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