no. 1
A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 123-145.

This paper is dedicated to a new way of presenting the Tikhonov regularization in the form of a mixed formulation. Such formulation is well adapted to the regularization of linear ill-posed partial differential equations because when it comes to discretization, the mixed formulation enables us to use some standard finite elements. As an application of our theory, we consider an inverse obstacle problem in an acoustic waveguide. In order to solve it we use the so-called “exterior approach”, which couples the mixed formulation of Tikhonov regularization and a level set method. Some 2d numerical experiments show the feasibility of our approach.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2018008
Classification : 35J05, 35R25, 35R30, 35R35
Mots clés : Inverse obstacle problem, acoustic waveguide, Tikhonov regularization, mixed formulation, quasi-reversibility, level set method
Bourgeois, Laurent 1 ; Recoquillay, Arnaud 1

1 
@article{M2AN_2018__52_1_123_0,
     author = {Bourgeois, Laurent and Recoquillay, Arnaud},
     title = {A mixed formulation of the {Tikhonov} regularization and its application to inverse {PDE} problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {123--145},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {1},
     year = {2018},
     doi = {10.1051/m2an/2018008},
     zbl = {1397.35074},
     mrnumber = {3808155},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2018008/}
}
TY  - JOUR
AU  - Bourgeois, Laurent
AU  - Recoquillay, Arnaud
TI  - A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 123
EP  - 145
VL  - 52
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2018008/
DO  - 10.1051/m2an/2018008
LA  - en
ID  - M2AN_2018__52_1_123_0
ER  - 
%0 Journal Article
%A Bourgeois, Laurent
%A Recoquillay, Arnaud
%T A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 123-145
%V 52
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2018008/
%R 10.1051/m2an/2018008
%G en
%F M2AN_2018__52_1_123_0
Bourgeois, Laurent; Recoquillay, Arnaud. A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 123-145. doi : 10.1051/m2an/2018008. http://www.numdam.org/articles/10.1051/m2an/2018008/

[1] V. Baronian, L. Bourgeois and A. Recoquillay, Imaging an acoustic waveguide from surface data in the time domain. Wave Motion 66 (2016) 68–87. | DOI | MR | Zbl

[2] E. Bécache, L. Bourgeois, L. Franceschini and J. Dardé, Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1D case. Inverse Probl. Imaging 9 (2015) 971–1002. | MR | Zbl

[3] F.B. Belgacem, Why is the Cauchy problem severely ill-posed? Inverse Probl. 23 (2007) 823–836. | DOI | MR | Zbl

[4] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Probl. 21 (2005) 1087–1104. | DOI | MR | Zbl

[5] L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem. Inverse Probl. Imaging 4 (2010) 351–377. | DOI | MR | Zbl

[6] L. Bourgeois and J. Dardé, The “exterior approach” to solve the inverse obstacle problem for the Stokes system. Inverse Probl. Imaging 8 (2014) 23–51. | DOI | MR | Zbl

[7] L. Bourgeois and J. Dardé, The “exterior approach” applied to the inverse obstacle problem for the heat equation. SIAM J. Numer. Anal. 55 (2017) 1820–1842. | DOI | MR | Zbl

[8] L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: a modal formulation. Inverse Probl. 24 (2008) 015018. | DOI | MR | Zbl

[9] L. Bourgeois, C. Chambeyron and S. Kusiak, Locating an obstacle in a 3d finite depth ocean using the convex scattering support. Special Issue: The Seventh International Conference on Mathematical and Numerical Aspects of Waves (WAVES05). J. Comput. Appl. Math. 204 (2007) 387–399. | DOI | MR | Zbl

[10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). | MR | Zbl

[11] M. Burger and S. J. Osher, A survey on level set methods for inverse problems and optimal design. Eur. J. Appl. Math. 16 (2005) 263–301. | DOI | MR | Zbl

[12] E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput. 35 (2013) A2752–A2780. | DOI | MR | Zbl

[13] E. Burman, The elliptic Cauchy problem revisited: control of boundary data in natural norms. C. R. Math. Acad. Sci. Paris 355 (2017) 479–484. | DOI | MR | Zbl

[14] E. Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem. Math. Comp. 86 (2017) 75–96. | DOI | MR | Zbl

[15] A. Charalambopoulos, D. Gintides, K. Kiriaki and A. Kirsch, The factorization method for an acoustic wave guide, in Mathematical Methods in Scattering Theory and Biomedical Engineering. World Sci. Publ., Hackensack, NJ (2006) 120–127. | DOI | MR | Zbl

[16] N. Cîndea and A. Münch, Inverse problems for linear hyperbolic equations using mixed formulations. Inverse Probl. 31 (2015) 075001. | DOI | MR | Zbl

[17] C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. 30 (2007/08) 1–23. | DOI | MR | Zbl

[18] J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems. Inverse Probl. Imaging 10 (2016) 379–407. | DOI | MR | Zbl

[19] J. Dardé, A. Hannukainen and N. Hyvönen, An Hdiv-based mixed quasi-reversibility method for solving elliptic Cauchy problems. SIAM J. Numer. Anal. 51 (2013) 2123–2148. | DOI | MR | Zbl

[20] A. Ern and J.-L. Guermond, in Theory and Practice of Finite Elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). | DOI | MR | Zbl

[21] A. Henrot and M. Pierre, in Variation et optimisation de formes: une analyse géométrique. [A geometric analysis]. Vol. 48 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin (2005). | DOI | MR | Zbl

[22] V. Isakov, in Inverse Problems for Partial Differential Equations. Vol. 127 of Applied Mathematical Sciences, second edn. Springer, New York (2006). | MR | Zbl

[23] K. Ito and B. Jin, Inverse Problems: Tikhonov Theory and Algorithms. Vol. 22 of Series on Applied Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2015). | MR | Zbl

[24] A. Kirsch, in An Introduction to the Mathematical Theory of Inverse Problems. Vol. 120 Applied Mathematical Sciences. Springer-Verlag, New York (1996). | DOI | MR | Zbl

[25] M.V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM J. Appl. Math. 51 (1991) 1653–1675. | DOI | MR | Zbl

[26] R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications, in Travaux et Recherches Mathématiques. No. 15, Dunod, Paris (1967). | MR | Zbl

[27] K. Liu, Y. Xu and J. Zou, Imaging wave-penetrable objects in a finite depth ocean. Appl. Math. Comput. 235 (2014) 364–376. | MR | Zbl

[28] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000). | MR | Zbl

[29] P. Monk and V. Selgas, An inverse acoustic waveguide problem in the time domain. Inverse Probl. 32 (2016) 055001. | DOI | MR | Zbl

[30] P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering. Proc. Am. Math. Soc. 132 (2004) 1351–1354. | DOI | MR | Zbl

Cité par Sources :