The "exterior approach" to solve the inverse obstacle problem for the Stokes system

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February 2014, 8(1): 23-51. doi: 10.3934/ipi.2014.8.23

The "exterior approach" to solve the inverse obstacle problem for the Stokes system

Laurent Bourgeois ^1, and Jérémi Dardé ^2,

1.	Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, France
2.	Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9, France

Received January 2013 Revised June 2013 Published March 2014

Abstract
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We apply an ``exterior approach" based on the coupling of a method of quasi-reversibility and of a level set method in order to recover a fixed obstacle immersed in a Stokes flow from boundary measurements. Concerning the method of quasi-reversibility, two new mixed formulations are introduced in order to solve the ill-posed Cauchy problems for the Stokes system by using some classical conforming finite elements. We provide some proofs for the convergence of the quasi-reversibility methods on the one hand and of the level set method on the other hand. Some numerical experiments in $2D$ show the efficiency of the two mixed formulations and of the exterior approach based on one of them.

Keywords: quasi-reversibility method, level set method, inverse obstacle problem., Stokes system, finite element method.

Mathematics Subject Classification: 35A15, 35M30, 35R25, 35R30, 35R35, 65M6.

Citation: Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems and Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23

References:

[1]	C. Fabre and G. Lebeau, Prolongement unique des solutions de Stokes, Commun. Partial Differ. Eq., 21 (1996), 573-596. doi: 10.1080/03605309608821198.
[2]	C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273-1290. doi: 10.3934/dcds.2010.28.1273.
[3]	M. Boulakia, A.-C. Egloffe and C. Grandmont, Stability estimates for a Robin coefficient in the two-dimensional Stokes system, Mathematical Control and Related Fields, 3 (2013), 21-49. doi: 10.3934/mcrf.2013.3.21.
[4]	A. L. Bukhgeim, Extension of solutions of elliptic equations from discrete sets, J. Inverse Ill-Posed Probl., 1 (1993), 17-32. doi: 10.1515/jiip.1993.1.1.17.
[5]	J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.
[6]	A. Ben Abda, I. Ben Saad and M. Hassine, Data completion for the Stokes system, CRAS Mécanique, 337 (2009), 703-708.
[7]	C. Alvarez, C. Conca, L. Fritz and O. Kavian, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003.
[8]	A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015. doi: 10.1088/0266-5611/26/12/125015.
[9]	N. F. M. Martins and A. L. Silvestre, An iterative MFS approach for the detection of immersed obstacles, Engineering Analysis with Boundary Elements, 32 (2008), 517-524. doi: 10.1016/j.enganabound.2007.10.011.
[10]	C. Alvarez, C. Conca, R. Lecaros and J. H. Ortega, On the identification of a rigid body immersed in a fluid: A numerical approach, Engineering Analysis with Boundary Elements, 32 (2008), 919-925. doi: 10.1016/j.enganabound.2007.02.007.
[11]	M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660.
[12]	F. Caubet, M. Dambrine, D. Kateb and C. D. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Problems and Imaging, 7 (2013), 123-157. doi: 10.3934/ipi.2013.7.123.
[13]	A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control and Optimization, 48 (2009), 2871-2900. doi: 10.1137/070704332.
[14]	F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems, 28 (2012), 105007. doi: 10.1088/0266-5611/28/10/105007.
[15]	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.
[16]	J. Dardé, The exterior approach: A new framework to solve inverse obstacle problems, Inverse Problems, 28 (2012), 015008. doi: 10.1088/0266-5611/28/1/015008.
[17]	C. Conca, P. Cumsille, J. Ortega and L. Rosier, On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Problems, 24 (2008), 045001. doi: 10.1088/0266-5611/24/4/045001.
[18]	C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010. doi: 10.1088/0266-5611/26/9/095010.
[19]	C. Conca, E. Schwindt and T. Takahashi, On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems, 28 (2012), 015005. doi: 10.1088/0266-5611/28/1/015005.
[20]	L. Bourgeois and J. Dardé, About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case, Applicable Analysis, 90 (2011), 1481-1497. doi: 10.1080/00036811.2010.549481.
[21]	H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Dunod, Paris, 1983.
[22]	R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967.
[23]	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: 10.1137/0151085.
[24]	P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978.
[25]	W. Ming and J. Xu, The Morley element for fourth order elliptic equations in any dimensions, Numerische Mathematik, 103 (2006), 155-169. doi: 10.1007/s00211-005-0662-x.
[26]	G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.
[27]	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.
[28]	J. Dardé, A. Hannukaiinen and N. Hyvönen, An $H_{ d i v}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Num. Anal., 51 (2013), 2123-2148.
[29]	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. doi: 10.1088/0266-5611/26/9/095016.
[30]	I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels, Dunod, 1974.
[31]	S. Osher and J. A. Sethian, Front propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.
[32]	A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, Springer, Paris, 2005.
[33]	F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New-York, 1991. doi: 10.1007/978-1-4612-3172-1.
[34]	F. Hecht, A. Le Hyaric, J. Morice, K. Ohtsuka and O. Pironneau, Freefem++ Manual, http://www.freefem.org/ff++/ftp/freefem++doc.pdf, 2012.
[35]	V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, 1979.

show all references

References:

[1]	C. Fabre and G. Lebeau, Prolongement unique des solutions de Stokes, Commun. Partial Differ. Eq., 21 (1996), 573-596. doi: 10.1080/03605309608821198.
[2]	C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273-1290. doi: 10.3934/dcds.2010.28.1273.
[3]	M. Boulakia, A.-C. Egloffe and C. Grandmont, Stability estimates for a Robin coefficient in the two-dimensional Stokes system, Mathematical Control and Related Fields, 3 (2013), 21-49. doi: 10.3934/mcrf.2013.3.21.
[4]	A. L. Bukhgeim, Extension of solutions of elliptic equations from discrete sets, J. Inverse Ill-Posed Probl., 1 (1993), 17-32. doi: 10.1515/jiip.1993.1.1.17.
[5]	J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.
[6]	A. Ben Abda, I. Ben Saad and M. Hassine, Data completion for the Stokes system, CRAS Mécanique, 337 (2009), 703-708.
[7]	C. Alvarez, C. Conca, L. Fritz and O. Kavian, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003.
[8]	A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015. doi: 10.1088/0266-5611/26/12/125015.
[9]	N. F. M. Martins and A. L. Silvestre, An iterative MFS approach for the detection of immersed obstacles, Engineering Analysis with Boundary Elements, 32 (2008), 517-524. doi: 10.1016/j.enganabound.2007.10.011.
[10]	C. Alvarez, C. Conca, R. Lecaros and J. H. Ortega, On the identification of a rigid body immersed in a fluid: A numerical approach, Engineering Analysis with Boundary Elements, 32 (2008), 919-925. doi: 10.1016/j.enganabound.2007.02.007.
[11]	M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660.
[12]	F. Caubet, M. Dambrine, D. Kateb and C. D. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Problems and Imaging, 7 (2013), 123-157. doi: 10.3934/ipi.2013.7.123.
[13]	A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control and Optimization, 48 (2009), 2871-2900. doi: 10.1137/070704332.
[14]	F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems, 28 (2012), 105007. doi: 10.1088/0266-5611/28/10/105007.
[15]	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.
[16]	J. Dardé, The exterior approach: A new framework to solve inverse obstacle problems, Inverse Problems, 28 (2012), 015008. doi: 10.1088/0266-5611/28/1/015008.
[17]	C. Conca, P. Cumsille, J. Ortega and L. Rosier, On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Problems, 24 (2008), 045001. doi: 10.1088/0266-5611/24/4/045001.
[18]	C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010. doi: 10.1088/0266-5611/26/9/095010.
[19]	C. Conca, E. Schwindt and T. Takahashi, On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems, 28 (2012), 015005. doi: 10.1088/0266-5611/28/1/015005.
[20]	L. Bourgeois and J. Dardé, About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case, Applicable Analysis, 90 (2011), 1481-1497. doi: 10.1080/00036811.2010.549481.
[21]	H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Dunod, Paris, 1983.
[22]	R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967.
[23]	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: 10.1137/0151085.
[24]	P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978.
[25]	W. Ming and J. Xu, The Morley element for fourth order elliptic equations in any dimensions, Numerische Mathematik, 103 (2006), 155-169. doi: 10.1007/s00211-005-0662-x.
[26]	G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.
[27]	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.
[28]	J. Dardé, A. Hannukaiinen and N. Hyvönen, An $H_{ d i v}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Num. Anal., 51 (2013), 2123-2148.
[29]	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. doi: 10.1088/0266-5611/26/9/095016.
[30]	I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels, Dunod, 1974.
[31]	S. Osher and J. A. Sethian, Front propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.
[32]	A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, Springer, Paris, 2005.
[33]	F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New-York, 1991. doi: 10.1007/978-1-4612-3172-1.
[34]	F. Hecht, A. Le Hyaric, J. Morice, K. Ohtsuka and O. Pironneau, Freefem++ Manual, http://www.freefem.org/ff++/ftp/freefem++doc.pdf, 2012.
[35]	V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, 1979.

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