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 and 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), 383-398. doi: 10.1088/0266-5611/21/1/023.

[2]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.

[3]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66. doi: 10.1088/0266-5611/22/3/R01.

[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), 1039-1053. doi: 10.1137/0145064.

[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), 506-523. doi: 10.1137/0146034.

[6]

D. Colton and B. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253.

[7]

M. Di Cristo and J. Sun, An inverse scattering problem for a partially coated buried obstacle, Inverse Problems, 22 (2006), 2331-2350. doi: 10.1088/0266-5611/22/6/025.

[8]

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

[9]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.

[10]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299.

[11]

A. Kirsch and R. Kress, An optimization method in inverse acoustic scattering, in "Boundary Elements IX, Vol. 3" (eds. C. A. Brebbia, et al.) (Stuttgart, 1987), Comput. Mech., Southampton, (1987), 3-18.

[12]

A. Kirsch, R. Kress, P. Monk and A. Zinn, Two methods for solving the inverse acoustic scattering problem, Inverse Problems, 4 (1988), 749-770.

[13]

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

[14]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91-S104. doi: 10.1088/0266-5611/19/6/056.

[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), 112-147. doi: 10.1137/S1052623496303470.

[16]

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

[17]

P. Monk and J. Sun, Inverse scattering using finite elements and gap reciprocity, Inverse Prob. Imaging, 1 (2007), 643-660. doi: 10.3934/ipi.2007.1.643.

[18]

R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems, 10 (1994), 431-447.

[19]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17 pp. doi: 10.1088/0266-5611/27/3/035005.

[20]

H. Qin and D. Colton, The inverse scattering problem for cavities, Applied Numerical Mathematics, 62 (2012), 699-708. doi: 10.1016/j.apnum.2010.10.011.

[21]

H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Advances in Computational Mathematics, 36 (2012), 157-174. doi: 10.1007/s10444-011-9179-2.

[22]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17 pp. doi: 10.1088/0266-5611/27/12/125002.

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), 383-398. doi: 10.1088/0266-5611/21/1/023.

[2]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Second edition, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998.

[3]

D. Colton and R. Kress, Using fundamental solutions in inverse scattering, Inverse Problems, 22 (2006), R49-R66. doi: 10.1088/0266-5611/22/3/R01.

[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), 1039-1053. doi: 10.1137/0145064.

[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), 506-523. doi: 10.1137/0146034.

[6]

D. Colton and B. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253.

[7]

M. Di Cristo and J. Sun, An inverse scattering problem for a partially coated buried obstacle, Inverse Problems, 22 (2006), 2331-2350. doi: 10.1088/0266-5611/22/6/025.

[8]

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

[9]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.

[10]

A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), 285-299.

[11]

A. Kirsch and R. Kress, An optimization method in inverse acoustic scattering, in "Boundary Elements IX, Vol. 3" (eds. C. A. Brebbia, et al.) (Stuttgart, 1987), Comput. Mech., Southampton, (1987), 3-18.

[12]

A. Kirsch, R. Kress, P. Monk and A. Zinn, Two methods for solving the inverse acoustic scattering problem, Inverse Problems, 4 (1988), 749-770.

[13]

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

[14]

R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Problems, 19 (2003), S91-S104. doi: 10.1088/0266-5611/19/6/056.

[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), 112-147. doi: 10.1137/S1052623496303470.

[16]

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

[17]

P. Monk and J. Sun, Inverse scattering using finite elements and gap reciprocity, Inverse Prob. Imaging, 1 (2007), 643-660. doi: 10.3934/ipi.2007.1.643.

[18]

R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Problems, 10 (1994), 431-447.

[19]

H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17 pp. doi: 10.1088/0266-5611/27/3/035005.

[20]

H. Qin and D. Colton, The inverse scattering problem for cavities, Applied Numerical Mathematics, 62 (2012), 699-708. doi: 10.1016/j.apnum.2010.10.011.

[21]

H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Advances in Computational Mathematics, 36 (2012), 157-174. doi: 10.1007/s10444-011-9179-2.

[22]

F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Problems, 27 (2011), 125002, 17 pp. doi: 10.1088/0266-5611/27/12/125002.

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