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An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

Published online by Cambridge University Press:  20 August 2015

Houde Han ,
Leevan Ling  and
Tomoya Takeuchi
Houde Han*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Leevan Ling*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Tomoya Takeuchi*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8212, USA
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]

Abstract

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Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

Keywords

65N1265N1565N21Inverse problemstabilityerror boundsTikhonov regularizationmethod of fundamental solution
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 878 - 896
Copyright
Copyright © Global Science Press Limited 2011

References

[1]Inglese, G., An inverse problem in corrosion detection, Inverse. Probl., 13 (4) (1997), 977–994.CrossRefGoogle Scholar
[2]Buttazzo, G., and Kohn, R. V., Reinforcement by a thin layer with oscillating thickness, Appl. Math. Opt., 16 (3) (1987), 247–261.Google Scholar
[3]Inglese, G., and Santosa, F., An Inverse Problem in Corrosion Detection, No. 75, Pubblicazioni dellIstituto di Analisi Globale ed Applicazioni del CNR, Firenze, 1995.Google Scholar
[4]Protter, M. H., and Weinberger, H. F., Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984, corrected reprint of the 1967 original.Google Scholar
[5]Alessandrini, G., Stable determination of a crack from boundary measurements, Proc. Roy. Soc. Edinburgh. Sec. A., 123(3) (1993), 497–516.Google Scholar
[6]Al-Najem, N. M., Osman, A. M., El-Refaee, M. M., and Khanafer, K. M., Two dimensional steady-state inverse heat conduction problems, Int. Commun. Heat. Mass. Trans., 25(4) (1998), 541–550.Google Scholar
[7]Colli Franzone, P., and Magenes, E., On the inverse potential problem of electrocardiology, Calcolo., 16(4) (1979), 459–538.Google Scholar
[8]Han, H., The finite element method in the family of improperly posed problems, Math. Comput., 38(157) (1982), 55–65.Google Scholar
[9]Han, H., and Reinhardt, H.-J., Some stability estimates for Cauchy problems for elliptic equations, J. Inverse. Ill-Pose. Probl., 5(5) (1997), 437–454.Google Scholar
[10]Falk, R. S., and Monk, P. B., Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation, Math. Comput., 47(175) (1986), 135–149.Google Scholar
[11]Ang, D. D., Nghia, N. H., and Tam, N. C., Regularized solutions of Cauchy problem for the laplace equation in an irregular layer: a three dimensional case, Acta. Math. Vietnamica., 23 (1998), 65–74.Google Scholar
[12]Berntsson, F., and Eldén, L., Numerical solution of a Cauchy problem for the Laplace equation, Inverse. Probl., 17(4) (2001), 839–853.CrossRefGoogle Scholar
[13]Reinhardt, H.-J., Han, H., and Háo, D. N., Stability and regularization of a discrete approximation to the Cauchy problem for Laplace’s equation, SIAM J. Numer. Anal., 36(3) (1999), 890–905.Google Scholar
[14]Cheng, J., Hon, Y. C., Wei, T., and Yamamoto, M., Numerical computation of a Cauchy problem for Laplace’s equation, ZAMM Z. Angew. Math. Mech., 81(10) (2001), 665–674.3.0.CO;2-V>CrossRefGoogle Scholar
[15]Hon, Y. C., and Wei, T., Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation, Inverse. Probl., 17(2) (2001), 261–271.CrossRefGoogle Scholar
[16]Wei, T., Hon, Y. C., and Ling, L., Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. Anal. Bound. Elem., 31(4) (2007), 373–385.CrossRefGoogle Scholar
[17]Takeuchi, T., and Yamamoto, M., Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation, SIAM J. Sci. Comput., 31(1) (2008), 112–142.Google Scholar
[18]Leitão, A., An iterative method for solving elliptic Cauchy problems, Numer. Funct. Anal. Opt., 21(5-6) (2000), 715–742.Google Scholar
[19]Wei, T., Qin, H. H., and Shi, R., Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation, Inverse. Probl., 24(3) (2008), 035003.Google Scholar
[20]Yang, X., Choulli, M., and Cheng, J., An iterative BEM for the inverse problem of detecting corrosion in a pipe, Num. Math., J. China. Univ., 14(3) (2005), 252–266.Google Scholar
[21]Han, H., Yan, M., and Wu, C., An energy regularization method for the backward diffusion problem and its applications to image deblurring, Commun. Comput. Phys., 4(1) (2008), 177–197.Google Scholar
[22]Baumeister, J., Stable Solution of Inverse Problems, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1987.Google Scholar
[23]Engl, H. W., Hanke, M., and Neubauer, A., Regularization of Inverse Problems, Vol. 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996.Google Scholar
[24]Groetsch, C. W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Vol. 105 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984.Google Scholar
[25]Groetsch, C. W., Inverse Problems in the Mathematical Sciences, Vieweg Mathematics for Scientists and Engineers, Friedr. Vieweg & Sohn, Braunschweig, 1993.Google Scholar
[26]Hofmann, B., On the Generalized Discrepancy Principle in Regularization, Vol. 48, Wissenschaftliche Informationen, 1984.Google Scholar
[27]Hofmann, B., Regularization for Applied Inverse and Ill-Posed Problems, Vol. 85 Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 1986.Google Scholar
[28]Vasin, V. V., Some tendencies in the Tikhonov regularization of ill-posed problems, J. Inverse. Ill-Pose. Probl., 14(8) (2006), 813–840.CrossRefGoogle Scholar
[29]Ito, K., and Kunisch, K., On the choice of the regularization parameter in nonlinear inverse problems, SIAM J. Opt., 2(3) (1992), 376–404.Google Scholar
[30]Kunisch, K., and Zou, J., Iterative choices of regularization parameters in linear inverse problems, Inverse. Probl., 14(5) (1998), 1247–1264.Google Scholar
[31]Xie, J., and Zou, J., An improved model function method for choosing regularization parameters in linear inverse problems, Inverse. Probl., 18(3) (2002), 631–643.Google Scholar
[32]Fan, C. M., Chen, C. S., and Monroe, J., The method of fundamental solutions for solving convection-diffusion equations with variable coefficients, Adv. Appl. Math. Mech., 1(2) (2009), 215–230.Google Scholar
[33]Karageorghis, A., and Lesnic, D., The method of fundamental solutions for steady-state heat conduction in nonlinear materials, Commun. Comput. Phys., 4(4) (2008), 911–928.Google Scholar
[34]Hon, Y., and Wei, T., A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28(5) (2004), 489–495.CrossRefGoogle Scholar
[35]Hon, Y., and Wei, T., The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. Model. Eng. Sci., 7(2) (2005), 119–132.Google Scholar
[36]Ling, L., and Takeuchi, T., Point sources identification problems for heat equations, Commun. Comput. Phys., 5(5) (2009), 897–913.Google Scholar
[37]Ling, L., Yamamoto, M., Hon, Y. C., and Takeuchi, T., Identification of source locations in two-dimensional heat equations, Inverse. Probl., 22(4) (2006), 1289–1305.Google Scholar