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

Mark Asch; Marion Darbas; Jean-Baptiste Duval

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 1016-1034
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Asch, Mark, Darbas, Marion, and 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 17.4 (2011): 1016-1034. <http://eudml.org/doc/221917>.

@article{Asch2011,
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. },
author = {Asch, Mark, Darbas, Marion, Duval, Jean-Baptiste},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Wave equation; exact controllability; inverse problem; finite elements; Fourier inversion; conductivity imperfections; finite element solution; exact controllability method},
language = {eng},
month = {11},
number = {4},
pages = {1016-1034},
publisher = {EDP Sciences},
title = {Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume},
url = {http://eudml.org/doc/221917},
volume = {17},
year = {2011},
}

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
DA - 2011/11//
PB - EDP Sciences
VL - 17
IS - 4
SP - 1016
EP - 1034
AB - 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.
LA - eng
KW - Wave equation; exact controllability; inverse problem; finite elements; Fourier inversion; conductivity imperfections; finite element solution; exact controllability method
UR - http://eudml.org/doc/221917
ER -

References

top
  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.  Zbl1028.74021
  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.  Zbl1028.35159
  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.  Zbl1082.78006
  4. H. Ammari and H. Kang, Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences162. Springer-Verlag, New York (2007).  Zbl1220.35001
  5. H. Ammari, S. Moskow and M. Vogelius, Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter. ESAIM: COCV62 (2002) 94–106.  
  6. H. Ammari, P. Calmon and E. Iakovleva, Direct elastic imaging of a small inclusion. SIAM J. Imaging Sci.1 (2008) 169–187.  Zbl1179.35341
  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.  Zbl1149.78005
  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.  Zbl1187.92058
  9. H. Ammari, E. Bossy, V. Jugnon and H. Kang, Mathematical Modelling in Photo-Acoustic Imaging. SIAM Rev. (to appear).  Zbl1257.74091
  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 .  Zbl1230.35143URIhttp://www.cmap.polytechnique.fr/~ammari/preprints.html
  11. M. Asch and G. Lebeau, Geometrical aspects of exact boundary controllability for the wave equation – A numerical study. ESAIM: COCV3 (1998) 163–212.  Zbl1052.93501
  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.  Zbl1174.35113
  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.  Zbl1189.93022
  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, (2001).  URIhttp://www.mcs.anl.gov/petsc
  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.  Zbl0786.93009
  16. E.O. Brigham. The fast Fourier transform and its applications. Prentice Hall, New Jersey (1988).  
  17. Y. Capdebosq and M.S. Vogelius, A review of some recent work on impedance imaging for inhomogeneities of low volume fraction, in Contemporary Mathematics362, C. Conca, R. Manasevich, G. Uhlmann and M.S. Vogelius Eds., AMS (2004) 69–88.  Zbl1072.35198
  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.  Zbl1102.93004
  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.  Zbl1139.93005
  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.  Zbl0916.35132
  21. P.G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and Its Applications4. North-Holland Publishing Company (1978).  Zbl0383.65058
  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).  
  24. R. Glowinski, Ensuring well posedness by analogy; Stokes problem and boundary control for the wave equation. J. Comput. Phys.103 (1992) 189–221.  Zbl0763.76042
  25. R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numer.4 (1995) 159–328.  Zbl0838.93014
  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.  Zbl0699.65055
  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.  Zbl1159.93006
  28. J.A. Infante and E. Zuazua, Boundary observability for the space discretization of the one-dimensional wave equation. ESAIM: M2AN33 (1999) 407–438.  
  29. G. Lebeau and M. Nodet, Experimental study of the HUM control operator for linear waves. Experimental Mathematics19 (2010) 93–120.  Zbl1190.35011
  30. J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité exacte. Masson, Paris (1988).  Zbl0653.93002
  31. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer (1997).  Zbl1151.65339
  32. M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. ESAIM: M2AN34 (2000) 723–748.  Zbl0971.78004
  33. W.L. Wood, Practical time-stepping schemes. Oxford Applied Mathematics and Computing Science Series, Clarendon Press, Oxford (1990).  Zbl0694.65043
  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.  Zbl0939.93016
  35. E. Zuazua, Propagation, observation and control of waves approximated by finite difference methods. SIAM Rev.47 (2005) 197–243.  Zbl1077.65095

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.