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Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map
PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem
1. | Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shan'xi, China, China |
2. | LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China |
3. | LSEC, NCMIS, Institute of Systems Science, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China |
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Problems, 19 (2003), 973-984.
doi: 10.1088/0266-5611/19/4/312. |
[3] |
G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion, J. Comp. Appl. Math., 198 (2007), 307-320.
doi: 10.1016/j.cam.2005.06.048. |
[4] |
S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Problems, 12 (1996), 553-563.
doi: 10.1088/0266-5611/12/5/002. |
[5] |
M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336.
doi: 10.1088/0266-5611/22/4/012. |
[6] |
F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory, Inverse Problems, 21 (2005), 1915-1936.
doi: 10.1088/0266-5611/21/6/008. |
[7] |
M. Berggren, Approximation of very weak solutions to boundary value problems, SIAM J. Numer. Anal., 42 (2004), 860-877.
doi: 10.1137/S0036142903382048. |
[8] |
L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016, 21pp.
doi: 10.1088/0266-5611/26/9/095016. |
[9] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, $3^{th}$ edition, Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[10] |
M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal., 35 (2001), 595-605.
doi: 10.1051/m2an:2001128. |
[11] |
E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611.
doi: 10.1137/050626600. |
[12] |
A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems, 22 (2006), 1191-1206.
doi: 10.1088/0266-5611/22/4/005. |
[13] |
J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, Z. Angew. Math. Mech., 81 (2001), 665-674.
doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V. |
[14] |
P. G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.
doi: 10.1137/1.9780898719208. |
[15] |
A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570.
doi: 10.1088/0266-5611/17/3/313. |
[16] |
F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology, Calcolo, 16 (1979), 459-538.
doi: 10.1007/BF02576643. |
[17] |
K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains, SIAM J. Control Optim., 48 (2009), 2798-2819.
doi: 10.1137/080735369. |
[18] |
H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861-884.
doi: 10.1081/NFA-100108313. |
[19] |
D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim., 12 (1991), 299-314.
doi: 10.1080/01630569108816430. |
[20] |
A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements, J. Ind. Univ. Math., 38 (1989), 527-556.
doi: 10.1512/iumj.1989.38.38025. |
[21] |
A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation, Tr. Mosk. Mat. Obs., 52 (1991), 139-176. |
[22] |
H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations, J. Inverse Ill-Posed Problems, 5 (1997), 437-454.
doi: 10.1515/jiip.1997.5.5.437. |
[23] |
W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675.
doi: 10.1088/0266-5611/22/5/008. |
[24] |
W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations, Inverse Problems, 23 (2007), 2401-2415.
doi: 10.1088/0266-5611/23/6/008. |
[25] |
M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.
doi: 10.1007/s10589-005-4559-5. |
[26] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009. |
[27] |
G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994.
doi: 10.1088/0266-5611/13/4/006. |
[28] |
V. Isakov, Inverse Problems for Partial Differential Equations, $2^{nd}$ edition, Springer, New York, 2006. |
[29] |
M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation, Appl. Anal., 81 (2002), 1065-1083.
doi: 10.1080/0003681021000029819. |
[30] |
S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed problems, 3 (1995), 21-46.
doi: 10.1515/jiip.1995.3.1.21. |
[31] |
M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675.
doi: 10.1137/0151085. |
[32] |
R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667.
doi: 10.1002/cpa.3160380513. |
[33] |
V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45-52. |
[34] |
R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications], Travaux et Recherches Mathématiques, No. 15, Dunod, Paris, 1967. |
[35] |
R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().
|
[36] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971. |
[37] |
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York-Heidelberg, 1972. |
[38] |
M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting, in Industrial Engineering and Systems Management (IESM), Proceedings of 2013 International Conference on, Rabat, Morocco, 2013, p431. |
[39] |
S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces, Numer. Funct. Anal. Optim., 21 (2000), 901-916.
doi: 10.1080/01630560008816993. |
[40] |
N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995. |
[41] |
A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington D.C., 1977. |
[42] |
W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body, Inverse Problems, 17 (2001), 1957-1975.
doi: 10.1088/0266-5611/17/6/325. |
[43] |
K. Yosida, Functional Analysis, $5^{th}$ edition, Springer-Verlag, Berlin-New York, 1978. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Problems, 19 (2003), 973-984.
doi: 10.1088/0266-5611/19/4/312. |
[3] |
G. Alessandrini and E. Sincich, Solving elliptic Cauchy problems and the identification of nonlinear corrosion, J. Comp. Appl. Math., 198 (2007), 307-320.
doi: 10.1016/j.cam.2005.06.048. |
[4] |
S. Andrieux and A. A. Ben, Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Problems, 12 (1996), 553-563.
doi: 10.1088/0266-5611/12/5/002. |
[5] |
M. Azaïez, F. B. Belgacem and H. E. Fekih, On Cauchy's problem: II. Completion, regularization and approximation, Inverse Problems, 22 (2006), 1307-1336.
doi: 10.1088/0266-5611/22/4/012. |
[6] |
F. B. Belgacem and H. E. Fekih, On Cauchy's problem: I. A variational Steklov-Poincaré theory, Inverse Problems, 21 (2005), 1915-1936.
doi: 10.1088/0266-5611/21/6/008. |
[7] |
M. Berggren, Approximation of very weak solutions to boundary value problems, SIAM J. Numer. Anal., 42 (2004), 860-877.
doi: 10.1137/S0036142903382048. |
[8] |
L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016, 21pp.
doi: 10.1088/0266-5611/26/9/095016. |
[9] |
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, $3^{th}$ edition, Springer, New York, 2008.
doi: 10.1007/978-0-387-75934-0. |
[10] |
M. Brühl, M. Hanke and M. Pidcock, Crack detection using electrostatic measurements, M2AN Math. Model. Numer. Anal., 35 (2001), 595-605.
doi: 10.1051/m2an:2001128. |
[11] |
E. Casas and J. P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611.
doi: 10.1137/050626600. |
[12] |
A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems, 22 (2006), 1191-1206.
doi: 10.1088/0266-5611/22/4/005. |
[13] |
J. Cheng, Y. Hon, T. Wei and M. Yamamoto, Numerical computation of a Cauchy problem for Laplace's equation, Z. Angew. Math. Mech., 81 (2001), 665-674.
doi: 10.1002/1521-4001(200110)81:10<665::AID-ZAMM665>3.0.CO;2-V. |
[14] |
P. G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.
doi: 10.1137/1.9780898719208. |
[15] |
A. Cimetière, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization, Inverse Problems, 17 (2001), 553-570.
doi: 10.1088/0266-5611/17/3/313. |
[16] |
F. P. Colli and E. Magenes, On the inverse potential problem of electrocardiology, Calcolo, 16 (1979), 459-538.
doi: 10.1007/BF02576643. |
[17] |
K. Deckelnick, A. Günther and M. Hinze, Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains, SIAM J. Control Optim., 48 (2009), 2798-2819.
doi: 10.1137/080735369. |
[18] |
H. W. Engl and A. Leitão, A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim., 22 (2001), 861-884.
doi: 10.1081/NFA-100108313. |
[19] |
D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim., 12 (1991), 299-314.
doi: 10.1080/01630569108816430. |
[20] |
A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements, J. Ind. Univ. Math., 38 (1989), 527-556.
doi: 10.1512/iumj.1989.38.38025. |
[21] |
A. V. Fursikov, The Cauchy problem for a second-order elliptic equation in a conditionally well-posed formulation, Tr. Mosk. Mat. Obs., 52 (1991), 139-176. |
[22] |
H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations, J. Inverse Ill-Posed Problems, 5 (1997), 437-454.
doi: 10.1515/jiip.1997.5.5.437. |
[23] |
W. Han, W. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Problems, 22 (2006), 1659-1675.
doi: 10.1088/0266-5611/22/5/008. |
[24] |
W. Han, J. Huang, K. Kazmi and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations, Inverse Problems, 23 (2007), 2401-2415.
doi: 10.1088/0266-5611/23/6/008. |
[25] |
M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.
doi: 10.1007/s10589-005-4559-5. |
[26] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Springer, New York, 2009. |
[27] |
G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994.
doi: 10.1088/0266-5611/13/4/006. |
[28] |
V. Isakov, Inverse Problems for Partial Differential Equations, $2^{nd}$ edition, Springer, New York, 2006. |
[29] |
M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation, Appl. Anal., 81 (2002), 1065-1083.
doi: 10.1080/0003681021000029819. |
[30] |
S. I. Kabanikhin and A. L. Karchevsky, Optimizational method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed problems, 3 (1995), 21-46.
doi: 10.1515/jiip.1995.3.1.21. |
[31] |
M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675.
doi: 10.1137/0151085. |
[32] |
R. V. Kohn and M. S. Vogelius, Determining conductivity by boundary measurements: II. Interior results, Comm. Pure Appl. Math., 38 (1985), 643-667.
doi: 10.1002/cpa.3160380513. |
[33] |
V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, U.S.S.R. Comput. Math. and Math. Phys., 31 (1991), 45-52. |
[34] |
R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications], Travaux et Recherches Mathématiques, No. 15, Dunod, Paris, 1967. |
[35] |
R. Li and W. Liu, The C++ software library AFEPack., Available from: , ().
|
[36] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971. |
[37] |
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York-Heidelberg, 1972. |
[38] |
M. Maher, K. Moez and T. Anis, A Cauchy problem for an inverse problem in image inpainting, in Industrial Engineering and Systems Management (IESM), Proceedings of 2013 International Conference on, Rabat, Morocco, 2013, p431. |
[39] |
S. Pereverzev and E. Schock, Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces, Numer. Funct. Anal. Optim., 21 (2000), 901-916.
doi: 10.1080/01630560008816993. |
[40] |
N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin, 1995. |
[41] |
A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems, Winston and Sons, Washington D.C., 1977. |
[42] |
W. Weikl, H. Andra and E. Schnack, An alternating iterative algorithm for the reconstruction of internal cracks in a three-dimensional solid body, Inverse Problems, 17 (2001), 1957-1975.
doi: 10.1088/0266-5611/17/6/325. |
[43] |
K. Yosida, Functional Analysis, $5^{th}$ edition, Springer-Verlag, Berlin-New York, 1978. |
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