An inverse initial boundary value problem for the wave equation in the presence of imperfections of small volume

SIAM Journal on Control and Optimization

Volumn 41, Issue 4, 2003, Pages 1194-1211

  Ammari, Habib ^a  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

Geometric control; Inverse problem; Reconstruction; Wave equation

Indexed keywords

APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; FUNCTIONS; LINEAR SYSTEMS; THEOREM PROVING; WAVE EQUATIONS;

GEOMETRIC CONTROL;

INVERSE PROBLEMS;

EID: 0038679242     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012901384247     Document Type: Article

References (34)

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 (2001), pp. 94-106.
2. H. AMMARI, Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements, J. Math. Anal. Appl., submitted; also available online from http://www.cmap.polytechnique/̃ammari/̃preprints.
3. H. AMMARI AND H. KANG, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter, SIAM J. Math. Anal., submitted; also available online from http://www.cmap.polytechnique/̃ammari/̃preprints.
4. H. AMMARI, S. MOSKOW, AND M. VOGELIUS, Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter, ESAIM Control Optim. Calc. Var., to appear; also available online from http://www.cmap.polytechnique/̃ammari/̃preprints.
5. H. AMMARI AND J. K. SEO, A new algorithm for the reconstruction of conductivity inhomogeneities, J. Amer. Math. Soc., submitted; also available online from http://www.cmap.polytechnique/̃ammari/̃preprints.
6. H. AMMARI, M. VOGELIUS, AND D. VOLKOV, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations, J. Math. Pures Appl. (9), 80 (2001), pp. 769-814.
7. 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), pp. 1024-1065.
8. M. I. BELISHEV AND YA KURYLEV, Boundary control, wave field continuation and inverse problems for the wave equation, Comput. Math. Appl., 22 (1991), pp. 27-52.
9. E. BERETTA, A. MUKHERJEE, AND M. VOGELIUS, Asymptotic formuli for steady state voltage potentials in the presence of conductivity imperfection of small area, Z. Angew. Math. Phys., 52 (2001), pp. 543-572.
10. G. BRUCKNER AND M. YAMAMOTO, Determination of point wave sources by pointwise observations: Stability and reconstruction, Inverse Problems, 16 (2000), pp. 723-748.
11. A. P. CALDERÓN, On an inverse boundary value problem, in Proceedings of a Seminar on Numerical Analysis and its Applications to Continuum Physics, Sociedade Brasileira de Matemática, Rio de Janeiro, Brazil, 1980, pp. 65-73.
12. D. J. CEDIO-FENGYA, S. MOSKOW, AND M. VOGELIUS, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computional reconstruction, Inverse Problems, 14 (1998), pp. 553-595.
13. I. DAUBECHIES, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, SIAM, Philadelphia, 1992.
14. L. C. EVANS, Partial Differential Equations, Grad. Stud. Math. 19, AMS, Providence, RI, 1998.
15. A. FRIEDMAN AND M. VOGELIUS, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal., 105 (1989), pp. 299-326.
16. M. GRASSELLI AND M. YAMAMOTO, Identifying a spatial body force in linear elastodynamics via traction measurements, SIAM J. Control Optim., 36 (1998), pp. 1190-1206.
17. V. ISAKOV, Inverse Source Problems, AMS, Providence, RI, 1990.
18. R. E. KLEINMAN AND T. B. A. SENIOR, Rayleigh scattering, in Low and High Frequency Asymptotics, V. K. Varadan and V. V. Varadan, eds., North-Holland, Amsterdam, 1986, pp. 1-70.
19. J.-L. LIONS, Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Masson, Paris, 1988.
20. J.-L. LIONS AND E. MAGENES, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, 1972.
21. A. I. NACHMAN, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), pp. 71-96.
22. S. NICAISE, Exact boundary controllability of Maxwell's equations in heteregeneous media and an application to an inverse source problem, SIAM J. Control Optim., 38 (2000), pp. 1145-1170.
23. J.-P. FUEL AND M. YAMAMOTO, Applications de la contrôlabilité exacte à quelques problèmes inverses hyperboliques, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), pp. 1171-1176.
24. J.-P. FUEL AND M. YAMAMOTO, On a global estimate in a linear inverse hyperbolic problem, Inverse Problems, 12 (1996), pp. 995-1002.
25. J.-P. PUEL AND M. YAMAMOTO, Smoothing property in multidimensional inverse hyperbolic problems: Applications to uniqueness and stability, J. Inverse Ill-Posed Prob., 4 (1996), pp. 283-296.
26. J.-P. PUEL AND M. YAMAMOTO, Generic well-posedness in a multidimensional inverse hyperbolic problem, J. Inverse Ill-Posed Prob., 5 (1997), pp. 55-83.
27. RAKESH AND W. SYMES, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), pp. 87-96.
28. V. G. ROMANOV AND S. I. KABANIKHIN, Inverse Problems for Maxwell's Equations, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994.
29. J. SYLVESTER AND G. UHLMANN, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), pp. 153-169.
30. Z. SUN, On the continuous dependence for an inverse initial boundary value problem for the wave equation, J. Math. Anal. Appl., 150 (1990), pp. 188-204.
31. M. YAMAMOTO, Well-posedness of some inverse hyperbolic problems by the Hilbert uniqueness method, J. Inverse Ill-Posed Prob., 2 (1994), pp. 349-368.
32. M. YAMAMOTO, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), pp. 481-496.
33. M. YAMAMOTO, Determination of forces in vibrations of beams and plates by pointwise and line observations, J. Inverse Ill-Posed Prob., 4 (1996), pp. 437-457.
34. M. VOGELIUS AND D. VOLKOV, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 723-748.