 2019, 13 (1) : 159-175. doi: 10.3934/ipi.2019009
[17]	Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205
[18]	Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435
[19]	Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059
[20]	Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977

2018 Impact Factor: 1.469

American Institute of Mathematical Sciences