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Some geometric inverse problems for the linear wave equation
1. | University of Sevilla, Dpto. E.D.A.N., Aptdo. 1160, 41080, Sevilla, Spain, Spain |
References:
[1] |
G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements, Inverse Problems, 18 (2002), 1333-1353.
doi: 10.1088/0266-5611/18/5/308. |
[2] |
G. Alessandrini, A. Morassi and E. Rosset, Edi Size estimates, in Inverse Problems: Theory and Applications (Cortona/Pisa, 2002), Contemp. Math., 333, Amer. Math. Soc., Providence, RI, 2003, 1-33.
doi: 10.1090/conm/333/05951. |
[3] |
C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552.
doi: 10.1088/0266-5611/21/5/003. |
[4] |
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. |
[5] |
S. Andrieux, A. Ben Abda and M. Jaoua, On the inverse emergent plane crack problem, Math. Methods Appl. Sci., 21 (1998), 895-906.
doi: 10.1002/(SICI)1099-1476(19980710)21:10<895::AID-MMA975>3.0.CO;2-1. |
[6] |
S. Auliac, Développement d'outils d'optimisation pour FreeFem++, These, Université Pierre et Marie Curie - Paris VI, 2014. Available from: http://tel.archives-ouvertes.fr/tel-01001631. |
[7] |
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 Optim., 48 (): 2871.
doi: 10.1137/070704332. |
[8] |
E. G. Birgin and J. M. Martínez, Improving ultimate convergence of an augmented Lagrangian method, Optim. Methods Softw., 23 (2008), 177-195.
doi: 10.1080/10556780701577730. |
[9] |
M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), R1-R50.
doi: 10.1088/0266-5611/21/2/R01. |
[10] |
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, 18pp.
doi: 10.1088/0266-5611/24/4/045001. |
[11] |
A. R. Conn, N. I. M. Gould and P. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds, SIAM J. Numer. Anal., 28 (1991), 545-572.
doi: 10.1137/0728030. |
[12] |
A. Doubova, E. Fernández-Cara, M. González-Burgos and J. H. Ortega, A geometric inverse problem for the Boussinesq system, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1213-1238.
doi: 10.3934/dcdsb.2006.6.1213. |
[13] |
A. Doubova, E. Fernández-Cara and J. H. Ortega, On the identification of a single body immersed in a Navier-Stokes fluid, European J. Appl. Math., 18 (2007), 57-80.
doi: 10.1017/S0956792507006821. |
[14] |
E. Fernández-Cara and F. Maestre, On some inverse problems arising in elastography, Inverse Problems, 28 (2012), 085001, 15pp.
doi: 10.1088/0266-5611/28/8/085001. |
[15] |
D. E. Finkel, DIRECT Optimization Algorithm User Guide, Center for Research in Scientific Computation, North Carolina State University, (2003). |
[16] |
F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265. |
[17] |
L. Hörmander, Linear Partial Differential Operators, Third revised printing, Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York Inc., New York, 1969. |
[18] |
V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 2006. |
[19] |
L. Ji, R. McLaughlin, D. Renzi and J.-R. Yoon, Interior elastodynamics inverse problems: Shear wave speed reconstruction in transient elastography, Inverse Problems, 19 (2003), S1-S29.
doi: 10.1088/0266-5611/19/6/051. |
[20] |
S. G. Johnson, The NLopt Nonlinear-optimization Package, 2011,, , ().
|
[21] |
D. R. Jones, C. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl., 79 (1993), 157-181.
doi: 10.1007/BF00941892. |
[22] |
P. Kaelo and M. M. Ali, Some variants of the controlled random search algorithm for global optimization, J. Optim. Theory Appl., 130 (2006), 253-264.
doi: 10.1007/s10957-006-9101-0. |
[23] |
O. Kavian, Lectures on parameter identification, in Three Courses on Partial Differential Equations, IRMA Lect. Math. Theor. Phys., 4, de Gruyter, Berlin, 2003, 125-162. |
[24] |
J.-L. Lions, Contrôlabilité Exacte [Exact Controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch, Masson, Paris, 1988. |
[25] |
A. E. Martínez-Castro, I. H. Faris and R. Gallego, Identification of cavities in a three-dimensional layer by minimization of an optimal cost functional expansion, CMES Comput. Model. Eng. Sci., 87 (2012), 177-206. |
[26] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer-Verlag, New York, 1999.
doi: 10.1007/b98874. |
[27] |
J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi and X. Li, Elastography: A quantitative method for imaging the elasticity of biological tissues, Ultrasonic Imaging, 13 (1991), 111-134.
doi: 10.1016/0161-7346(91)90079-W. |
[28] |
A. Y. Petrov, J. G. Chase, M. Sellier and P. D. Docherty, Non-identifiability of the Rayleigh damping material model in magnetic resonance elastography, Math. Biosci., 246 (2013), 191-201.
doi: 10.1016/j.mbs.2013.08.012. |
[29] |
W. L. Price, A controlled random search procedure for global optimisation, The Computer Journal, 20 (1977), 367-370.
doi: 10.1093/comjnl/20.4.367. |
[30] |
W. L. Price, Global optimization by controlled random search, J. Optim. Theory Appl., 40 (1983), 333-348.
doi: 10.1007/BF00933504. |
show all references
References:
[1] |
G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements, Inverse Problems, 18 (2002), 1333-1353.
doi: 10.1088/0266-5611/18/5/308. |
[2] |
G. Alessandrini, A. Morassi and E. Rosset, Edi Size estimates, in Inverse Problems: Theory and Applications (Cortona/Pisa, 2002), Contemp. Math., 333, Amer. Math. Soc., Providence, RI, 2003, 1-33.
doi: 10.1090/conm/333/05951. |
[3] |
C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552.
doi: 10.1088/0266-5611/21/5/003. |
[4] |
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. |
[5] |
S. Andrieux, A. Ben Abda and M. Jaoua, On the inverse emergent plane crack problem, Math. Methods Appl. Sci., 21 (1998), 895-906.
doi: 10.1002/(SICI)1099-1476(19980710)21:10<895::AID-MMA975>3.0.CO;2-1. |
[6] |
S. Auliac, Développement d'outils d'optimisation pour FreeFem++, These, Université Pierre et Marie Curie - Paris VI, 2014. Available from: http://tel.archives-ouvertes.fr/tel-01001631. |
[7] |
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 Optim., 48 (): 2871.
doi: 10.1137/070704332. |
[8] |
E. G. Birgin and J. M. Martínez, Improving ultimate convergence of an augmented Lagrangian method, Optim. Methods Softw., 23 (2008), 177-195.
doi: 10.1080/10556780701577730. |
[9] |
M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse Problems, 21 (2005), R1-R50.
doi: 10.1088/0266-5611/21/2/R01. |
[10] |
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, 18pp.
doi: 10.1088/0266-5611/24/4/045001. |
[11] |
A. R. Conn, N. I. M. Gould and P. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds, SIAM J. Numer. Anal., 28 (1991), 545-572.
doi: 10.1137/0728030. |
[12] |
A. Doubova, E. Fernández-Cara, M. González-Burgos and J. H. Ortega, A geometric inverse problem for the Boussinesq system, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1213-1238.
doi: 10.3934/dcdsb.2006.6.1213. |
[13] |
A. Doubova, E. Fernández-Cara and J. H. Ortega, On the identification of a single body immersed in a Navier-Stokes fluid, European J. Appl. Math., 18 (2007), 57-80.
doi: 10.1017/S0956792507006821. |
[14] |
E. Fernández-Cara and F. Maestre, On some inverse problems arising in elastography, Inverse Problems, 28 (2012), 085001, 15pp.
doi: 10.1088/0266-5611/28/8/085001. |
[15] |
D. E. Finkel, DIRECT Optimization Algorithm User Guide, Center for Research in Scientific Computation, North Carolina State University, (2003). |
[16] |
F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265. |
[17] |
L. Hörmander, Linear Partial Differential Operators, Third revised printing, Die Grundlehren der mathematischen Wissenschaften, Band 116, Springer-Verlag New York Inc., New York, 1969. |
[18] |
V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 2006. |
[19] |
L. Ji, R. McLaughlin, D. Renzi and J.-R. Yoon, Interior elastodynamics inverse problems: Shear wave speed reconstruction in transient elastography, Inverse Problems, 19 (2003), S1-S29.
doi: 10.1088/0266-5611/19/6/051. |
[20] |
S. G. Johnson, The NLopt Nonlinear-optimization Package, 2011,, , ().
|
[21] |
D. R. Jones, C. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl., 79 (1993), 157-181.
doi: 10.1007/BF00941892. |
[22] |
P. Kaelo and M. M. Ali, Some variants of the controlled random search algorithm for global optimization, J. Optim. Theory Appl., 130 (2006), 253-264.
doi: 10.1007/s10957-006-9101-0. |
[23] |
O. Kavian, Lectures on parameter identification, in Three Courses on Partial Differential Equations, IRMA Lect. Math. Theor. Phys., 4, de Gruyter, Berlin, 2003, 125-162. |
[24] |
J.-L. Lions, Contrôlabilité Exacte [Exact Controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch, Masson, Paris, 1988. |
[25] |
A. E. Martínez-Castro, I. H. Faris and R. Gallego, Identification of cavities in a three-dimensional layer by minimization of an optimal cost functional expansion, CMES Comput. Model. Eng. Sci., 87 (2012), 177-206. |
[26] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer-Verlag, New York, 1999.
doi: 10.1007/b98874. |
[27] |
J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi and X. Li, Elastography: A quantitative method for imaging the elasticity of biological tissues, Ultrasonic Imaging, 13 (1991), 111-134.
doi: 10.1016/0161-7346(91)90079-W. |
[28] |
A. Y. Petrov, J. G. Chase, M. Sellier and P. D. Docherty, Non-identifiability of the Rayleigh damping material model in magnetic resonance elastography, Math. Biosci., 246 (2013), 191-201.
doi: 10.1016/j.mbs.2013.08.012. |
[29] |
W. L. Price, A controlled random search procedure for global optimisation, The Computer Journal, 20 (1977), 367-370.
doi: 10.1093/comjnl/20.4.367. |
[30] |
W. L. Price, Global optimization by controlled random search, J. Optim. Theory Appl., 40 (1983), 333-348.
doi: 10.1007/BF00933504. |
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