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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-3172-1.

[9]

H. Brezis, Analyse fonctionnelle, Théorie et applications,, Masson, (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). 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. 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). doi: 10.1088/0266-5611/26/8/085001.

[13]

P.-G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978).

[14]

G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems,, Elect. J. of Diff. Eqns., 8 (1994), 1.

[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.

[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.

[17]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements,, Springer, (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). doi: 10.1137/110855703.

[19]

L. Hörmander, Linear Partial Differential Operators,, Springer Verlag, (1976).

[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.

[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.

[22]

V. Isakov, Inverse obstacle problems,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123002.

[23]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems,, Applied Mathematical Sciences, (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, (2004). doi: 10.1515/9783110915549.

[25]

R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications,, Dunod, (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. doi: 10.1007/s00607-004-0109-8.

[27]

J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Vol. 2, Dunod, (1968).

[28]

L. E. Payne, Improperly Posed Problems in Partial Differential Equations,, SIAM, (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. 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, 5 (2012), 337.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-3172-1.

[9]

H. Brezis, Analyse fonctionnelle, Théorie et applications,, Masson, (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). 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. 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). doi: 10.1088/0266-5611/26/8/085001.

[13]

P.-G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (1978).

[14]

G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well-posed problems,, Elect. J. of Diff. Eqns., 8 (1994), 1.

[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.

[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.

[17]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements,, Springer, (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). doi: 10.1137/110855703.

[19]

L. Hörmander, Linear Partial Differential Operators,, Springer Verlag, (1976).

[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.

[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.

[22]

V. Isakov, Inverse obstacle problems,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123002.

[23]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems,, Applied Mathematical Sciences, (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, (2004). doi: 10.1515/9783110915549.

[25]

R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications,, Dunod, (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. doi: 10.1007/s00607-004-0109-8.

[27]

J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Vol. 2, Dunod, (1968).

[28]

L. E. Payne, Improperly Posed Problems in Partial Differential Equations,, SIAM, (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. 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, 5 (2012), 337.

[1]

Laurent Bourgeois, Jérémi Dardé. A quasi-reversibility approach to solve the inverse obstacle problem. Inverse Problems & Imaging, 2010, 4 (3) : 351-377. doi: 10.3934/ipi.2010.4.351

[2]

Jérémi Dardé. Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems. Inverse Problems & Imaging, 2016, 10 (2) : 379-407. doi: 10.3934/ipi.2016005

[3]

Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277

[4]

Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073

[5]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[6]

Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643

[7]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[8]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617

[9]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

[10]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154

[11]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489

[12]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[13]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469

[14]

Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2018, 22 (11) : 1-16. doi: 10.3934/dcdsb.2018109

[15]

Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459

[16]

Roman Chapko, B. Tomas Johansson. An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions. Inverse Problems & Imaging, 2008, 2 (3) : 317-333. doi: 10.3934/ipi.2008.2.317

[17]

Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

[18]

Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131

[19]

Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025

[20]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (6)

[Back to Top]