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Numerical solution of an inverse initial boundary value problem forthe wave equation in the presence of conductivity imperfections ofsmall volume

Published online by Cambridge University Press:  06 August 2010

Mark Asch ,
Marion Darbas  and
Jean-Baptiste Duval
Mark Asch
Affiliation:
LAMFA-CNRS UMR 6140, Université de Picardie Jules Verne, 80039 Amiens, France. mark.asch@u-picardie.fr
Marion Darbas
Affiliation:
LAMFA-CNRS UMR 6140, Université de Picardie Jules Verne, 80039 Amiens, France. mark.asch@u-picardie.fr
Jean-Baptiste Duval
Affiliation:
LAMFA-CNRS UMR 6140, Université de Picardie Jules Verne, 80039 Amiens, France. mark.asch@u-picardie.fr
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Abstract

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.

Keywords

Wave equationexact controllabilityinverse problemfinite elementsFourier inversion
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1016 - 1034
Copyright
© EDP Sciences, SMAI, 2010

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References

Alves, C. and Ammari, H., Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium. SIAM J. Appl. Math. 62 (2002) 94–106. CrossRef
Ammari, H., 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. CrossRef
Ammari, H., Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements. J. Math. Anal. Appl. 282 (2003) 479–494. CrossRef
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).
Ammari, H., Moskow, S. and Vogelius, M., Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter. ESAIM: COCV 62 (2002) 94–106.
Ammari, H., Calmon, P. and Iakovleva, E., Direct elastic imaging of a small inclusion. SIAM J. Imaging Sci. 1 (2008) 169–187. CrossRef
Ammari, H., Kang, H., Kim, E., Louati, K. and Vogelius, M., MUSIC-type, A algorithm for detecting internal corrosion from electrostatic boundary measurements. Numer. Math. 108 (2008) 501–528. CrossRef
Ammari, H., Capdeboscq, Y., Kang, H. and Kozhemyak, A., Mathematical models and reconstruction methods in magneto-acoustic imaging. Eur. J. Appl. Math. 20 (2009) 303–317. CrossRef
H. Ammari, E. Bossy, V. Jugnon and H. Kang, Mathematical Modelling in Photo-Acoustic Imaging. SIAM Rev. (to appear).
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.
Asch, M. and Lebeau, G., Geometrical aspects of exact boundary controllability for the wave equation – A numerical study. ESAIM: COCV 3 (1998) 163–212. CrossRef
Asch, M. and Mefire, S.M., Numerical localizations of 3D imperfections from an asymptotic formula for perturbations in the electric fields. J. Comput. Math. 26 (2008) 149–195.
Asch, M. and Münch, A., Uniformly controllable schemes for the wave equation on the unit square. J. Optim. Theory Appl. 143 (2009) 417–438. CrossRef
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).
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. CrossRef
E.O. Brigham. The fast Fourier transform and its applications. Prentice Hall, New Jersey (1988).
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.
Castro, C. and Micu, S., Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102 (2006) 413–462. CrossRef
Castro, C., Micu, S. and Münch, A., 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. CrossRef
Cedio-Fengya, D.J., Moskow, S. and Vogelius, M., Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inv. Probl. 14 (1998) 553–595. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and Its Applications 4. North-Holland Publishing Company (1978).
J.-B. Duval, Identification dynamique de petites imperfections. Ph.D. Thesis, Université de Picardie Jules Verne, France (2009).
L.C. Evans, Partial Differential Equations, Grad. Stud. Math. 19. AMS, Providence (1998).
R. Glowinski, Ensuring well posedness by analogy; Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189–221.
Glowinski, R. and Lions, J.-L., Exact and approximate controllability for distributed parameter systems. Acta Numer. 4 (1995) 159–328. CrossRef
Glowinski, R., Li, C.H. and Lions, J.-L., 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. CrossRef
Ignat, L.I. and Zuazua, E., Convergence of a two-grid method algorithm for the control of the wave equation. J. Eur. Math. Soc. 11 (2009) 351–391. CrossRef
Infante, J.A. and Zuazua, E., Boundary observability for the space discretization of the one-dimensional wave equation. ESAIM: M2AN 33 (1999) 407–438. CrossRef
Lebeau, G. and Nodet, M., Experimental study of the HUM control operator for linear waves. Experimental Mathematics 19 (2010) 93–120. CrossRef
J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité exacte. Masson, Paris (1988).
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997).
Vogelius, M. and Volkov, D., Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. ESAIM: M2AN 34 (2000) 723–748. CrossRef
W.L. Wood, Practical time-stepping schemes. Oxford Applied Mathematics and Computing Science Series, Clarendon Press, Oxford (1990).
Zuazua, E., 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. CrossRef
Zuazua, E., Propagation, observation and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. CrossRef