Abstract
We consider the inverse scattering problem of determining the shape of a crack with impedance boundary condition on one side from the complex conjugate of point sources placed on a closed curve which contains the crack. The near field factorization method is established to reconstruct the crack from the measurements on the same curve. Then, we deduce an inversion algorithm and present some numerical examples to show the viability of our method. So far as we know, when the incident waves are the point sources, the near field operator cannot be directly decomposed into the form that the factorization method required. However, we overcome this difficulty with the complex conjugate of point sources which are recently used in [Inverse Problems 30 (2014), Article ID 095005], [Inverse Problems 30 (2014), Article ID 045008] for the justification of the factorization method from near field data.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11401241
Award Identifier / Grant number: 61374085
Award Identifier / Grant number: 11171127
Award Identifier / Grant number: CZQ15020
Funding statement: The research is supported in part by Program for Changjiang Scholars and Innovative Research Team in University No. IRT13066. The first author is supported by NSFC grant No. 11401241, the second author is supported by NSFC grant No. 61374085, and the third author is supported by NSFC grant No. 11171127 and CZQ15020.
References
[1]
Angell T. and Kleinman R.,
The Helmholtz equation with
[2] Arens T., Gintides D. and Lechleiter A., Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM J. Appl. Math. 71 (2011), 753–772. 10.1137/100806333Search in Google Scholar
[3] Boukari Y. and Haddar H., The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging 4 (2013), 1123–1138. 10.3934/ipi.2013.7.1123Search in Google Scholar
[4] Cakoni F. and Colton D., The linear sampling method for cracks, Inverse Problems 19 (2003), 279–295. 10.1088/0266-5611/19/2/303Search in Google Scholar
[5] Cakoni F., Colton D. and Meng S., The inverse scattering problem for a penetrable cavity with internal measurements, Inverse Problems and Applications, Contemp. Math. 615, American Mathematical Society, Providence (2014), 71–88. 10.1090/conm/615/12246Search in Google Scholar
[6] Colton D. and Kress R., Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983. Search in Google Scholar
[7] Hu G., Yang J., Zhang B. and Zhang H., Near-field imaging of scattering obstacles with the factorization method, Inverse Problems 30 (2014), Article ID 095005. 10.1088/0266-5611/30/9/095005Search in Google Scholar
[8] Kirsch A., Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems 14 (1998), 1489–1512. 10.1088/0266-5611/14/6/009Search in Google Scholar
[9] Kirsch A., An integral equation for Maxwell’s equations in a layered medium with an application to the factorization method, J. Integral Equations Appl. 19 (2007), 333–357. 10.1216/jiea/1190905490Search in Google Scholar
[10] Kirsch A. and Gringerg N., The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. 10.1093/acprof:oso/9780199213535.001.0001Search in Google Scholar
[11] Kirsch A. and Ritter S., A linear sampling method for inverse scattering from an open arc, Inverse Problems 16 (2000), 89–105. 10.1088/0266-5611/16/1/308Search in Google Scholar
[12] Kress R., Frechet differentiability of the far field operator for scattering from a crack, J. Inverse Ill-Posed Probl. 3 (1995), 305–313. 10.1515/jiip.1995.3.4.305Search in Google Scholar
[13] Lechleiter A., The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging 3 (2009), 123–138. 10.3934/ipi.2009.3.123Search in Google Scholar
[14] Liu J. and Sini M., Reconstruction of cracks of different types from far-field measurements, Math. Methods Appl. Sci. 33 (2010), 950–973. 10.1002/mma.1203Search in Google Scholar
[15] Liu X., The factorization method for cavities, Inverse Problems 30 (2014), Article ID 015006. 10.1088/0266-5611/30/1/015006Search in Google Scholar
[16] Mclean W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. Search in Google Scholar
[17] Meng S., Haddar H. and Cakoni F., The factorization method for a cavity in an inhomogeneous medium, Inverse Problems 30 (2014), Article ID 045008. 10.1088/0266-5611/30/4/045008Search in Google Scholar
[18] Mönch L., On the inverse acoustic scattering problem by an open arc: On the sound-hard case, Inverse Problems 13 (1997), 1379–1392. 10.1088/0266-5611/13/5/017Search in Google Scholar
[19] Qin H. and Colton D., The inverse scattering problem for cavities, Appl. Numer. Math. 62 (2012), 699–708. 10.1016/j.apnum.2010.10.011Search in Google Scholar
[20] Zeev N. and Cakoni F., The identification for thin dielectric objects from far field or near field scattering data, SIAM J. Appl. Math. 69 (2009), 1024–1042. 10.1137/070711542Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- A finite element method for the inverse problem of boundary data recovery in an oxygen balance model
- On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation
- Application of the factorization method to retrieve a crack from near field data
- Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions
- The variational formulation of an inverse problem for multidimensional nonlinear time-dependent Schrödinger equation
- Integral identity for a class of ill-posed problems generated by a parabolic equation
- A meshless method to the solution of an ill-posed problem
- About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication
- Error analysis for the operator marching method applied to range dependent waveguides
Articles in the same Issue
- Frontmatter
- A finite element method for the inverse problem of boundary data recovery in an oxygen balance model
- On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation
- Application of the factorization method to retrieve a crack from near field data
- Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions
- The variational formulation of an inverse problem for multidimensional nonlinear time-dependent Schrödinger equation
- Integral identity for a class of ill-posed problems generated by a parabolic equation
- A meshless method to the solution of an ill-posed problem
- About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication
- Error analysis for the operator marching method applied to range dependent waveguides