American Institute of Mathematical Sciences

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

The problem of detecting corrosion by an electric measurement revisited

Mourad Choulli 1, and  Aymen Jbalia 2,

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.
Keywords: boundary measurement., detecting corrosion, Logarithmic stability estimate.
Mathematics Subject Classification: Primary: 35R3.
Citation: Mourad Choulli, Aymen Jbalia. The problem of detecting corrosion by an electric measurement revisited. Discrete & 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.  doi: 10.1088/0266-5611/19/4/312.  Google Scholar

[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.  doi: 10.3934/ipi.2013.7.341.  Google Scholar

[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.  doi: 10.1002/mma.2762.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.01.066.  Google Scholar

[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.  doi: 10.1051/m2an/2010016.  Google Scholar

[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.  doi: 10.1088/0266-5611/20/1/003.  Google Scholar

[7]

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

[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.  doi: 10.1142/S0218202508002620.  Google Scholar

[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.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

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.  doi: 10.1088/0266-5611/19/4/312.  Google Scholar

[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.  doi: 10.3934/ipi.2013.7.341.  Google Scholar

[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.  doi: 10.1002/mma.2762.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2008.01.066.  Google Scholar

[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.  doi: 10.1051/m2an/2010016.  Google Scholar

[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.  doi: 10.1088/0266-5611/20/1/003.  Google Scholar

[7]

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

[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.  doi: 10.1142/S0218202508002620.  Google Scholar

[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.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[1]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[2]

Eva Sincich, Mourad Sini. Local stability for soft obstacles by a single measurement. Inverse Problems & Imaging, 2008, 2 (2) : 301-315. doi: 10.3934/ipi.2008.2.301
[3]

Jijun Liu, Gen Nakamura. Recovering the boundary corrosion from electrical potential distribution using partial boundary data. Inverse Problems & Imaging, 2017, 11 (3) : 521-538. doi: 10.3934/ipi.2017024
[4]

Max Fathi, Emanuel Indrei, Michel Ledoux. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6835-6853. doi: 10.3934/dcds.2016097
[5]

Michel Cristofol, Shumin Li, Eric Soccorsi. Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary. Mathematical Control & Related Fields, 2016, 6 (3) : 407-427. doi: 10.3934/mcrf.2016009
[6]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[7]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009
[8]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[9]

Shitao Liu. Recovery of the sound speed and initial displacement for the wave equation by means of a single Dirichlet boundary measurement. Evolution Equations & Control Theory, 2013, 2 (2) : 355-364. doi: 10.3934/eect.2013.2.355
[10]

L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630
[11]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[12]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176
[13]

Peter C. Gibson. On the measurement operator for scattering in layered media. Inverse Problems & Imaging, 2017, 11 (1) : 87-97. doi: 10.3934/ipi.2017005
[14]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229
[15]

Claire Chainais-Hillairet, Ingrid Lacroix-Violet. On the existence of solutions for a drift-diffusion system arising in corrosion modeling. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 77-92. doi: 10.3934/dcdsb.2015.20.77
[16]

Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81
[17]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[18]

Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems & Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135
[19]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
[20]

Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences