# American Institute of Mathematical Sciences

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

 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.
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 & 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.  doi: 10.1155/S1025583499000041.  Google Scholar [2] M. Bonnet, Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain,, Comput. Methods Appl. Mech. Engrg., 195 (2006), 5239.  doi: 10.1016/j.cma.2005.10.026.  Google Scholar [3] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 21 (2005), 1087.  doi: 10.1088/0266-5611/21/3/018.  Google Scholar [4] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains,, M2AN, 44 (2010), 715.  doi: 10.1051/m2an/2010016.  Google Scholar [5] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem,, Inverse Problems and Imaging, 4 (2010), 351.  doi: 10.3934/ipi.2010.4.351.  Google Scholar [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.  doi: 10.3934/ipi.2014.8.23.  Google Scholar [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).  doi: 10.1088/0266-5611/26/9/095016.  Google Scholar [8] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-3172-1.  Google Scholar [9] H. Brezis, Analyse fonctionnelle, Théorie et applications,, Masson, (1983).   Google Scholar [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).  doi: 10.1088/0266-5611/29/8/085009.  Google Scholar [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.  doi: 10.1088/0266-5611/14/4/006.  Google Scholar [12] Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085001.  Google Scholar [13] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978).   Google Scholar [14] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems,, Elect. J. of Diff. Eqns., 8 (1994), 1.   Google Scholar [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.  doi: 10.1137/06066970X.  Google Scholar [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.  doi: 10.1137/120895123.  Google Scholar [17] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements,, Springer, (2004).  doi: 10.1007/978-1-4757-4355-5.  Google Scholar [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).  doi: 10.1137/110855703.  Google Scholar [19] L. Hörmander, Linear Partial Differential Operators,, Springer Verlag, (1976).   Google Scholar [20] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/5/055010.  Google Scholar [21] M. Ikehata and M. Kawashita, The enclosure method for the heat equation,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/7/075005.  Google Scholar [22] V. Isakov, Inverse obstacle problems,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/12/123002.  Google Scholar [23] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-5338-9.  Google Scholar [24] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Inverse and Ill-Posed Problems),, VSP, (2004).  doi: 10.1515/9783110915549.  Google Scholar [25] R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications,, Dunod, (1967).   Google Scholar [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.  doi: 10.1007/s00607-004-0109-8.  Google Scholar [27] J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Vol. 2, Dunod, (1968).   Google Scholar [28] L. E. Payne, Improperly Posed Problems in Partial Differential Equations,, SIAM, (1975).   Google Scholar [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.  doi: 10.4171/JEMS/371.  Google Scholar [30] R. E. Puzyrev and A. A. Shlapunov, On an ill-posed problem for the heat equation,, Journal of Siberian Federal University, 5 (2012), 337.   Google Scholar

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.  doi: 10.1155/S1025583499000041.  Google Scholar [2] M. Bonnet, Topological sensitivity for 3D elastodynamic and acoustic inverse scattering in the time domain,, Comput. Methods Appl. Mech. Engrg., 195 (2006), 5239.  doi: 10.1016/j.cma.2005.10.026.  Google Scholar [3] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation,, Inverse Problems, 21 (2005), 1087.  doi: 10.1088/0266-5611/21/3/018.  Google Scholar [4] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of C1,1 domains,, M2AN, 44 (2010), 715.  doi: 10.1051/m2an/2010016.  Google Scholar [5] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem,, Inverse Problems and Imaging, 4 (2010), 351.  doi: 10.3934/ipi.2010.4.351.  Google Scholar [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.  doi: 10.3934/ipi.2014.8.23.  Google Scholar [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).  doi: 10.1088/0266-5611/26/9/095016.  Google Scholar [8] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (1991).  doi: 10.1007/978-1-4612-3172-1.  Google Scholar [9] H. Brezis, Analyse fonctionnelle, Théorie et applications,, Masson, (1983).   Google Scholar [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).  doi: 10.1088/0266-5611/29/8/085009.  Google Scholar [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.  doi: 10.1088/0266-5611/14/4/006.  Google Scholar [12] Q. Chen, H. Haddar, A. Lechleiter and P. Monk, A sampling method for inverse scattering in the time domain,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/8/085001.  Google Scholar [13] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978).   Google Scholar [14] G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems,, Elect. J. of Diff. Eqns., 8 (1994), 1.   Google Scholar [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.  doi: 10.1137/06066970X.  Google Scholar [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.  doi: 10.1137/120895123.  Google Scholar [17] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements,, Springer, (2004).  doi: 10.1007/978-1-4757-4355-5.  Google Scholar [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).  doi: 10.1137/110855703.  Google Scholar [19] L. Hörmander, Linear Partial Differential Operators,, Springer Verlag, (1976).   Google Scholar [20] M. Ikehata, The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval,, Inverse Problems, 26 (2010).  doi: 10.1088/0266-5611/26/5/055010.  Google Scholar [21] M. Ikehata and M. Kawashita, The enclosure method for the heat equation,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/7/075005.  Google Scholar [22] V. Isakov, Inverse obstacle problems,, Inverse Problems, 25 (2009).  doi: 10.1088/0266-5611/25/12/123002.  Google Scholar [23] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-5338-9.  Google Scholar [24] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Inverse and Ill-Posed Problems),, VSP, (2004).  doi: 10.1515/9783110915549.  Google Scholar [25] R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications,, Dunod, (1967).   Google Scholar [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.  doi: 10.1007/s00607-004-0109-8.  Google Scholar [27] J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Vol. 2, Dunod, (1968).   Google Scholar [28] L. E. Payne, Improperly Posed Problems in Partial Differential Equations,, SIAM, (1975).   Google Scholar [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.  doi: 10.4171/JEMS/371.  Google Scholar [30] R. E. Puzyrev and A. A. Shlapunov, On an ill-posed problem for the heat equation,, Journal of Siberian Federal University, 5 (2012), 337.   Google Scholar
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