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Convergence Estimates for Some Regularization Methods to Solve a Cauchy Problem of the Laplace Equation

Published online by Cambridge University Press:  28 May 2015

T. Wei ,
H. H. Qin  and
H. W. Zhang
T. Wei*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, P. R. China
H. H. Qin
Affiliation:
School of Mathematics and Statistics, Lanzhou University, P. R. China
H. W. Zhang
Affiliation:
School of Mathematics and Statistics, Lanzhou University, P. R. China
*
Corresponding author.Email address:tingwei@lzu.edu.cn
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Abstract

In this paper, we give a general proof on convergence estimates for some regularization methods to solve a Cauchy problem for the Laplace equation in a rectangular domain. The regularization methods we considered are: a non-local boundary value problem method, a boundary Tikhonov regularization method and a generalized method. Based on the conditional stability estimates, the convergence estimates for various regularization methods are easily obtained under the simple verifications of some conditions. Numerical results for one example show that the proposed numerical methods are effective and stable.

Keywords

65N1265N1565N2065N21Cauchy problemLaplace equationregularization methodsconvergence estimates
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 4 , November 2011 , pp. 459 - 477
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Alessandrini, G.. Stable determination of a crack from boundary measurements. Proc. Roy. Soc. Edinburgh Sect. A, 123(3):497–516, 1993.CrossRefGoogle Scholar
[2]Alessandrinil, G., Rondil, L., Rosset, E., and Vessella, S.. The stability for the Cauchy problem for elliptic equations. Inverse Problems, 25:123004, 2009.CrossRefGoogle Scholar
[3]Berntsson, F. and Eldén., L.Numerical solution of a Cauchy problem for the Laplace equation. Inverse Problems, 17(4):839–853, 2001. Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000).CrossRefGoogle Scholar
[4]Bourgeois, L.. A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Problems, 21(3):1087–1104, 2005.CrossRefGoogle Scholar
[5]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):665–674, 2001.3.0.CO;2-V>CrossRefGoogle Scholar
[6]Cheng, J. and Yamamoto, M.. One new strategy for a priori choice of regularizing parameters in Tikhonovąŕs regularization. Inverse Problems.Google Scholar
[7]Cheng, J. and Yamamoto, M.. Unique continuation on a line for harmonic functions. Inverse Problems, 14(4):869–882, 1998.CrossRefGoogle Scholar
[8]Colli Franzone, P. and Magenes, E.. On the inverse potential problem of electrocardiology. Calcolo, 16(4):459–538 (1980), 1979.Google Scholar
[9]Van Duc, N.Háo, D. N. and Lesnic, D.. A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Problems, 25:055002, 2009.Google Scholar
[10]Eldén, L. and Berntsson, F.. A stability estimate for a Cauchy problem for an elliptic partial differential equation. Inverse Problems, 21(5):1643–1653, 2005.CrossRefGoogle Scholar
[11]Engl, H. W., Hanke, M., and Neubauer, A.. Regularization of inverse problems, volume 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996.Google Scholar
[12]Engl, H. W. and Leitão., A.A Mann iterative regularization method for elliptic Cauchy problems. Numer. Funct. Anal. Optim., 22(7-8):861–884, 2001.CrossRefGoogle Scholar
[13]Hào, D. N. and Lesnic, D.. The Cauchy problem for Laplace’s equation via the conjugate gradient method. IMA J. Appl. Math., 65(2):199–217, 2000.CrossRefGoogle Scholar
[14]Hon, Y. C. and Wei, T.. Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation. Inverse Problems, 17(2):261–271, 2001.CrossRefGoogle Scholar
[15]Isakov, V.. Inverse problems for partial differential equations, volume 127 of Applied Mathematical Sciences. Springer-Verlag, New York, 1998.Google Scholar
[16]Kabanikhin, S. I.. Convergence rate estimation of gradient methods via conditional stability of inverse and ill-posed problems. J. Inverse Ill-Posed Probl., 13(3-6):259–264, 2005. Inverse problems: modeling and simulation.CrossRefGoogle Scholar
[17]Kabanikhin, S. I. and Schieck, M.. Impact of conditional stability: convergence rates for general linear regularization methods. J. Inverse Ill-Posed Probl., 16(3):267–282, 2008.CrossRefGoogle Scholar
[18]Kirsch., A.An introduction to the mathematical theory of inverse problems, volume 120 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.Google Scholar
[19]Klibanov, M. V. and Santosa, F.. A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM J. Appl. Math., 51(6):1653–1675, 1991.CrossRefGoogle Scholar
[20]Lesnic, D., Elliott, L., and Ingham, D. B.. An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation. Eng.Anal.Bound.Elem., 20:123–133, 1997.Google Scholar
[21]Payne, L. E.. Bounds in the Cauchy problem for the Laplace equation. Arch. Rational Mech. Anal., 5:35–45 (1960), 1960.Google Scholar
[22]Qian, Z., Fu, C. L., and Li, Z. P.. Two regularization methods for a Cauchy problem for the Laplace equation. J. Math. Anal. Appl., 338(4) :479–489, 2008.Google Scholar
[23]Qin, H.H. and Wei, T.. Two regularization methods for the Cauchy problems of the Helmholtz equation. Applied Mathematical Modelling.Google Scholar
[24]Reinhardt, H. J., Han, H., and Hào, D. N.. Stability and regularization of a discrete approximation to the Cauchy problem for Laplace’s equation. SIAM J. Numer. Anal, 36(3):890–905 (electronic), 1999.CrossRefGoogle Scholar
[25]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(4):112–142, 2008.CrossRefGoogle Scholar
[26]Tikhonov, A.N. and Arsenin, VY.. Solutions of ill-posed problems. V H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977.Google Scholar
[27] H. W Zhang, Qin, H. H., and Wei, T. A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation. International Journal of Computer Mathematics, DOI: 10.1080/00207160.2010.482986.CrossRefGoogle Scholar