June  2017, 11(3): 521-538. doi: 10.3934/ipi.2017024

Recovering the boundary corrosion from electrical potential distribution using partial boundary data

1. 

School of Mathematics, Southeast University, Nanjing 210096, China

2. 

Department of Mathematics, Inha University, Incheon 22212, Korea

* Corresponding author: Prof. Dr. Jijun Liu, email: jjliu@seu.edu.cn

Received  January 2016 Revised  February 2017 Published  April 2017

Fund Project: This work is supported by NSFC grant No.11531005, No.11421110002, and the Fundamental Research Funds for the Central Universities (3207017455)

We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specified on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Hölder type based on some a priori assumptions on the unknown impedance and the electrical current input specified in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data.

Citation: 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
References:
[1]

G. AlessandriniL. D. 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. Google Scholar

[2]

F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Problems and Imaging, 1 (2007), 229-245. doi: 10.3934/ipi.2007.1.229. Google Scholar

[3]

F. CakoniY. Q. Hu and R. Kress, Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging, Inverse Problems, 30 (2014), 105009(19pp). doi: 10.1088/0266-5611/30/10/105009. Google Scholar

[4]

F. CakoniR. Kress and C. Schuft, Simultaneous reconstruction of shape and impedance in corrosion detection, Methods and Applications of Analysis, 17 (2010), 357-377. doi: 10.4310/MAA.2010.v17.n4.a3. Google Scholar

[5]

F. CakoniR. Kress and C. Schuft, Integral equations for shape and impedance reconstruction in corrosion detection, Inverse Problems, 26 (2010), 095012(24pp). doi: 10.1088/0266-5611/26/9/095012. Google Scholar

[6]

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

[7]

J. Cheng and M. Yamamoto, One new strategy for a-priori choice of regularizing parameters in Tikhonov regularization, Inverse Problems, 16 (2000), L31-L38. doi: 10.1088/0266-5611/16/4/101. Google Scholar

[8]

H. Eckel and R. Kress, Nonlinear integral equations for the inverse electrical impedance problem, Inverse Problems, 23 (2007), 475-491. doi: 10.1088/0266-5611/23/2/002. Google Scholar

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM Classics in Applied Mathematics Series, 2011. doi: 10.1137/1.9781611972030.ch1. Google Scholar

[10]

G. H. Hu and M. Yamamoto, Hölder stability estimate of Robin coeffcient in corrosion detection with a single boundary measurement, Inverse Problems, 31 (2015), 115009, 20pp. doi: 10.1088/0266-5611/31/11/115009. Google Scholar

[11]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, 2006. Google Scholar

[12]

P. G. Kaup and F. Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, J. Nondestruct. Eval., 14 (1995), 127-136. doi: 10.1007/BF01183118. Google Scholar

[13]

P. G. KaupF. Santosa and M. Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data, Inverse Problems, 12 (1996), 279-293. doi: 10.1088/0266-5611/12/3/008. Google Scholar

[14]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675. doi: 10.1137/0151085. Google Scholar

[15]

R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002. Google Scholar

[16]

M. M. Lavrentev, V. G. Romanov and S. P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis, AMS, Trans. Math. Monographs, Vol. 64, Providance, Rhode Island, 1986. Google Scholar

[17]

L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Ration Mech. Anal., 5 (1960), 35-45. doi: 10.1007/BF00252897. Google Scholar

[18]

W. Rundell, Recovering an obstacle and its impedance from Cauchy data, Inverse Problems, 24 (2008), 045003(22p). doi: 10.1088/0266-5611/24/4/045003. Google Scholar

[19]

T. Shigeta and D. L. Young, Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points, J. Comput. Phys., 228 (2009), 1903-1915. doi: 10.1016/j.jcp.2008.11.018. Google Scholar

[20]

T. WeiY. C. Hon and L. V. Ling, Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Engineering Analysis with Boundary Elements, 31 (2007), 373-385. doi: 10.1016/j.enganabound.2006.07.010. Google Scholar

[21]

J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1987. doi: 10.1017/CBO9781139171755. Google Scholar

show all references

References:
[1]

G. AlessandriniL. D. 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. Google Scholar

[2]

F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Problems and Imaging, 1 (2007), 229-245. doi: 10.3934/ipi.2007.1.229. Google Scholar

[3]

F. CakoniY. Q. Hu and R. Kress, Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging, Inverse Problems, 30 (2014), 105009(19pp). doi: 10.1088/0266-5611/30/10/105009. Google Scholar

[4]

F. CakoniR. Kress and C. Schuft, Simultaneous reconstruction of shape and impedance in corrosion detection, Methods and Applications of Analysis, 17 (2010), 357-377. doi: 10.4310/MAA.2010.v17.n4.a3. Google Scholar

[5]

F. CakoniR. Kress and C. Schuft, Integral equations for shape and impedance reconstruction in corrosion detection, Inverse Problems, 26 (2010), 095012(24pp). doi: 10.1088/0266-5611/26/9/095012. Google Scholar

[6]

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

[7]

J. Cheng and M. Yamamoto, One new strategy for a-priori choice of regularizing parameters in Tikhonov regularization, Inverse Problems, 16 (2000), L31-L38. doi: 10.1088/0266-5611/16/4/101. Google Scholar

[8]

H. Eckel and R. Kress, Nonlinear integral equations for the inverse electrical impedance problem, Inverse Problems, 23 (2007), 475-491. doi: 10.1088/0266-5611/23/2/002. Google Scholar

[9]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM Classics in Applied Mathematics Series, 2011. doi: 10.1137/1.9781611972030.ch1. Google Scholar

[10]

G. H. Hu and M. Yamamoto, Hölder stability estimate of Robin coeffcient in corrosion detection with a single boundary measurement, Inverse Problems, 31 (2015), 115009, 20pp. doi: 10.1088/0266-5611/31/11/115009. Google Scholar

[11]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Springer, 2006. Google Scholar

[12]

P. G. Kaup and F. Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, J. Nondestruct. Eval., 14 (1995), 127-136. doi: 10.1007/BF01183118. Google Scholar

[13]

P. G. KaupF. Santosa and M. Vogelius, Method for imaging corrosion damage in thin plates from electrostatic data, Inverse Problems, 12 (1996), 279-293. doi: 10.1088/0266-5611/12/3/008. Google Scholar

[14]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675. doi: 10.1137/0151085. Google Scholar

[15]

R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002. Google Scholar

[16]

M. M. Lavrentev, V. G. Romanov and S. P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis, AMS, Trans. Math. Monographs, Vol. 64, Providance, Rhode Island, 1986. Google Scholar

[17]

L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Ration Mech. Anal., 5 (1960), 35-45. doi: 10.1007/BF00252897. Google Scholar

[18]

W. Rundell, Recovering an obstacle and its impedance from Cauchy data, Inverse Problems, 24 (2008), 045003(22p). doi: 10.1088/0266-5611/24/4/045003. Google Scholar

[19]

T. Shigeta and D. L. Young, Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points, J. Comput. Phys., 228 (2009), 1903-1915. doi: 10.1016/j.jcp.2008.11.018. Google Scholar

[20]

T. WeiY. C. Hon and L. V. Ling, Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Engineering Analysis with Boundary Elements, 31 (2007), 373-385. doi: 10.1016/j.enganabound.2006.07.010. Google Scholar

[21]

J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1987. doi: 10.1017/CBO9781139171755. Google Scholar

Figure 1.  Geometric configuration for our direct problem (left) and the extension for dealing with the corner point A (right).
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