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.
Mots clés : 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},
pages = {1016--1034},
publisher = {EDP-Sciences},
volume = {17},
number = {4},
year = {2011},
doi = {10.1051/cocv/2010031},
mrnumber = {2859863},
zbl = {1254.35238},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010031/}
}
TY - JOUR AU - Asch, Mark AU - Darbas, Marion AU - Duval, Jean-Baptiste TI - Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 1016 EP - 1034 VL - 17 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010031/ DO - 10.1051/cocv/2010031 LA - en ID - COCV_2011__17_4_1016_0 ER -
%0 Journal Article %A Asch, Mark %A Darbas, Marion %A Duval, Jean-Baptiste %T Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 1016-1034 %V 17 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010031/ %R 10.1051/cocv/2010031 %G en %F 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, Tome 17 (2011) no. 4, pp. 1016-1034. doi : 10.1051/cocv/2010031. http://www.numdam.org/articles/10.1051/cocv/2010031/
[1] and , Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium. SIAM J. Appl. Math. 62 (2002) 94-106. | MR | Zbl
[2] , 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 | Zbl
[3] , Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements. J. Math. Anal. Appl. 282 (2003) 479-494. | MR | Zbl
[4] and , Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences 162. Springer-Verlag, New York (2007). | MR | Zbl
[5] , and , Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter. ESAIM: COCV 62 (2002) 94-106. | Zbl
[6] , and , Direct elastic imaging of a small inclusion. SIAM J. Imaging Sci. 1 (2008) 169-187. | MR | Zbl
[7] , , , and , A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements. Numer. Math. 108 (2008) 501-528. | MR | Zbl
[8] , , and , Mathematical models and reconstruction methods in magneto-acoustic imaging. Eur. J. Appl. Math. 20 (2009) 303-317. | MR | Zbl
[9] , , and , Mathematical Modelling in Photo-Acoustic Imaging. SIAM Rev. (to appear). | MR
[10] , , , and , Transient wave imaging with limited-view data. SIAM J. Imaging Sci. (submitted) preprint available from http://www.cmap.polytechnique.fr/~ammari/preprints.html. | MR | Zbl
[11] and , Geometrical aspects of exact boundary controllability for the wave equation - A numerical study. ESAIM: COCV 3 (1998) 163-212. | Numdam | MR | Zbl
[12] and , Numerical localizations of 3D imperfections from an asymptotic formula for perturbations in the electric fields. J. Comput. Math. 26 (2008) 149-195. | MR | Zbl
[13] and , Uniformly controllable schemes for the wave equation on the unit square. J. Optim. Theory Appl. 143 (2009) 417-438. | MR | Zbl
[14] , , , , , , and , PETSc Web page, http://www.mcs.anl.gov/petsc (2001).
[15] , and , Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR | Zbl
[16] . The fast Fourier transform and its applications. Prentice Hall, New Jersey (1988). | Zbl
[17] and , 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 | Zbl
[18] and , 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 | Zbl
[19] , and , 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 | Zbl
[20] , and , Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inv. Probl. 14 (1998) 553-595. | MR | Zbl
[21] , The finite element method for elliptic problems, Studies in Mathematics and Its Applications 4. North-Holland Publishing Company (1978). | MR | Zbl
[22] , Identification dynamique de petites imperfections. Ph.D. Thesis, Université de Picardie Jules Verne, France (2009).
[23] , Partial Differential Equations, Grad. Stud. Math. 19. AMS, Providence (1998). | MR | Zbl
[24] , Ensuring well posedness by analogy; Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189-221. | MR | Zbl
[25] and , Exact and approximate controllability for distributed parameter systems. Acta Numer. 4 (1995) 159-328. | MR | Zbl
[26] , and , 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 | Zbl
[27] and , Convergence of a two-grid method algorithm for the control of the wave equation. J. Eur. Math. Soc. 11 (2009) 351-391. | MR | Zbl
[28] and , Boundary observability for the space discretization of the one-dimensional wave equation. ESAIM: M2AN 33 (1999) 407-438. | Numdam | MR | Zbl
[29] and , Experimental study of the HUM control operator for linear waves. Experimental Mathematics 19 (2010) 93-120. | MR | Zbl
[30] , Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité exacte. Masson, Paris (1988). | MR | Zbl
[31] and , Numerical Approximation of Partial Differential Equations. Springer (1997). | MR | Zbl
[32] and , 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 | Zbl
[33] , Practical time-stepping schemes. Oxford Applied Mathematics and Computing Science Series, Clarendon Press, Oxford (1990). | MR | Zbl
[34] , 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 | Zbl
[35] , Propagation, observation and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. | MR | Zbl
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