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

Eliane  Bécache; Laurent Bourgeois; Lucas  Franceschini; Jérémi  Dardé

doi:10.3934/ipi.2015.9.971

Article Contents

2015, Volume 9, Issue 4: 971-1002. Doi: 10.3934/ipi.2015.9.971

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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
2.
Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9

Received: November 2014

Revised: June 2015

Published: November 2015

Abstract / Introduction Related Papers Cited by

Abstract

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:

Backward heat equation,
heat/wave equation with lateral Cauchy data,
inverse obstacle problem,
quasi-reversibility method,
let-set method,
finite element method,
mixed formulation.

Mathematics Subject Classification: Primary: 35R25, 35R30, 35R35; Secondary: 65M60.

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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.