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Nomonotone spectral gradient method for sparse recovery
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] | |
[2] |
G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse Problems, 19 (2003), 973.
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.
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.
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.
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.
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.
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).
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, (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.
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.
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.
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.
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, (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.
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.
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.
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.
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.
doi: 10.1080/01630569108816430. |
[20] |
A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements,, J. Ind. Univ. Math., 38 (1989), 527.
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.
|
[22] |
H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations,, J. Inverse Ill-Posed Problems, 5 (1997), 437.
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.
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.
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.
doi: 10.1007/s10589-005-4559-5. |
[26] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).
|
[27] |
G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977.
doi: 10.1088/0266-5611/13/4/006. |
[28] |
V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (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.
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.
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.
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.
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.
|
[34] |
R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications],, Travaux et Recherches Mathématiques, (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, (1971).
|
[37] |
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (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), (2013). |
[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.
doi: 10.1080/01630560008816993. |
[40] |
N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).
|
[41] |
A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (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.
doi: 10.1088/0266-5611/17/6/325. |
[43] |
show all references
References:
[1] | |
[2] |
G. Alessandrini, L. D. Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse Problems, 19 (2003), 973.
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.
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.
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.
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.
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.
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).
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, (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.
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.
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.
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.
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, (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.
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.
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.
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.
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.
doi: 10.1080/01630569108816430. |
[20] |
A. Friedman and M. S. Vogelius, Determining cracks by boundary measurements,, J. Ind. Univ. Math., 38 (1989), 527.
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.
|
[22] |
H. Han and H. J. Reinhardt, Some stability estimates for Cauchy problems for elliptic equations,, J. Inverse Ill-Posed Problems, 5 (1997), 437.
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.
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.
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.
doi: 10.1007/s10589-005-4559-5. |
[26] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints,, Springer, (2009).
|
[27] |
G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977.
doi: 10.1088/0266-5611/13/4/006. |
[28] |
V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (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.
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.
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.
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.
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.
|
[34] |
R. Lattès and J.-L. Lions, Méthode de Quasi-réversibilité et Applications, (French) [The Quasi-Reversibility Methods and Applications],, Travaux et Recherches Mathématiques, (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, (1971).
|
[37] |
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Vol. II, (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), (2013). |
[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.
doi: 10.1080/01630560008816993. |
[40] |
N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations,, Akademie Verlag, (1995).
|
[41] |
A. N. Tikhonov and V. A. Arsenin, Solutions of Ill-Posed Problems,, Winston and Sons, (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.
doi: 10.1088/0266-5611/17/6/325. |
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