Acta mathematica scientia, Series B >
A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION
Received date: 2015-01-20
Revised date: 2014-08-26
Online published: 2015-11-01
Supported by
The project is supported by the National Natural Science Foundation of China (11171136, 11261032), the Distinguished Young Scholars Fund of Lan Zhou University of Technology (Q201015), the basic scientific research business expenses of Gansu province college and the Natural Science Foundation of Gansu province (1310RJYA021).
In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.
Fan YANG , Chuli FU , Xiaoxiao LI . A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION[J]. Acta mathematica scientia, Series B, 2015 , 35(6) : 1339 -1348 . DOI: 10.1016/S0252-9602(15)30058-8
[1] Tikhonov A N, Arsenin V Y. Solutions of Ill-posed Problems. Washington: Winston and Sons, 1977
[2] Lavrent'v M M, Romanov V G, Shishat·skiï S P. Ill-posed Problems of Mathematical Physics and Analysis. Providence RI: American Mathematical Society, 1986
[3] Gorenflo R. Funktionentheoretische Bestimmung des Aussenfeldes zu einer zweidimensionalen magnetohy- drostatischen Konguration. Z Angew Math Phys, 1965, 16: 279-290
[4] Colli-Franzone P, Guerri L, Tentoni S, Viganotti C, Baruffi S, Spaggiari S, Taccardi B. A mathematical procedure for solving the inverse potential problem of electrocardiography. Analysis of the time-space accuracy from in vitro experimental data. Math Biosci, 1985, 77: 353-396
[5] Johnson C R. Computational and numerical methods for bioelectric field problems. Crit Rev Biomed Eng, 1997, 25: 1-81
[6] Alessandrini G. Stable determination of a crack from boundary measurements. Proc R Soc Edinburgh A, 1993, 123: 497-516
[7] Hadamard J. Lecture on the Cauchy Problem in Linear Partial Differential Equations. New Haven: Yale University Press, 1923
[8] Fu C L, Li H F, Qian Z, Xiong X T. Fourier regularization method for solving a Cauchy problem for the Laplace equation. Inverse Probl Sci Eng, 2008, 16(2): 159-169
[9] Xiong X T, Fu C L. Central difference regularization method for the Cauchy problem of the Laplace's equation. Appl Math Comput, 2006, 181: 675-684
[10] Klibanov M V, Santosa F. A computational quasi-reversibility method for Cauchy problems for Laplace's equation. SIAM J Appl Math, 1991, 51: 1653-1675
[11] Lattès R, Lions J L. The Method of Quasi-reversibility: Applications to Partial Differential Equations. New York: Elsevier, 1969
[12] Qian Z, Fu C L, Li Z P. Two regularization methods for a Cauchy problem for the Laplace equation. J Math Anal Appl, 2008, 338: 479-489
[13] Qian Z, Fu C L, Xiong X T. Fourth-order modified method for the Cauchy problem for the Laplace equation. J Comput Appl Math 2006, 192: 205-218
[14] Ang D D, Nghia N H, Tam N C. Regularized solutions of a Cauchy problem for the Laplace equation in an irregular layer: a three-dimensional model. Acta Math Vietnam, 1998, 23: 65-74
[15] H`ao D N, Lesnic D. The Cauchy problem for Laplace's equation via the conjugate gradient method. IMA J Appl Math, 2000, 65: 199-217
[16] Cheng J, Hon Y C, Wei T, Yamamoto M. Numerical computation of a Cauchy problem for Laplace's equation. ZAMMZ Angew Math Mech, 2001, 81(10): 665-674
[17] Hon Y C, Wei T. Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation. Inverse Prob, 2001, 17: 261-271
[18] Wei T, Hon Y C, Cheng J. Computation for multidimensional Cauchy problem. SIAM J Control Optim, 2003, 42(2): 381-396
[19] Vani C, Avudainayagam A. Regularized solution of the Cauchy problem for the Laplace equation using Meyer wavelets. Math Comput Model, 2002, 36: 1151-1159
[20] Qiu C Y, Fu C L. Wavelets and regularization of the Cauchy problem for the Laplace equation. J Math Anal Appl, 2008, 338: 1440-1447
[21] Li Z P, Fu C L. A mollification method for a Cauchy problem for the Laplace equation. Appl Math Comput, 2011, 217: 9209-9218
[22] Wei T, Zhou D Y. Convergence analysis for the Cauchy problem of Laplace's equation by a regularized method of fundamental solutions. Adv Comput Math, 2010, 33(4): 491-510
[23] Fu C L, Ma Y J, Cheng H, Zhang Y X. The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data. Appl Math Model, 2013, 37(14/15): 7764-7777
[24] Carasso A. Determining surface temperature from interior observations. SIAM J Appl Math, 1982, 42: 558-574
[25] Zhao Z Y, Meng Z H. A modified Tikhonov regularization method for a backward heat equation. Inverse Probl Sci Eng, 2011, 19: 1175-1182
[26] Fu C L. Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation. J Comput Appl Math, 2004, 167: 449-463
[27] Feng X L, Fu C L, Cheng H. A regularization method for solving the Cauchy problem for the Helmholtz equation. Appl Math Model, 2011, 35: 3301-3315
[28] Cheng W, Fu C L, Qian Z. A modified Tikhonov regularization method for a spherically symmetric three- dimensional inverse heat conduction problem. Math Comput Simul, 2007, 75: 97-112
[29] Cheng W, Fu C L, Qian Z. Two regularization methods for a spherically symmetric inverse heat conduction problem. Appl Math Model, 2008, 32: 432-442
[30] Yang F, Fu C L. Two regularization methods for identification of the heat source depending only on spatial variable for the heat equation. J Inv Ill-Posed Problems, 2009, 17: 815-830
[31] Yang F, Fu C L, Li X X. Identifying an unknown source in space-fractional diffusion equation. Acta Math Sci, 2014, 34B(4): 1012-1024
[32] Kirsch A. An Introduction to the Mathematical Theory of Inverse Problems. Berlin: Springer-Verlag, 1996
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