Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 1016-1034

We consider the numerical solution, in two- and three-dimensional bounded domains, of the inverse problem for identifying the location of small-volume, conductivity imperfections in a medium with homogeneous background. A dynamic approach, based on the wave equation, permits us to treat the important case of “limited-view” data. Our numerical algorithm is based on the coupling of a finite element solution of the wave equation, an exact controllability method and finally a Fourier inversion for localizing the centers of the imperfections. Numerical results, in 2- and 3-D, show the robustness and accuracy of the approach for retrieving randomly placed imperfections from both complete and partial boundary measurements.

DOI : https://doi.org/10.1051/cocv/2010031
Classification:  35R30,  35L05,  65M60
Keywords: wave equation, exact controllability, inverse problem, finite elements, Fourier inversion
@article{COCV_2011__17_4_1016_0,
     author = {Asch, Mark and Darbas, Marion and Duval, Jean-Baptiste},
     title = {Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     pages = {1016-1034},
     doi = {10.1051/cocv/2010031},
     zbl = {1254.35238},
     mrnumber = {2859863},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_4_1016_0}
}
Asch, Mark; Darbas, Marion; Duval, Jean-Baptiste. Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1016-1034. doi : 10.1051/cocv/2010031. http://www.numdam.org/item/COCV_2011__17_4_1016_0/

[1] C. Alves and H. Ammari, Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium. SIAM J. Appl. Math. 62 (2002) 94-106. | MR 1857537 | Zbl 1028.74021

[2] H. Ammari, An inverse initial boundary value problem for the wave equation in the presence of imperfections of small volume. SIAM J. Control Optim. 41 (2002) 1194-1211. | MR 1972508 | Zbl 1028.35159

[3] H. Ammari, Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements. J. Math. Anal. Appl. 282 (2003) 479-494. | MR 1989105 | Zbl 1082.78006

[4] H. Ammari and H. Kang, Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences 162. Springer-Verlag, New York (2007). | MR 2327884 | Zbl 1220.35001

[5] H. Ammari, S. Moskow and M. Vogelius, Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter. ESAIM: COCV 62 (2002) 94-106. | Zbl 1028.74021

[6] H. Ammari, P. Calmon and E. Iakovleva, Direct elastic imaging of a small inclusion. SIAM J. Imaging Sci. 1 (2008) 169-187. | MR 2486036 | Zbl 1179.35341

[7] H. Ammari, H. Kang, E. Kim, K. Louati and M. Vogelius, A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements. Numer. Math. 108 (2008) 501-528. | MR 2369202 | Zbl 1149.78005

[8] H. Ammari, Y. Capdeboscq, H. Kang and A. Kozhemyak, Mathematical models and reconstruction methods in magneto-acoustic imaging. Eur. J. Appl. Math. 20 (2009) 303-317. | MR 2511278 | Zbl 1187.92058

[9] H. Ammari, E. Bossy, V. Jugnon and H. Kang, Mathematical Modelling in Photo-Acoustic Imaging. SIAM Rev. (to appear). | MR 2736968

[10] H. Ammari, M. Asch, L.G. Bustos, V. Jugnon and H. Kang, Transient wave imaging with limited-view data. SIAM J. Imaging Sci. (submitted) preprint available from http://www.cmap.polytechnique.fr/~ammari/preprints.html. | MR 2861101 | Zbl 1230.35143

[11] M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation - A numerical study. ESAIM: COCV 3 (1998) 163-212. | Numdam | MR 1624783 | Zbl 1052.93501

[12] M. Asch and S.M. Mefire, Numerical localizations of 3D imperfections from an asymptotic formula for perturbations in the electric fields. J. Comput. Math. 26 (2008) 149-195. | MR 2395589 | Zbl 1174.35113

[13] M. Asch and A. Münch, Uniformly controllable schemes for the wave equation on the unit square. J. Optim. Theory Appl. 143 (2009) 417-438. | MR 2558639 | Zbl 1189.93022

[14] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L. Curfman Mcinnes, B.F. Smith and H. Zhang, PETSc Web page, http://www.mcs.anl.gov/petsc (2001).

[15] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR 1178650 | Zbl 0786.93009

[16] E.O. Brigham. The fast Fourier transform and its applications. Prentice Hall, New Jersey (1988). | Zbl 0375.65052

[17] Y. Capdebosq and M.S. Vogelius, A review of some recent work on impedance imaging for inhomogeneities of low volume fraction, in Contemporary Mathematics 362, C. Conca, R. Manasevich, G. Uhlmann and M.S. Vogelius Eds., AMS (2004) 69-88. | MR 2091492 | Zbl 1072.35198

[18] C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102 (2006) 413-462. | MR 2207268 | Zbl 1102.93004

[19] C. Castro, S. Micu and A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Num. Anal. 28 (2008) 186-214. | MR 2387911 | Zbl 1139.93005

[20] D.J. Cedio-Fengya, S. Moskow and M. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inv. Probl. 14 (1998) 553-595. | MR 1629995 | Zbl 0916.35132

[21] P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and Its Applications 4. North-Holland Publishing Company (1978). | MR 520174 | Zbl 0383.65058

[22] J.-B. Duval, Identification dynamique de petites imperfections. Ph.D. Thesis, Université de Picardie Jules Verne, France (2009).

[23] L.C. Evans, Partial Differential Equations, Grad. Stud. Math. 19. AMS, Providence (1998). | MR 1625845 | Zbl 0902.35002

[24] R. Glowinski, Ensuring well posedness by analogy; Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189-221. | MR 1196839 | Zbl 0763.76042

[25] R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer. 4 (1995) 159-328. | MR 1352473 | Zbl 0838.93014

[26] R. Glowinski, C.H. Li and J.-L. Lions, A numerical approach to the exact controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods. Jpn. J. Appl. Math. 7 (1990) 1-76. | MR 1039237 | Zbl 0699.65055

[27] L.I. Ignat and E. Zuazua, Convergence of a two-grid method algorithm for the control of the wave equation. J. Eur. Math. Soc. 11 (2009) 351-391. | MR 2486937 | Zbl 1159.93006

[28] J.A. Infante and E. Zuazua, Boundary observability for the space discretization of the one-dimensional wave equation. ESAIM: M2AN 33 (1999) 407-438. | Numdam | MR 1700042 | Zbl 0947.65101

[29] G. Lebeau and M. Nodet, Experimental study of the HUM control operator for linear waves. Experimental Mathematics 19 (2010) 93-120. | MR 2649987 | Zbl 1190.35011

[30] J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité exacte. Masson, Paris (1988). | MR 931277 | Zbl 0653.93002

[31] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997). | MR 1299729 | Zbl 0803.65088

[32] M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. ESAIM: M2AN 34 (2000) 723-748. | Numdam | MR 1784483 | Zbl 0971.78004

[33] W.L. Wood, Practical time-stepping schemes. Oxford Applied Mathematics and Computing Science Series, Clarendon Press, Oxford (1990). | MR 1079731 | Zbl 0694.65043

[34] E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78 (1999) 523-563. | MR 1697041 | Zbl 0939.93016

[35] E. Zuazua, Propagation, observation and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. | MR 2179896 | Zbl 1077.65095