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Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case
1. | Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, France, France |
2. | Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9, France |
References:
[1] |
K. A. Ames and L. E. Payne, Continuous dependence on modeling for some well-posed perturbations of backward heat equation, J. of Inequal. & Appl., 3 (1999), 51-64.
doi: 10.1155/S1025583499000041. |
[2] |
M. Bonnet, Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain, Comput. Methods Appl. Mech. Engrg., 195 (2006), 5239-5254.
doi: 10.1016/j.cma.2005.10.026. |
[3] |
L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104.
doi: 10.1088/0266-5611/21/3/018. |
[4] |
L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains, M2AN, 44 (2010), 715-735.
doi: 10.1051/m2an/2010016. |
[5] |
L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Problems and Imaging, 4 (2010), 351-377.
doi: 10.3934/ipi.2010.4.351. |
[6] |
L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system, Inverse Problems and Imaging, 8 (2014), 23-51.
doi: 10.3934/ipi.2014.8.23. |
[7] |
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. |
[8] |
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New-York, 1991.
doi: 10.1007/978-1-4612-3172-1. |
[9] |
H. Brezis, Analyse fonctionnelle, Théorie et applications, Masson, Paris, 1983. |
[10] |
M. de Buhan and M. Kray, A new approach to solve the inverse scattering problem for waves: combining the TRAC and the adaptive inverse methods, Inverse Problems, 29 (2013), 085009 (24pp).
doi: 10.1088/0266-5611/29/8/085009. |
[11] |
R. Chapko, R. Kress and J-R Yoon, On the numerical solution of an inverse boundary value problem for the heat equation, Inverse Problems, 14 (1998), 853-867.
doi: 10.1088/0266-5611/14/4/006. |
[12] |
Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001 (17pp).
doi: 10.1088/0266-5611/26/8/085001. |
[13] |
P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. |
[14] |
G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Elect. J. of Diff. Eqns., 8 (1994), 1-9. |
[15] |
C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comp., 30 (2008), 1-23.
doi: 10.1137/06066970X. |
[16] |
J. Dardé, A. Hannukainen and N. Hyvönen, An $H_\text{div}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Num. Anal., 51 (2013), 2123-2148.
doi: 10.1137/120895123. |
[17] |
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer, New-York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
[18] |
H. Harbrecht and J. Tausch, On the numerical solution of a shape optimization problem for the heat equation, SIAM J. Sci. Comput., 35 (2013), A104-A121.
doi: 10.1137/110855703. |
[19] |
L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. |
[20] |
M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval, Inverse Problems, 26 (2010), 055010 (20pp).
doi: 10.1088/0266-5611/26/5/055010. |
[21] |
M. Ikehata and M. Kawashita, The enclosure method for the heat equation, Inverse Problems, 25 (2009), 075005 (10pp).
doi: 10.1088/0266-5611/25/7/075005. |
[22] |
V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002 (18pp).
doi: 10.1088/0266-5611/25/12/123002. |
[23] |
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-5338-9. |
[24] |
M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Inverse and Ill-Posed Problems), VSP, Utrecht, 2004.
doi: 10.1515/9783110915549. |
[25] |
R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967. |
[26] |
C. D. Lines and S. N. Chandler-Wilde, A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.
doi: 10.1007/s00607-004-0109-8. |
[27] |
J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Vol. 2 Dunod, Paris, 1968. |
[28] |
L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, 1975. |
[29] |
K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its application, J. Eur. Math. Soc., 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[30] |
R. E. Puzyrev and A. A. Shlapunov, On an ill-posed problem for the heat equation, Journal of Siberian Federal University, Mathematical & Physics, 5 (2012), 337-348. |
show all references
References:
[1] |
K. A. Ames and L. E. Payne, Continuous dependence on modeling for some well-posed perturbations of backward heat equation, J. of Inequal. & Appl., 3 (1999), 51-64.
doi: 10.1155/S1025583499000041. |
[2] |
M. Bonnet, Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain, Comput. Methods Appl. Mech. Engrg., 195 (2006), 5239-5254.
doi: 10.1016/j.cma.2005.10.026. |
[3] |
L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104.
doi: 10.1088/0266-5611/21/3/018. |
[4] |
L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains, M2AN, 44 (2010), 715-735.
doi: 10.1051/m2an/2010016. |
[5] |
L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Problems and Imaging, 4 (2010), 351-377.
doi: 10.3934/ipi.2010.4.351. |
[6] |
L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system, Inverse Problems and Imaging, 8 (2014), 23-51.
doi: 10.3934/ipi.2014.8.23. |
[7] |
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. |
[8] |
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New-York, 1991.
doi: 10.1007/978-1-4612-3172-1. |
[9] |
H. Brezis, Analyse fonctionnelle, Théorie et applications, Masson, Paris, 1983. |
[10] |
M. de Buhan and M. Kray, A new approach to solve the inverse scattering problem for waves: combining the TRAC and the adaptive inverse methods, Inverse Problems, 29 (2013), 085009 (24pp).
doi: 10.1088/0266-5611/29/8/085009. |
[11] |
R. Chapko, R. Kress and J-R Yoon, On the numerical solution of an inverse boundary value problem for the heat equation, Inverse Problems, 14 (1998), 853-867.
doi: 10.1088/0266-5611/14/4/006. |
[12] |
Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain, Inverse Problems, 26 (2010), 085001 (17pp).
doi: 10.1088/0266-5611/26/8/085001. |
[13] |
P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. |
[14] |
G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems, Elect. J. of Diff. Eqns., 8 (1994), 1-9. |
[15] |
C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comp., 30 (2008), 1-23.
doi: 10.1137/06066970X. |
[16] |
J. Dardé, A. Hannukainen and N. Hyvönen, An $H_\text{div}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Num. Anal., 51 (2013), 2123-2148.
doi: 10.1137/120895123. |
[17] |
A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer, New-York, 2004.
doi: 10.1007/978-1-4757-4355-5. |
[18] |
H. Harbrecht and J. Tausch, On the numerical solution of a shape optimization problem for the heat equation, SIAM J. Sci. Comput., 35 (2013), A104-A121.
doi: 10.1137/110855703. |
[19] |
L. Hörmander, Linear Partial Differential Operators, Springer Verlag, Berlin-New York, 1976. |
[20] |
M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval, Inverse Problems, 26 (2010), 055010 (20pp).
doi: 10.1088/0266-5611/26/5/055010. |
[21] |
M. Ikehata and M. Kawashita, The enclosure method for the heat equation, Inverse Problems, 25 (2009), 075005 (10pp).
doi: 10.1088/0266-5611/25/7/075005. |
[22] |
V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002 (18pp).
doi: 10.1088/0266-5611/25/12/123002. |
[23] |
A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Applied Mathematical Sciences, 120, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-5338-9. |
[24] |
M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Inverse and Ill-Posed Problems), VSP, Utrecht, 2004.
doi: 10.1515/9783110915549. |
[25] |
R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967. |
[26] |
C. D. Lines and S. N. Chandler-Wilde, A time domain point source method for inverse scattering by rough surfaces, Computing, 75 (2005), 157-180.
doi: 10.1007/s00607-004-0109-8. |
[27] |
J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Vol. 2 Dunod, Paris, 1968. |
[28] |
L. E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Philadelphia, 1975. |
[29] |
K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its application, J. Eur. Math. Soc., 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[30] |
R. E. Puzyrev and A. A. Shlapunov, On an ill-posed problem for the heat equation, Journal of Siberian Federal University, Mathematical & Physics, 5 (2012), 337-348. |
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