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A a posteriori regularization for the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data. (English) Zbl 1443.65295

Summary: In this paper the Cauchy problem for the Helmholtz equation with inhomogeneous Neumann data is considered. This problem is severely ill-posed, the solution does not depend continuously on the data. An approximate method based on the a posteriori Fourier regularization in the frequency space is analyzed. Some crucial information about the regularization parameter hidden in the a posteriori choice rule are found, and some sharp error estimates between the exact solution and its regularization approximate solution are proved. Numerical examples show the effectiveness of the method. A comparison of numerical effect between the a posteriori and the a priori Fourier method is also taken into account.

MSC:

65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R25 Ill-posed problems for PDEs
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References:

[1] Regińska, T.; Regiński, K., Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Prob., 22, 975-989 (2006) · Zbl 1099.35160
[2] Isakov, V., Inverse Problems for Partial Differential Equation (1998), Springer: Springer New York · Zbl 0908.35134
[3] Fu, C. L.; Feng, X. L.; Qian, Z., The Fourier regularization for solving the Cauchy problem for the Helmholtz equation, Appl. Numer. Math., 59, 2625-2640 (2009) · Zbl 1169.65333
[4] Xiong, X. T.; Fu, C. L., Two approximate methods of a Cauchy problem for the Helmholtz equation, Comput. Appl. Math., 26, 285-307 (2007) · Zbl 1182.35237
[5] Regińska, T.; Tautenhahn, U., Conditional stability estimates and regularization with applications to Cauchy problems for the Helmholtz equation, Numer. Funct. Anal. Optim., 30, 9-10, 1065-1097 (2009) · Zbl 1181.47009
[6] Qin, H. H.; Wei, T., Modified regularization method for the Cauchy problem of the Helmholtz equation, Appl. Math. Model., 33, 2334-2348 (2009) · Zbl 1185.65203
[7] Qin, H. H.; Wei, T.; Shi, R., Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math., 224, 39-53 (2009) · Zbl 1158.65072
[8] Qian, A. L.; Xiong, X. T.; Wu, Y. J., On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math., 233, 1969-1979 (2010) · Zbl 1185.65171
[9] Xiong, X. T., A regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math., 233, 1723-1732 (2010) · Zbl 1186.65127
[10] Fu, C. L.; Qian, Z., Numerical pseudodifferential operator and Fourier regularization, Adv. Comput. Math., 33, 4, 449-470 (2010) · Zbl 1207.65167
[11] Fu, C. L.; Zhang, Y. X.; Cheng, H.; Ma, Y. J., The a posteriori Fourier method for solving ill-posed problems, Inverse Prob., 28, 095002 (2012), (26pp) · Zbl 1253.35210
[12] Feng, X. L.; Fu, C. L.; Cheng, H., A regularization method for solving the Cauchy problem for the Helmholtz equation, Appl. Math. Model., 35, 3301-3315 (2011) · Zbl 1221.65295
[13] Eldén, L.; Berntsson, F.; Regińska, T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput., 21, 6, 2187-2205 (2000) · Zbl 0959.65107
[14] 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., 37, 7764-7777 (2013) · Zbl 1438.35452
[15] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0859.65054
[16] Hào, D. N.; Sahli, H., On a class of severely ill-posed problems, Vietnam J. Math., 32, 143-152 (2004) · Zbl 1082.47044
[17] Fu, C. L.; Xiong, X. T.; Qian, Z., Fourier regularization for a backward heat equation, J. Math. Anal. Appl., 331, 1, 472-480 (2007) · Zbl 1146.35420
[18] Fu, C. L.; Li, H. F.; Qian, Z.; Xiong, X. T., Fourier regularization method for solving a Cauchy problem for the Laplace equation, Inverse Prob. Sci. Eng., 16, 2, 159-169 (2008) · Zbl 1258.65094
[19] Fu, C. L.; Dou, F. F.; Feng, X. L.; Qian, Z., A simple regularization method for stable analytic continuation, Inverse Prob., 24, 065003 (2008), (15pp) · Zbl 1160.30023
[20] Dou, F. F.; Fu, C. L.; Yang, F. L., Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation, J. Comput. Appl. Math., 230, 2, 728-737 (2009) · Zbl 1219.65100
[21] Morozov, V. A., Choice of parameter for the solution of functional equations by the regularization method, Sov. Math. Dokl., 8, 1000-1003 (1967) · Zbl 0189.47501
[22] Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0865.35004
[23] Hanke, M.; Hansen, D. C., Regularization methods for large-scale problems, Surv. Math. Ind., 3, 253-315 (1993) · Zbl 0805.65058
[24] Tautenhahn, U., Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19, 377-398 (1998) · Zbl 0907.65049
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