Advanced Search
Article Contents
Article Contents

The problem of detecting corrosion by an electric measurement revisited

Abstract Related Papers Cited by
  • We establish a logarithmic stability estimate for the problem of detecting corrosion by a single electric measurement. We give a proof based on an adaptation of the method initiated in [3] for solving the inverse problem of recovering the surface impedance of an obstacle from the scattering amplitude. The key idea consists in estimating accurately a lower bound of the local $L^2$-norm at the boundary, of the solution of the boundary value problem used in modeling the problem of detection corrosion by an electric measurement.
    Mathematics Subject Classification: Primary: 35R30.


    \begin{equation} \\ \end{equation}
  • [1]

    G. Alessandrini, L. Del Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse problems, 19 (2003), 973-984.doi: 10.1088/0266-5611/19/4/312.


    G. Alessandrini, E. Sincich and S. Vessella, Stable determination of surface impedance on a rough obstacle by far field data, Inverse Problems and Imaging, 7 (2013), 341-351.doi: 10.3934/ipi.2013.7.341.


    M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude, Math. Methods Appl. Sci., 36 (2013), 2429-2448.doi: 10.1002/mma.2762.


    M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math Anal. Appl., 343 (2008), 328-336.doi: 10.1016/j.jmaa.2008.01.066.


    L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of $C^{1,1}$ domains, Math. Model. Numer. Anal., 44 (2010), 715-735.doi: 10.1051/m2an/2010016.


    S. Chaabane, I. Fellah, M. Jaoua and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems, 20 (2004), 47-59.doi: 10.1088/0266-5611/20/1/003.


    S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements, Inverse Problems, 15 (1999), 1425-1438.doi: 10.1088/0266-5611/15/6/303.


    J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123.doi: 10.1142/S0218202508002620.


    J. Cheng, M. Choulli and X. Yang, An iterative BEM for the inverse problem of detecting corrosion in a pipe, Numer. Math. J. Chinese Univ., 14 (2005), 252-266.


    M. Choulli, Stability estimates for an inverse elliptic problem, J. Inverse Ill-Posed Probl., 10 (2002), 601-610.doi: 10.1515/jiip.2002.10.6.601.


    M. Choulli, An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-Posed Probl., 12 (2004), 349-367.doi: 10.1515/1569394042248247.


    M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, SMAI-Springer, Berlin, 2009.doi: 10.1007/978-3-642-02460-3.


    M. Choulli and F. Triki, New stability estimates for the inverse medium problem with internal data, SIAM J. Math. Anal., 47 (2015), 1778-1799.doi: 10.1137/140988577.


    M. Choulli, Applications of elliptic Carleman inequalities, to appear in BCAM SpringerBriefs.


    D. Fasino and G. Inglese, An inverse Robin problem for Laplace's equation: Theoretical and numerical methods}, Inverse Problems, 15 (1999), 41-48.doi: 10.1088/0266-5611/15/1/008.


    G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994.doi: 10.1088/0266-5611/13/4/006.


    E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements, SIAM J. Math. Anal., 38 (2006), 434-451.doi: 10.1137/050631513.


    E. Sincich, Lipschitz stability for the inverse Robin problem, Inverse Problems, 23 (2007), 1311-1326.doi: 10.1088/0266-5611/23/3/027.


    E. Sincich, Smoothness dependent stability in corrosion detection, J. Math. Anal. Appl., 426 (2015), 364-379.doi: 10.1016/j.jmaa.2014.10.036.

  • 加载中

Article Metrics

HTML views() PDF downloads(62) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint