Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
EnglishFrançais
East Asian Journal on Applied Mathematics East Asian Journal on Applied Mathematics

Article contents

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
E-mail address:
g_ybma@stu.edu.cn
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

45Q05 65N21 Robin inverse problem ill-posedness boundary integral equation regularisation conjugate gradient method
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.
If you should have access and can't see this content please contact technical support.

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).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, 639–649 (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, 229–245 (2007).Google 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).CrossRefGoogle 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).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, 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).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, 105–118 (1999).CrossRefGoogle Scholar
[15]Fang, W. and Zeng, X., A direct soluton of the Robin inverse problem, J. Integral Equations Appl. 4, 545–557 (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, 977–994 (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, 433–453 (2007).CrossRefGoogle 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).CrossRefGoogle Scholar
[20]Jin, B. and Zou, J., Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim. 48, 1977–2002 (2009).CrossRefGoogle 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.CrossRefGoogle 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).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. 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).CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 21 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 25th January 2021. This data will be updated every 24 hours.

2 Cited by
Hostname: page-component-898fc554b-gwqw7 Total loading time: 0.349 Render date: 2021-01-25T15:28:00.295Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Conjugate Gradient Method for Estimation of Robin Coefficients
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Conjugate Gradient Method for Estimation of Robin Coefficients
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Conjugate Gradient Method for Estimation of Robin Coefficients
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *