References
- Inglese G. An inverse problem in corrosion detection. Inverse Probl. 1997;13:977–994. doi: 10.1088/0266-5611/13/4/006
- Busenberg S, Fang W. Identification of semiconductor contact resistivity. Quart J Appl Math. 1991;49:639–649. doi: 10.1090/qam/1134746
- Osman AM, Beck JV. Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients. J Thermophys Heat Transf. 1989;3:146–152. doi: 10.2514/3.141
- Chaabane S, Jaoua M. Identification of Robin coefficients by the means of boundary measurements. Inverse Probl. 1999;15:1425–1438. doi: 10.1088/0266-5611/15/6/303
- Colton D, Kirsch A. The determination of the surface impedance of an obstacle from measurements of the far field pattern. SIAM J Appl Math. 1981;41:8–15. doi: 10.1137/0141002
- Jin B, Zou J. Numerical estimation of the Robin coefficient in a stationary diffusion equation. IMA J Numer Anal. 2010;30:677–701. doi: 10.1093/imanum/drn066
- Baratchart L, Bourgeois L, Leblond J. Uniqueness results for inverse Robin problems with bounded coefficient. J Funct Anal. 2016;270:2508–2542. doi: 10.1016/j.jfa.2016.01.011
- Chaabane S, Fellah I, Jaoua M, et al. Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems. Inverse Probl. 2004;20:47–59. doi: 10.1088/0266-5611/20/1/003
- Chaabane S, Elhechmi C, Jaoua M. A stable recovery method for the Robin inverse problem. Math Comput Simul. 2004;66:367–383. doi: 10.1016/j.matcom.2004.02.016
- Chaabane S, Ferchichi J, Kunisch K. Differentiability properties of the L1-tracking functional and application to the Robin inverse problem. Inverse Probl. 2004;20:1083–1097. doi: 10.1088/0266-5611/20/4/006
- Fang W, Lu M. A fast collocation method for an inverse boundary value problem. Int J Numer Methods Eng. 2004;59:1563–1585. doi: 10.1002/nme.928
- Lin F, Fang W. A linear integral equation approach to the Robin inverse problem. Inverse Probl. 2005;21:1757–1772. doi: 10.1088/0266-5611/21/5/015
- Xu YF, Zou J. Analysis of an adaptive finite element method for recovering the Robin coefficient. SIAM J Control Optim. 2015;53:622–644. doi: 10.1137/130941742
- Chantasiriwan S. Inverse determination of steady-state heat transfer coefficient. Int Commun Heat Mass Transf. 2000;27:1155–1164. doi: 10.1016/S0735-1933(00)00202-5
- Andrieux S, Nouri-Baranger T, Ben Abda A. Solving Cauchy problems by minimizing an energy-like functional. Inverse Probl. 2006;22:115–133. doi: 10.1088/0266-5611/22/1/007
- 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. 2008;197:902–920. doi: 10.1016/j.cma.2007.08.022
- Andrieux S, Ben Abda A, Nouri-Baranger T. Data completion via an energy error functional. C R Mécanique. 2005;333:171–177. doi: 10.1016/j.crme.2004.10.005
- Blum J. Numerical simulation and optimal control in plasma physics with application to Tokamaks. Chichester: Wiley; 1989.
- Bourgeois L. Optimal control and inverse problems in plasticity [Ph. D thesis]. Paris: Ecole Polytechnique; 1989
- 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
- Ben Belgacem F, El Fekih H. On Cauchy's problem: I. A variational Steklov-poincare theory. Inverse Probl. 2005;21:1915–1936. doi: 10.1088/0266-5611/21/6/008
- Ben Belgacem F. Why is the Cauchy problem severely ill-posed? Inverse Probl. 2007;23:823–836. doi: 10.1088/0266-5611/23/2/020
- Lavrent'ev MM. On the Cauchy problem for the Laplace equation. Izv Akad Nauk SSSR, Ser Matem. 1956;20:819–842.
- Payne LE. Bounds in the Cauchy problem for the Laplace equation. Arch Rational Mech Anal. 1960;5:35–45. doi: 10.1007/BF00252897
- Alessandrini G, Rondi L, Rosset E, et al. The stability for the Cauchy problem for elliptic equations. Inverse Probl. 2009;25:123004. doi: 10.1088/0266-5611/25/12/123004
- Bourgeois L, Dardé J. A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noise data. Inverse Probl. 2010;26:095016. doi: 10.1088/0266-5611/26/9/095016
- Klibanov MV, Santosa F. A computational quasi-reversibility method for Cauchy problems for Laplaces equation. SIAM J Appl Math. 1991;51:1653–1675. doi: 10.1137/0151085
- Jourhmane M, Nachaoui A. An alternating method for an inverse Cauchy problem. Numer Algorithms. 1999;21:247–260. doi: 10.1023/A:1019134102565
- Leitão A. An iterative method for solving elliptic Cauchy problems. Numer Funct Anal Opt. 2000;21:715–742. doi: 10.1080/01630560008816982
- Ben Belgacem F, Du DT, Jelassi F. Extended-domain-Lavrentiev's regularization of the data completion problem. Inverse Probl. 2011;27:045005.
- Ben Belgacem F, El Fekih H, Jelassi F. The Lavrentiev regularization of the data completion problem. Inverse Probl. 2008;24:045009. doi: 10.1088/0266-5611/24/4/045009
- Fu CL, Li HF, Qian Z, et al. Fourier regularization method for solving a Cauchy problem for the Laplace equation. Inverse Probl Sci Eng. 2008;16:159–169. doi: 10.1080/17415970701228246
- Qian Z, Fu CL, Li ZP. Two regularization methods for a Cauchy problem for the Laplace equation. J Math Anal Appl. 2008;338:479–489. doi: 10.1016/j.jmaa.2007.05.040
- Chakib A, Nachaoui A. Convergence analysis for finite element approximation to an inverse Cauchy problem. Inverse Probl. 2006;22:1191–1206. doi: 10.1088/0266-5611/22/4/005
- Xiong XT, Fu CL. Central difference regularization method for the Cauchy problem of Laplace's equation. Appl Math Comput. 2006;181:675–684.
- Belaid LJ, Ben Abda A, Malki NA. The Cauchy problem for the Laplace equation and application to image inpainting. ISRN Math Anal. 2011;2011. Articl ID 150979
- Cheng F, Hon YC, Wei T, et al. Numerical computation of a Cauchy problem for Laplace's equation. Z Angew Math Mech. 2001;81:665–674. doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V
- Zhang H. Modified quasi-boundary value method for Cauchy problems of elliptic equation with variable coefficients. Electron J Differ Equ. 2011;106:1–10.
- Azaïez M, Ben Belgacem F, El Fekih H. On Cauchy's problem: II. Completion, regularization and approximation. Inverse Probl. 2005;22:1307–1336. doi: 10.1088/0266-5611/22/4/012
- Berntsson F, Eldén L. Numerical solution of a Cauchy problem for the Laplace equation. Inverse Probl. 2001;17:839–853. doi: 10.1088/0266-5611/17/4/316
- Cimetière A, Delvare F, Jaoua M, et al. Solution of the Cauchy problem using iterated Tikhonov regularization. Inverse Probl. 2001;17:553–570. doi: 10.1088/0266-5611/17/3/313
- Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Probl. 2014;30:055002. doi: 10.1088/0266-5611/30/5/055002
- Cheng XL, Gong RF, Han W. 2016 A coupled complex boundary method for the Cauchy problem. Inverse Probl Sci Eng. 2016;24:1510–1527. doi: 10.1080/17415977.2015.1130040
- Gong RF, Cheng XL, Han W. A new coupled complex boundary method for bioluminescence tomography. Commun Comput Phys. 2016;19:226–250. doi: 10.4208/cicp.230115.150615a
- Adams RA. Sobolev spaces. New York: Academic Press; 1975.
- Atkinson K, Han W. Theoretical numerical analysis: a functional analysis framework. 3rd ed. New York: Springer-Verlag; 2009.
- Glashoff K, Gustafson SA. Linear optimization and approximation: an introduction to the theoretical analysis and numerical treatment of semi-infinite programs. New York: Springer-Verlag; 1983.
- Isakov V. Inverse problems for partial differential equations. New York: Springer-Verlag; 1998.
- Dautray R, Lions JL. Mathematical analysis and numerical methods for science and technology. Vol 2. Berlin: Springer; 1988.