June  2016, 9(3): 643-650. doi: 10.3934/dcdss.2016018

The problem of detecting corrosion by an electric measurement revisited

1. 

Institut Élie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine-Metz, Boulevard des Aiguillettes, BP 70239, 54506 Vandoeuvre les Nancy cedex, Ile du Saulcy, 57045 Metz cedex 01, France

2. 

Faculté des Sciences de Bizerte, Département des Mathématiques, 7021 Jarzouna Bizerte, Tunisia

Received  February 2015 Revised  September 2015 Published  April 2016

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.
Citation: Mourad Choulli, Aymen Jbalia. The problem of detecting corrosion by an electric measurement revisited. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 643-650. doi: 10.3934/dcdss.2016018
References:
[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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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

[12]

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

[13]

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.

[14]

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

[15]

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.

[16]

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

[17]

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.

[18]

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

[19]

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

show all references

References:
[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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10]

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.

[11]

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

[12]

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

[13]

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.

[14]

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

[15]

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.

[16]

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

[17]

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.

[18]

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

[19]

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

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