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November  2015, 9(4): 971-1002. doi: 10.3934/ipi.2015.9.971

Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case

Eliane Bécache 1, , Laurent Bourgeois 1, , Lucas Franceschini 1, and  Jérémi Dardé 2,

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

Received  November 2014 Revised  June 2015 Published  October 2015

In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical Lagrange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations.
Keywords: heat/wave equation with lateral Cauchy data, quasi-reversibility method, let-set method, inverse obstacle problem, Backward heat equation, mixed formulation., finite element method.
Mathematics Subject Classification: Primary: 35R25, 35R30, 35R35; Secondary: 65M6.
Citation: Eliane Bécache, Laurent Bourgeois, Lucas Franceschini, Jérémi Dardé. Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case. Inverse Problems and Imaging, 2015, 9 (4) : 971-1002. doi: 10.3934/ipi.2015.9.971
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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