Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-14T17:33:36.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Conjugate Gradient Method for Estimation of Robin Coefficients

Published online by Cambridge University Press:  28 May 2015

Yan-Bo Ma and
Fu-Rong Lin
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 author. Email Address: g_ybma@stu.edu.cn
Corresponding author. Email Address: frlin@stu.edu.cn
Get access

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.

Keywords

45Q0565N21Robin inverse problemill-posednessboundary integral equationregularisationconjugate gradient method

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 2 , May 2014, pp. 189 - 204
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Alessandrini, G., Piero, L. Del and Rondi, L., Stable determination of corrosion by single electrostatic boundary measurement, Inverse Problems 19, 973–984 (2003).Google Scholar
[2]Atkinson, K., The Numerical Solution of Integral Equation ofthe Second Kind. Cambridge University Press, 1997.Google Scholar
[3]Busenberg, S. and Fang, W., Identification of semiconductor contact resistivity, Quart. Appl. Math. 49, 639–649 (1991).Google Scholar
[4]Cakoni, F. and Kress, R., Integral equations for inverse problems in corrosion detection from partial Cauchydata, Inverse Problems and Imaging 1, 229–245 (2007).CrossRefGoogle Scholar
[5]Chaabane, S., Elhechmi, C. and Jaoua, M., Astable recoverymethod for the Robin inverse problem, Mathematics and Computers in Simulation 66, 367–383 (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, 47–59 (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, 1083–1097 (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).Google Scholar
[9]Chaabane, S. and Jaoua, M., Identification of Robin coefficients by means of boundary measurements, Inverse Problems 15, 1425–1438 (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, 699–709 (1992).Google Scholar
[12]Fang, W. and Lu, M., A fast wavelet collocation method for an inverse boundary value problem, Int. J. Numer. Methods Eng. 59, 1563–1585 (2004).CrossRefGoogle Scholar
[13]Fasino, D. and Inglese, G., An inverse problem for Laplace's equation: theoretical results and numerical methods, Inverse Problems 15, 41–48 (1999).Google 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, 105–118 (1999).Google Scholar
[15]Fang, W. and Zeng, X., A direct soluton of the Robin inverse problem, J. Integral Equations Appl. 4, 545–557 (2009).Google 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, 977–994 (1997).Google Scholar
[18]Jin, B., Conjugate gradient method for the Robin inverse problem associated with the Laplace equation, Int. J. Numer. Meth. Engng. 71, 433–453 (2007).Google Scholar
[19]Jin, B. and Zou, J., Inversion of Robin coefficient by a spectral stochastic finite element approach, J. Comp. Phys. 227, 3282–3306 (2008).Google Scholar
[20]Jin, B. and Zou, J., Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim. 48, 1977–2002 (2009).Google Scholar
[21]Kaup, P.G. and Santosa, F., Nondestructive evaluation ofcorrosion damage using electrostatic measurements, J. Nondestruct. Eval. 14, 127–136 (1995).CrossRefGoogle Scholar
[22]Kress, R., Linear Integral Equations (2nd edition). Springer, New York, 1999.Google Scholar
[23]Lin, F.R. and Fang, W., Alinear integral equation approach to the Robin inverse problem, Inverse Problems 21, 1757–1772 (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, 512–524 (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, 105–108 (1985).Google 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. 127–222, 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, 656–679 (1998).Google Scholar