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Conjugate Gradient Method for Estimation of Robin Coefficients

Published online by Cambridge University Press:  28 May 2015

Yan-Bo Ma
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, P. R. China
Fu-Rong Lin
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, P. R. China
Corresponding
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Abstract

We consider a Robin inverse problem associated with the Laplace equation, which is a severely ill-posed and nonlinear. We formulate the problem as a boundary integral equation, and introduce a functional of the Robin coefficient as a regularisation term. A conjugate gradient method is proposed for solving the consequent regularised nonlinear least squares problem. Numerical examples are presented to illustrate the effectiveness of the proposed method.

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Research Article
Copyright
Copyright © Global-Science Press 2014

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References

[1]Alessandrini, G., Piero, L. Del and Rondi, L., Stable determination of corrosion by single electrostatic boundary measurement, Inverse Problems 19, 973984 (2003).CrossRefGoogle Scholar
[2]Atkinson, K., The Numerical Solution of Integral Equation ofthe Second Kind. Cambridge University Press, 1997.CrossRefGoogle Scholar
[3]Busenberg, S. and Fang, W., Identification of semiconductor contact resistivity, Quart. Appl. Math. 49, 639649 (1991).CrossRefGoogle Scholar
[4]Cakoni, F. and Kress, R., Integral equations for inverse problems in corrosion detection from partial Cauchydata, Inverse Problems and Imaging 1, 229245 (2007).Google Scholar
[5]Chaabane, S., Elhechmi, C. and Jaoua, M., Astable recoverymethod for the Robin inverse problem, Mathematics and Computers in Simulation 66, 367383 (2004).CrossRefGoogle Scholar
[6]Chaabane, S., Fellah, I., Jaoua, M. and Leblond, J., Logarithmic stabilityestimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems 20, 4759 (2004).CrossRefGoogle Scholar
[7]Chaabane, S., Ferchichi, J. and Kunisch, K., Differentiability properties of L1-tracking functional and application to the Robin inverse problem, Inverse Problems 20, 10831097 (2004).CrossRefGoogle Scholar
[8]Chaabane, S., Feki, I. and Mars, N., Numerical reconstruction of a piecewise constant Robin parameter in the two- or three-dimensional case, Inverse Problems 28, 065016, 19pp. (2012).CrossRefGoogle Scholar
[9]Chaabane, S. and Jaoua, M., Identification of Robin coefficients by means of boundary measurements, Inverse Problems 15, 14251438 (1999).CrossRefGoogle Scholar
[10]Delves, L.M. and Mohamed, J.L., Computational Methods for Integral Equations. Cambridge University Press, 1985.CrossRefGoogle Scholar
[11]Fang, W. and Cumberbatch, E., Inverse problems for MOSFET contact resistivity, SIAM J. Appl. Math. 52, 699709 (1992).CrossRefGoogle Scholar
[12]Fang, W. and Lu, M., A fast wavelet collocation method for an inverse boundary value problem, Int. J. Numer. Methods Eng. 59, 15631585 (2004).CrossRefGoogle Scholar
[13]Fasino, D. and Inglese, G., An inverse problem for Laplace's equation: theoretical results and numerical methods, Inverse Problems 15, 4148 (1999).CrossRefGoogle Scholar
[14]Fasino, D. and Inglese, G., Discrete methods in the study of an inverse problem for Laplace's equation, SIAM J. Numer. Anal. 19, 105118 (1999).CrossRefGoogle Scholar
[15]Fang, W. and Zeng, X., A direct soluton of the Robin inverse problem, J. Integral Equations Appl. 4, 545557 (2009).CrossRefGoogle Scholar
[16]Gilbarg, D.A. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order. Springer Verlag, Berlin, 2001.Google Scholar
[17]Inglese, G., An inverse problem in corrosion detection, Inverse Problems 13, 977994 (1997).CrossRefGoogle Scholar
[18]Jin, B., Conjugate gradient method for the Robin inverse problem associated with the Laplace equation, Int. J. Numer. Meth. Engng. 71, 433453 (2007).CrossRefGoogle Scholar
[19]Jin, B. and Zou, J., Inversion of Robin coefficient by a spectral stochastic finite element approach, J. Comp. Phys. 227, 32823306 (2008).CrossRefGoogle Scholar
[20]Jin, B. and Zou, J., Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim. 48, 19772002 (2009).CrossRefGoogle Scholar
[21]Kaup, P.G. and Santosa, F., Nondestructive evaluation ofcorrosion damage using electrostatic measurements, J. Nondestruct. Eval. 14, 127136 (1995).CrossRefGoogle Scholar
[22]Kress, R., Linear Integral Equations (2nd edition). Springer, New York, 1999.CrossRefGoogle Scholar
[23]Lin, F.R. and Fang, W., Alinear integral equation approach to the Robin inverse problem, Inverse Problems 21, 17571772 (2005).CrossRefGoogle Scholar
[24]Loh, W.H., Swirhun, S.E., Schreyer, T.A., Swanson, R.M. and Saraswat, K.C., Modeling and measurement of contact resistances, IEEE Transactions on Electron Devices 34, 512524 (1987).CrossRefGoogle Scholar
[25]Loh, W.H., Saraswat, K. and Dutton, R.W., Analysis and scaling of Kelvin resistors for extraction of specific contact resistivity, IEEE Electron. Device Lett. 6, 105108 (1985).CrossRefGoogle Scholar
[26]Mazya, V.G., Boundary integral equations analysis IV, in Encyclopaedia of Mathematical Sciences, Volume 27, (Mazya, V.G. and Nikol'skii, S.M., Eds.). Springer, New York, pp. 127222, 1991.Google Scholar
[27]Nocedal, J. and Wright, S., Numerical Optimization. Springer, New York, 2006.Google Scholar
[28]Santosa, F., Vogelius, M. and Xu, J.M., An effective nonlinear boundary condition for corroding surface identification of damage based on steady state electric data, Z. Angew. Math. Phys. 49, 656679 (1998).CrossRefGoogle Scholar

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