A regularization method for solving the Cauchy problem for the Helmholtz equation. (English) Zbl 1221.65295
Summary: We investigate a Cauchy problem associated with Helmholtz-type equation in an infinite “strip”. This problem is well known to be severely ill-posed. The optimal error bound for the problem with only nonhomogeneous Neumann data is deduced, which is independent of the selected regularization methods. A framework of a modified Tikhonov regularization in conjunction with the Morozov’s discrepancy principle is proposed, it may be useful to the other linear ill-posed problems and helpful for the other regularization methods. Some sharp error estimates between the exact solutions and their regularization approximation are given. Numerical tests are also provided to show that the modified Tikhonov method works well.
MSC:
| 65N20 | Numerical methods for ill-posed problems for boundary value problems involving PDEs |
Keywords:
Tikhonov regularization; Helmholtz equation; optimal error bound; a priori strategy; a posteriori; Morozov’s discrepancy principleReferences:
| [1] | Alessandrini, G.; Rondi, L.; Rosset, E.; Vessella, S., The stability for the Cauchy problem for elliptic equations, Inverse Probl., 25, 123004 (2009), 47p · Zbl 1190.35228 |
| [2] | Eldén, L.; Berntsson, F., A stability estimate for a Cauchy problem for an elliptic partial differential equation, Inverse Probl., 21, 1643-1653 (2005) · Zbl 1086.35115 |
| [3] | Isakov, V., Inverse Problems for Partial Differential Equations. Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, vol. 127 (2006), Springer: Springer New York · Zbl 1092.35001 |
| [4] | Payne, L. E., Improved stability estimates for classes of ill-posed Cauchy problems, Appl. Anal. Int. J., 19, 63-64 (1985) · Zbl 0563.35080 |
| [5] | Hào, D. N.; Lesnic, D., The Cauchy problem for Laplace’s equation using the conjugate gradient method, IMA J. Appl. Math., 65, 199-217 (2000) · Zbl 0967.35032 |
| [6] | Reinhardt, H. J.; Han, H.; Hào, D. N., Stability and regularization of a discrete approximation to the Cauchy problem of Laplace’s equation, SIAM J. Numer. Anal., 36, 890-905 (1999) · Zbl 0928.35184 |
| [7] | Cheng, J.; Yamamoto, M., Unique continuation on a line for harmonic functions, Inverse Probl., 14, 869-882 (1998) · Zbl 0905.31001 |
| [8] | Hon, Y. C.; Wei, T., Backus-Gilbert algorithm for the Cauchy problem of Laplace equation, Inverse Probl., 17, 261-271 (2001) · Zbl 0980.35167 |
| [9] | Xiong, X. T., Central difference regularization method for the Cauchy problem of Laplace’s equation, Appl. Math. Comput., 181, 675-684 (2006) · Zbl 1148.65314 |
| [10] | 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 |
| [11] | Qian, Z.; Fu, C. L.; Xiong, X. T., Fourth-order modified method for the Cauchy problem for the Laplace equation, J. Comput. Appl. Math., 192, 205-218 (2006) · Zbl 1093.65107 |
| [12] | Beskos, D. E., Boundary element method in dynamic analysis: part II (1986-1996), ASME Appl. Mech. Rev., 50, 149-197 (1997) |
| [13] | Chen, J. T.; Wong, F. C., Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition, J. Sound. Vib., 217, 75-95 (1998) |
| [14] | Harari, I.; Barbone, P. E.; Slavutin, M.; Shalom, R., Boundary infinite elements for the Helmholtz equation in exterior domains, Int. J. Numer. Meth. Eng., 41, 1105-1131 (1998) · Zbl 0911.76035 |
| [15] | Hall, W. S.; Mao, X. Q., A boundary element investigation of irregular frequencies in electromagnetic scattering, Eng. Anal. Bound. Elem., 16, 245-252 (1995) |
| [16] | Kraus, A. D.; Aziz, A.; Welty, J., Extended Surface Heat Transfer (2001), Wiley: Wiley New York |
| [17] | Debye, P.; Hückel, E., The theory of electrolytes. I. Lowering of freezing point and related phenomena, Physik. Zeitschrift, 24, 185-206 (1923) · JFM 49.0587.11 |
| [18] | Liang, J.; Subramaniam, S., Computation of molecular electrostatics with boundary element methods, Biophys. J., 73, 1830-1841 (1997) |
| [19] | DeLillo, T.; Isakov, V.; Valdivia, N.; Wang, L., The detection of the source of acoustical noise in two dimensions, SIAM J. Appl. Math., 61, 2104-2121 (2001) · Zbl 0983.35149 |
| [20] | DeLillo, T.; Isakov, V.; Valdivia, N.; Wang, L., The detection of surface vibrations from interior acoustical pressure, Inverse Probl., 19, 507-524 (2003) · Zbl 1033.76053 |
| [21] | 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 |
| [22] | 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 |
| [23] | 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 |
| [24] | Qin, H. H.; Wei, T., Two regularization methods for the Cauchy problems of the Helmholtz equation, Appl. Math. Model., 34, 947-967 (2010) · Zbl 1185.35040 |
| [25] | Qin, H. H.; Wen, D. W., Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation, Appl. Math. Comput., 203, 617-628 (2009) · Zbl 1158.65070 |
| [26] | Regińska, T.; Regiński, K., Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Probl., 22, 975-989 (2006) · Zbl 1099.35160 |
| [27] | Regińska, T.; Tautenhahn, U., Conditional stability estimates and regularization with applications to cauchy problems for the Helmholtz equation, Numer. Funct. Anal. Optim., 30, 1065-1097 (2009) · Zbl 1181.47009 |
| [28] | Shi, R.; Wei, T.; Qin, H. H., A fourth-order modified method for the Cauchy problem of the modified Helmholtz equation, Numer. Math. Theor. Meth. Appl., 2, 326-340 (2009) · Zbl 1212.65355 |
| [29] | 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 |
| [30] | 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 |
| [31] | Wei, T.; H Qin, H.; Shi, R., Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation, Inverse Probl., 24, 1-18 (2008) · Zbl 1187.35286 |
| [32] | R. Shi, A Modified Method for the Cauchy Problems of the Helmholtz-type Equation, Master Thesis, Lanzhou University, PR China, 2008.; R. Shi, A Modified Method for the Cauchy Problems of the Helmholtz-type Equation, Master Thesis, Lanzhou University, PR China, 2008. |
| [33] | Jin, B. T.; Zheng, Y., Boundary knot method for some inverse problems associated with the Helmholtz equation, Int. J. Numer. Meth. Eng., 62, 1636-1651 (2005) · Zbl 1085.65104 |
| [34] | Jin, B. T.; Marin, L., The plane wave method for inverse problems associated with Helmholtz-type equations, Eng. Anal. Bound. Elem., 32, 223-240 (2008) · Zbl 1244.65163 |
| [35] | Johansson, B. T.; Marin, L., Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation, CMC: Comput. Mater. Con., 13, 153-190 (2010) · Zbl 1231.65237 |
| [36] | Marin, L.; Elliott, L.; Heggs, P. J.; Ingham, D. B.; Lesnic, D.; Wen, X., An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. Methods. Appl. Mech. Engrg., 192, 709-722 (2003) · Zbl 1022.78012 |
| [37] | Marin, L.; Elliott, L.; Heggs, P. J.; Ingham, D. B.; Lesnic, D.; Wen, X., Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Comput. Mech., 31, 367-377 (2003) · Zbl 1047.65097 |
| [38] | Marin, L.; Elliott, L.; Heggs, P. J.; Ingham, D. B.; Lesnic, D.; Wen, X., Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation, Int. J. Numer. Methods Eng., 60, 1933-1947 (2004) · Zbl 1062.78015 |
| [39] | Marin, L.; Elliott, L.; Heggs, P. J.; Ingham, D. B.; Lesnic, D.; Wen, X., BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method, Eng. Anal. Bound. Elem., 28, 1025-1034 (2004) · Zbl 1066.80009 |
| [40] | Marin, L.; Lesnic, D., The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations, Comput. Struct., 83, 267-278 (2005) · Zbl 1088.35079 |
| [41] | Marin, L., A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations, Appl. Math. Comput., 165, 355-374 (2005) · Zbl 1070.65115 |
| [42] | Marin, L., Boundary element-minimal error method for the Cauchy problem associated with Helmholtz-type equations, Comput. Mech., 44, 205-219 (2009) · Zbl 1185.74097 |
| [43] | Marin, L., An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation, Comput. Mech., 45, 665-677 (2010) · Zbl 1398.65278 |
| [44] | Carasso, A., Determining surface temperatures from interior observations, SIAM J. Appl. Math., 42, 558-574 (1982) · Zbl 0498.35084 |
| [45] | Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer Academic Publishes: Kluwer Academic Publishes Boston · Zbl 0859.65054 |
| [46] | Hohage, T., Regularization of exponentially ill-posed problems, Numer. Funct. Anal. Optim., 21, 439-464 (2000) · Zbl 0969.65041 |
| [47] | Mathé, P.; Pereverzev, S., Geometry of ill-posed problems in variable Hilbert scales, Inverse Probl., 19, 789-803 (2003) · Zbl 1026.65040 |
| [48] | Tautenhahn, U., Optimal stable solution of Cauchy problems of elliptic equations, J. Anal. Appl., 15, 961-984 (1996) · Zbl 0865.65076 |
| [49] | Tautenhahn, U., Optimality for ill-posed problems under general source conditions, Numer. Funct. Anal. Optim., 19, 377-398 (1998) · Zbl 0907.65049 |
| [50] | Feng, X. L.; Eldén, L.; Fu, C. L., Stability and regularization of a backward parabolic PDE with variable coefficients, J. Inverse Ill-posed Probl., 18, 217-243 (2010) · Zbl 1279.80002 |
| [51] | Hanke, M.; Hansen, P. C., Regularization methods for large-scale problems, Surv. Math. Ind., 3, 253-315 (1993) · Zbl 0805.65058 |
| [52] | Hanke, M., Conjugate Gradient Type Methods for Ill-posed Problems (1995), Longman Scientific and Technical: Longman Scientific and Technical Harlow · Zbl 0830.65043 |
| [53] | Murio, D. A., The Mollification Method and the Numerical Solution of Ill-Posed Problems [M] (1993), A Wiley-Interscience Publication, John Wiley and Sons Inc.: A Wiley-Interscience Publication, John Wiley and Sons Inc. New York |
| [54] | 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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
