American Institute of Mathematical Sciences

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May  2015, 9(2): 371-393. doi: 10.3934/ipi.2015.9.371

Some geometric inverse problems for the linear wave equation

Anna Doubova 1, and  Enrique Fernández-Cara 1,

1. 

University of Sevilla, Dpto. E.D.A.N., Aptdo. 1160, 41080, Sevilla, Spain, Spain

Received  June 2014 Revised  November 2014 Published  March 2015

In this paper we consider some geometric inverse problems for the linear wave equation. We prove uniqueness results, we present some reconstruction algorithms and we perform numerical experiments in dimensions one and two.
Keywords: Inverse problems, reconstruction, numerical solution, elastography., shape identification, Freefem++, wave equation.
Mathematics Subject Classification: 65M32, 35L05, 49K20, 49N4.
Citation: Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems and Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371
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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