Application of the factorization method to retrieve a crack from near field data.

*(English)*Zbl 06641005Summary: 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.

##### MSC:

35C15 | Integral representations of solutions to PDEs |

35Q60 | PDEs in connection with optics and electromagnetic theory |

78A45 | Diffraction, scattering |

PDF
BibTeX
XML
Cite

\textit{J. Guo} et al., J. Inverse Ill-Posed Probl. 24, No. 5, 527--541 (2016; Zbl 06641005)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Angell T. and Kleinman R., The Helmholtz equation with \({L^{2}}\)-boundary values, SIAM J. Math. Anal. 16 (1985), 259-278. · Zbl 0577.35026 |

[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. · Zbl 1228.35164 |

[3] | Boukari Y. and Haddar H., The factorization method applied to cracks with impedance boundary conditions, Inverse Probl. Imaging 4 (2013), 1123-1138. · Zbl 1302.35430 |

[4] | Cakoni F. and Colton D., The linear sampling method for cracks, Inverse Problems 19 (2003), 279-295. · Zbl 1171.35487 |

[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. · Zbl 1330.35524 |

[6] | Colton D. and Kress R., Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983. · Zbl 0522.35001 |

[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. · Zbl 1302.35284 |

[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. · Zbl 0919.35147 |

[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. · Zbl 1136.78310 |

[10] | Kirsch A. and Gringerg N., The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. |

[11] | Kirsch A. and Ritter S., A linear sampling method for inverse scattering from an open arc, Inverse Problems 16 (2000), 89-105. · Zbl 0968.35129 |

[12] | Kress R., Frechet differentiability of the far field operator for scattering from a crack, J. Inverse Ill-Posed Probl. 3 (1995), 305-313. · Zbl 0846.35146 |

[13] | Lechleiter A., The factorization method is independent of transmission eigenvalues, Inverse Probl. Imaging 3 (2009), 123-138. · Zbl 1184.78020 |

[14] | Liu J. and Sini M., Reconstruction of cracks of different types from far-field measurements, Math. Methods Appl. Sci. 33 (2010), 950-973. · Zbl 1193.35252 |

[15] | Liu X., The factorization method for cavities, Inverse Problems 30 (2014), Article ID 015006. · Zbl 1292.78014 |

[16] | Mclean W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. · Zbl 0948.35001 |

[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. · Zbl 1287.35102 |

[18] | Mönch L., On the inverse acoustic scattering problem by an open arc: On the sound-hard case, Inverse Problems 13 (1997), 1379-1392. · Zbl 0894.35079 |

[19] | Qin H. and Colton D., The inverse scattering problem for cavities, Appl. Numer. Math. 62 (2012), 699-708. · Zbl 1241.78027 |

[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. · Zbl 1173.35741 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.