Solving a nonlinear inverse Robin problem through a linear Cauchy problem. (English) Zbl 1447.65120
Summary: Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton’s law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton’s law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 65N20 | Numerical methods for ill-posed problems for boundary value problems involving PDEs |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 65F22 | Ill-posedness and regularization problems in numerical linear algebra |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
Keywords:
inverse Robin problem; Cauchy problem; coupled complex boundary method; Tikhonov regularizationReferences:
| [1] | Inglese, G., An inverse problem in corrosion detection, Inverse Probl, 13, 977-994 (1997) · Zbl 0882.35133 · doi:10.1088/0266-5611/13/4/006 |
| [2] | Busenberg, S.; Fang, W., Identification of semiconductor contact resistivity, Quart J Appl Math, 49, 639-649 (1991) · Zbl 0749.35062 · doi:10.1090/qam/1134746 |
| [3] | Osman, AM; Beck, JV., Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients, J Thermophys Heat Transf, 3, 146-152 (1989) · doi:10.2514/3.141 |
| [4] | Chaabane, S.; Jaoua, M., Identification of Robin coefficients by the means of boundary measurements, Inverse Probl, 15, 1425-1438 (1999) · Zbl 0943.35100 · doi:10.1088/0266-5611/15/6/303 |
| [5] | Colton, D.; Kirsch, A., The determination of the surface impedance of an obstacle from measurements of the far field pattern, SIAM J Appl Math, 41, 8-15 (1981) · Zbl 0464.35077 · doi:10.1137/0141002 |
| [6] | Jin, B.; Zou, J., Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J Numer Anal, 30, 677-701 (2010) · Zbl 1203.65232 · doi:10.1093/imanum/drn066 |
| [7] | Baratchart, L.; Bourgeois, L.; Leblond, J., Uniqueness results for inverse Robin problems with bounded coefficient, J Funct Anal, 270, 2508-2542 (2016) · Zbl 1439.35559 · doi:10.1016/j.jfa.2016.01.011 |
| [8] | Chaabane, S.; Fellah, I.; Jaoua, M., Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Probl, 20, 47-59 (2004) · Zbl 1055.35135 · doi:10.1088/0266-5611/20/1/003 |
| [9] | Chaabane, S.; Elhechmi, C.; Jaoua, M., A stable recovery method for the Robin inverse problem, Math Comput Simul, 66, 367-383 (2004) · Zbl 1054.65110 · doi:10.1016/j.matcom.2004.02.016 |
| [10] | Chaabane, S.; Ferchichi, J.; Kunisch, K., Differentiability properties of the \(####\)-tracking functional and application to the Robin inverse problem, Inverse Probl, 20, 1083-1097 (2004) · Zbl 1061.35163 · doi:10.1088/0266-5611/20/4/006 |
| [11] | Fang, W.; Lu, M., A fast collocation method for an inverse boundary value problem, Int J Numer Methods Eng, 59, 1563-1585 (2004) · Zbl 1059.65100 · doi:10.1002/nme.928 |
| [12] | Lin, F.; Fang, W., A linear integral equation approach to the Robin inverse problem, Inverse Probl, 21, 1757-1772 (2005) · Zbl 1086.35142 · doi:10.1088/0266-5611/21/5/015 |
| [13] | Xu, YF; Zou, J., Analysis of an adaptive finite element method for recovering the Robin coefficient, SIAM J Control Optim, 53, 622-644 (2015) · Zbl 1355.65148 · doi:10.1137/130941742 |
| [14] | Chantasiriwan, S., Inverse determination of steady-state heat transfer coefficient, Int Commun Heat Mass Transf, 27, 1155-1164 (2000) · doi:10.1016/S0735-1933(00)00202-5 |
| [15] | Andrieux, S.; Nouri-Baranger, T.; Ben Abda, A., Solving Cauchy problems by minimizing an energy-like functional, Inverse Probl, 22, 115-133 (2006) · Zbl 1089.35084 · doi:10.1088/0266-5611/22/1/007 |
| [16] | Andrieux, S.; Nouri-Baranger, T., An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity, Comput Methods Appl Mech Eng, 197, 902-920 (2008) · Zbl 1169.74335 · doi:10.1016/j.cma.2007.08.022 |
| [17] | Andrieux, S.; Ben Abda, A.; Nouri-Baranger, T., Data completion via an energy error functional, C R Mécanique, 333, 171-177 (2005) · Zbl 1223.80004 · doi:10.1016/j.crme.2004.10.005 |
| [18] | Blum, J., Numerical simulation and optimal control in plasma physics with application to Tokamaks (1989), Chichester: Wiley, Chichester · Zbl 0717.76009 |
| [19] | Bourgeois, L.Optimal control and inverse problems in plasticity [Ph. D thesis]. Paris: Ecole Polytechnique; 1989 |
| [20] | Colli Franzone, P, Guerri, L, Taccardi, B, et al. The direct and inverse potential problems in electrocardiology. Report number 222. Pavie: Laboratoire d’analyse numérique de Pavie; 1979 · Zbl 1081.92010 |
| [21] | Ben Belgacem, F.; El Fekih, H., On Cauchy’s problem: I. A variational Steklov-poincare theory, Inverse Probl, 21, 1915-1936 (2005) · Zbl 1112.35054 · doi:10.1088/0266-5611/21/6/008 |
| [22] | Ben Belgacem, F., Why is the Cauchy problem severely ill-posed?, Inverse Probl, 23, 823-836 (2007) · Zbl 1118.35060 · doi:10.1088/0266-5611/23/2/020 |
| [23] | Lavrent’ev, MM., On the Cauchy problem for the Laplace equation, Izv Akad Nauk SSSR, Ser Matem, 20, 819-842 (1956) · Zbl 0074.09004 |
| [24] | Payne, LE., Bounds in the Cauchy problem for the Laplace equation, Arch Rational Mech Anal, 5, 35-45 (1960) · Zbl 0094.29801 · doi:10.1007/BF00252897 |
| [25] | Alessandrini, G.; Rondi, L.; Rosset, E., The stability for the Cauchy problem for elliptic equations, Inverse Probl, 25 (2009) · Zbl 1190.35228 · doi:10.1088/0266-5611/25/12/123004 |
| [26] | Bourgeois, L.; Dardé, J., A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noise data, Inverse Probl, 26 (2010) · Zbl 1200.35315 · doi:10.1088/0266-5611/26/9/095016 |
| [27] | Klibanov, MV; Santosa, F., A computational quasi-reversibility method for Cauchy problems for Laplaces equation, SIAM J Appl Math, 51, 1653-1675 (1991) · Zbl 0769.35005 · doi:10.1137/0151085 |
| [28] | Jourhmane, M.; Nachaoui, A., An alternating method for an inverse Cauchy problem, Numer Algorithms, 21, 247-260 (1999) · Zbl 0944.65116 · doi:10.1023/A:1019134102565 |
| [29] | Leitão, A., An iterative method for solving elliptic Cauchy problems, Numer Funct Anal Opt, 21, 715-742 (2000) · Zbl 0969.65087 · doi:10.1080/01630560008816982 |
| [30] | Ben Belgacem, F.; Du, DT; Jelassi, F., Extended-domain-Lavrentiev’s regularization of the data completion problem, Inverse Probl, 27 (2011) · Zbl 1227.65088 |
| [31] | Ben Belgacem, F.; El Fekih, H.; Jelassi, F., The Lavrentiev regularization of the data completion problem, Inverse Probl, 24 (2008) · Zbl 1153.35399 · doi:10.1088/0266-5611/24/4/045009 |
| [32] | Fu, CL; Li, HF; Qian, Z., Fourier regularization method for solving a Cauchy problem for the Laplace equation, Inverse Probl Sci Eng, 16, 159-169 (2008) · Zbl 1258.65094 · doi:10.1080/17415970701228246 |
| [33] | Qian, Z.; Fu, CL; Li, ZP., Two regularization methods for a Cauchy problem for the Laplace equation, J Math Anal Appl, 338, 479-489 (2008) · Zbl 1132.35493 · doi:10.1016/j.jmaa.2007.05.040 |
| [34] | Chakib, A.; Nachaoui, A., Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Probl, 22, 1191-1206 (2006) · Zbl 1112.49027 · doi:10.1088/0266-5611/22/4/005 |
| [35] | Xiong, XT; Fu, CL., Central difference regularization method for the Cauchy problem of Laplace’s equation, Appl Math Comput, 181, 675-684 (2006) · Zbl 1148.65314 |
| [36] | Belaid, LJ; Ben Abda, A.; Malki, NA., The Cauchy problem for the Laplace equation and application to image inpainting, ISRN Math Anal, 2011 (2011) · Zbl 1238.35020 |
| [37] | Cheng, F.; Hon, YC; Wei, T., Numerical computation of a Cauchy problem for Laplace’s equation, Z Angew Math Mech, 81, 665-674 (2001) · Zbl 1002.70011 · doi:10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V |
| [38] | Zhang, H., Modified quasi-boundary value method for Cauchy problems of elliptic equation with variable coefficients, Electron J Differ Equ, 106, 1-10 (2011) · Zbl 1225.35081 |
| [39] | Azaïez, M.; Ben Belgacem, F.; El Fekih, H., On Cauchy’s problem: II. Completion, regularization and approximation, Inverse Probl, 22, 1307-1336 (2005) · Zbl 1113.35046 · doi:10.1088/0266-5611/22/4/012 |
| [40] | Berntsson, F.; Eldén, L., Numerical solution of a Cauchy problem for the Laplace equation, Inverse Probl, 17, 839-853 (2001) · Zbl 0993.65119 · doi:10.1088/0266-5611/17/4/316 |
| [41] | Cimetière, A.; Delvare, F.; Jaoua, M., Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Probl, 17, 553-570 (2001) · Zbl 0986.35128 · doi:10.1088/0266-5611/17/3/313 |
| [42] | Cheng, XL; Gong, RF; Han, W., A novel coupled complex boundary method for solving inverse source problems, Inverse Probl, 30 (2014) · Zbl 1290.35324 · doi:10.1088/0266-5611/30/5/055002 |
| [43] | Cheng, XL; Gong, RF; Han, W., 2016 A coupled complex boundary method for the Cauchy problem, Inverse Probl Sci Eng, 24, 1510-1527 (2016) · Zbl 1348.65157 · doi:10.1080/17415977.2015.1130040 |
| [44] | Gong, RF; Cheng, XL; Han, W., A new coupled complex boundary method for bioluminescence tomography, Commun Comput Phys, 19, 226-250 (2016) · Zbl 1373.74095 · doi:10.4208/cicp.230115.150615a |
| [45] | Adams, RA., Sobolev spaces (1975), New York: Academic Press, New York · Zbl 0314.46030 |
| [46] | Atkinson, K.; Han, W., Theoretical numerical analysis: a functional analysis framework (2009), New York: Springer-Verlag, New York · Zbl 1181.47078 |
| [47] | Glashoff, K.; Gustafson, SA., Linear optimization and approximation: an introduction to the theoretical analysis and numerical treatment of semi-infinite programs (1983), New York: Springer-Verlag, New York · Zbl 0526.90058 |
| [48] | Isakov, V., Inverse problems for partial differential equations (1998), New York: Springer-Verlag, New York · Zbl 0908.35134 |
| [49] | Dautray, R.; Lions, JL., Mathematical analysis and numerical methods for science and technology. Vol 2 (1988), Berlin: Springer, Berlin · Zbl 0664.47001 |
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