A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements.

*(English)*Zbl 1149.78005The authors establish an asymptotic representation formula for the steady state current perturbations caused by internal corrosive boundary parts of small surface measure. Here the corrosive boundary parts are a subset of the boundary of an inaccessible simply connected 2D region whose closure is a subset of a larger simply connected 2D region whose boundary is accessible and to which a voltage is applied. Based on this formula, the authors design a noniterative method of MUSIC (multiple signal classification) type to localize the corrosive parts from voltage-to-current observations. Numerical experiments are performed to test the viability of the algorithm. The algorithm is shown to perform well in the presence of relatively high noise ratios, provided the corrosion coefficients are sufficiently large.

Reviewer: Cornelis van der Mee (Cagliari)

##### MSC:

78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |

35R30 | Inverse problems for PDEs |

##### Keywords:

noniterative method of multiple signal classification type; electrostatic boundary measurements; voltage-to-current observation
PDF
BibTeX
XML
Cite

\textit{H. Ammari} et al., Numer. Math. 108, No. 4, 501--528 (2008; Zbl 1149.78005)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Ammari H. and Kang H. (2003). High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. SIAM J. Math. Anal. 34: 1152–1166 · Zbl 1036.35050 |

[2] | Banks H.T., Joyner M.L., Wincheski B. and Winfree W.P. (2002). Real time computational algorithms for eddy-current-based damage detection. Inverse Probl. 18: 795–823 · Zbl 1104.65321 |

[3] | Brühl M., Hanke M. and Vogelius M.S. (2003). A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. 93: 635–654 · Zbl 1016.65079 |

[4] | Buttazzo G. and Kohn R.V. (1988). Reinforcement by a thin layer with oscillating thickness. Appl. Math. Opt. 16: 247–261 |

[5] | Cheney M. (2001). The linear sampling method and the MUSIC algorithm. Inverse Probl. 17: 591–595 · Zbl 0988.00112 |

[6] | Colton D. and Kirsch A. (1996). A simple method for solving inverse scattering problems in the resonance region. Inverse Probl. 12: 383–393 · Zbl 0859.35133 |

[7] | Folland G.B. (1976). Introduction to Partial Differential Equations. Princeton University Press, Princeton · Zbl 0325.35001 |

[8] | Inglese G. (1997). An inverse problem in corrosion detection. Inverse Probl. 13: 977–994 · Zbl 0882.35133 |

[9] | Luong B. and Santosa F. (1998). Quantitative imaging of corrosion inplates by eddy current methods. SIAM J. Appl. Math. 58: 1509–1531 · Zbl 0926.35146 |

[10] | Kang H. and Seo J.K. (1996). Layer potential technique for the inverse conductivity problem. Inverse Probl. 12: 267–278 · Zbl 0857.35134 |

[11] | Kang, H., Seo, J.K.: Recent progress in the inverse conductivity problem with single measurement. In: Inverse Problems and Related Fields. CRC, Boca Raton, pp. 69–80 (2000) · Zbl 0963.35199 |

[12] | Kaup P. and Santosa F. (1995). Nondestructive evaluation of corrosion damage using electrostatic measurements. J. Nondestruct. Eval. 14: 127–136 |

[13] | Kaup P., Santosa F. and Vogelius M. (1996). A method for imaging corrosion damage in thin plates from electrostatic data. Inverse Probl. 12: 279–293 · Zbl 0851.35134 |

[14] | Kirsch A. (2002). The MUSIC algorithm and the factorisation method in inverse scattering theory for inhomogeneous media. Inverse Probl. 18: 1025–1040 · Zbl 1027.35158 |

[15] | Stein E.M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton · Zbl 0207.13501 |

[16] | Therrien C.W. (1992). Discrete Random Signals and Statistical Signal Processing. Prentice-Hall, Englewood Cliffs · Zbl 0747.94004 |

[17] | Vogelius M. and Xu J. (1998). A nonlinear elliptic boundary value problem related to corrosion modelling. Q. Appl. Math. 56: 479–505 · Zbl 0954.35067 |

[18] | Yang X., Choulli M. and Cheng J. (2005). An iterative BEM for the inverse problem of detecting corrosion in a pipe. Numer. Math. J. Chin. Univ. 14: 252–266 · Zbl 1121.35144 |

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.