February  2013, 7(1): 291-303. doi: 10.3934/ipi.2013.7.291

A decomposition method for an interior inverse scattering problem

1. 

Department of Mathematical Sciences, Delaware State University, Dover, DE 19901, United States, United States

2. 

Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, United States

Received  February 2012 Revised  May 2012 Published  February 2013

We consider an interior inverse scattering problem of reconstructing the shape of a cavity. The measurements are the scattered fields on a curve inside the cavity due to one point source. We employ the decomposition method to reconstruct the cavity and present some convergence results. Numerical examples are provided to show the viability of the method.
Citation: 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
References:
[1]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory,, Inverse Problems, 21 (2005), 383.  doi: 10.1088/0266-5611/21/1/023.  Google Scholar

[2]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Second edition, 93 (1998).   Google Scholar

[3]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/3/R01.  Google Scholar

[4]

D. Colton and P. Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region,, SIAM J. Appl. Math., 45 (1985), 1039.  doi: 10.1137/0145064.  Google Scholar

[5]

D. Colton and P. Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region. II,, SIAM J. Appl. Math., 46 (1986), 506.  doi: 10.1137/0146034.  Google Scholar

[6]

D. Colton and B. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA J. Appl. Math., 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar

[7]

M. Di Cristo and J. Sun, An inverse scattering problem for a partially coated buried obstacle,, Inverse Problems, 22 (2006), 2331.  doi: 10.1088/0266-5611/22/6/025.  Google Scholar

[8]

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

[9]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008).   Google Scholar

[10]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering,, Inverse Problems, 9 (1993), 285.   Google Scholar

[11]

A. Kirsch and R. Kress, An optimization method in inverse acoustic scattering,, in, (1987), 3.   Google Scholar

[12]

A. Kirsch, R. Kress, P. Monk and A. Zinn, Two methods for solving the inverse acoustic scattering problem,, Inverse Problems, 4 (1988), 749.   Google Scholar

[13]

R. Kress, "Uniqueness in Inverse Obstacle Scattering for Electromagnetic Waves,", Proceedings of the URSI General Assembly, (2002).   Google Scholar

[14]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares,, Inverse Problems, 19 (2003).  doi: 10.1088/0266-5611/19/6/056.  Google Scholar

[15]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions,, SIAM Journal of Optimization, 9 (1998), 112.  doi: 10.1137/S1052623496303470.  Google Scholar

[16]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000).   Google Scholar

[17]

P. Monk and J. Sun, Inverse scattering using finite elements and gap reciprocity,, Inverse Prob. Imaging, 1 (2007), 643.  doi: 10.3934/ipi.2007.1.643.  Google Scholar

[18]

R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scattering,, Inverse Problems, 10 (1994), 431.   Google Scholar

[19]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/3/035005.  Google Scholar

[20]

H. Qin and D. Colton, The inverse scattering problem for cavities,, Applied Numerical Mathematics, 62 (2012), 699.  doi: 10.1016/j.apnum.2010.10.011.  Google Scholar

[21]

H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition,, Advances in Computational Mathematics, 36 (2012), 157.  doi: 10.1007/s10444-011-9179-2.  Google Scholar

[22]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/12/125002.  Google Scholar

show all references

References:
[1]

D. Colton and H. Haddar, An application of the reciprocity gap functional to inverse scattering theory,, Inverse Problems, 21 (2005), 383.  doi: 10.1088/0266-5611/21/1/023.  Google Scholar

[2]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Second edition, 93 (1998).   Google Scholar

[3]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering,, Inverse Problems, 22 (2006).  doi: 10.1088/0266-5611/22/3/R01.  Google Scholar

[4]

D. Colton and P. Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region,, SIAM J. Appl. Math., 45 (1985), 1039.  doi: 10.1137/0145064.  Google Scholar

[5]

D. Colton and P. Monk, A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region. II,, SIAM J. Appl. Math., 46 (1986), 506.  doi: 10.1137/0146034.  Google Scholar

[6]

D. Colton and B. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA J. Appl. Math., 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar

[7]

M. Di Cristo and J. Sun, An inverse scattering problem for a partially coated buried obstacle,, Inverse Problems, 22 (2006), 2331.  doi: 10.1088/0266-5611/22/6/025.  Google Scholar

[8]

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

[9]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008).   Google Scholar

[10]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering,, Inverse Problems, 9 (1993), 285.   Google Scholar

[11]

A. Kirsch and R. Kress, An optimization method in inverse acoustic scattering,, in, (1987), 3.   Google Scholar

[12]

A. Kirsch, R. Kress, P. Monk and A. Zinn, Two methods for solving the inverse acoustic scattering problem,, Inverse Problems, 4 (1988), 749.   Google Scholar

[13]

R. Kress, "Uniqueness in Inverse Obstacle Scattering for Electromagnetic Waves,", Proceedings of the URSI General Assembly, (2002).   Google Scholar

[14]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares,, Inverse Problems, 19 (2003).  doi: 10.1088/0266-5611/19/6/056.  Google Scholar

[15]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions,, SIAM Journal of Optimization, 9 (1998), 112.  doi: 10.1137/S1052623496303470.  Google Scholar

[16]

W. McLean, "Strongly Elliptic Systems and Boundary Integral Equations,", Cambridge University Press, (2000).   Google Scholar

[17]

P. Monk and J. Sun, Inverse scattering using finite elements and gap reciprocity,, Inverse Prob. Imaging, 1 (2007), 643.  doi: 10.3934/ipi.2007.1.643.  Google Scholar

[18]

R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scattering,, Inverse Problems, 10 (1994), 431.   Google Scholar

[19]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/3/035005.  Google Scholar

[20]

H. Qin and D. Colton, The inverse scattering problem for cavities,, Applied Numerical Mathematics, 62 (2012), 699.  doi: 10.1016/j.apnum.2010.10.011.  Google Scholar

[21]

H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition,, Advances in Computational Mathematics, 36 (2012), 157.  doi: 10.1007/s10444-011-9179-2.  Google Scholar

[22]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/12/125002.  Google Scholar

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