Identification of small inhomogeneities: Asymptotic factorization

Mathematics of Computation

Volumn 76, Issue 259, 2007, Pages 1425-1448

  Ammari, Habib ^a   Griesmaier, Roland ^b   Hanke, Martin ^b  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b JOHANNES GUTENBERG UNIVERSITY   (Germany)

Author keywords

Asymptotic expansions; Electrical impedance tomography; Small conductivity inhomogeneities

Indexed keywords

EID: 41449098555     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-07-01946-1     Document Type: Article

References (37)

1. H. Ammari, E. Iakovleva, and D. Lesselier, Two numerical methods for recovering small electromagnetic inclusions from scattering amplitude at a fixed frequency, SIAM J. Sci. Comput. 27 (2005), 130-158. MR2201178 (2006h:35270)
2. _, A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency, SIAM Multiscale Model. Simul. 3 (2005), 597-628. MR2136165 (2006e:35336)
3. H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, MUSIC-type electromagnetic imaging of a collection of small three-dimensional bounded inclusions, SIAM J. Sci. Comput., to appear.
4. H. Ammari, E. Iakovleva, and S. Moskow, Recovery of small inhomogeneities from the scattering amplitude at a fixed frequency, SIAM J. Math. Anal. 34 (2003), 882-900. MR1969606 (2004a:78022)
5. 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. 34 (2003), no. 5, 1152-1166. MR2001663 (2004e:35027)
6. _, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Math., vol. 1846, Springer-Verlag, Berlin, 2004. MR2168949 (2006k:35295)
7. _, Boundary layer techniques for solving the Helmholtz equation in the presence of small inhomogeneities, J. Math. Anal. Appl. 296 (2004), 190-208. MR2070502 (2005c:35054)
8. H. Ammari and A. Khelifi, Electromagnetic scattering by small dielectric inhomogeneities, J. Math. Pures Appl. 82 (2003), no. 7, 749-842. MR2005296 (2005a:35262)
9. H. Ammari, S. Moskow, and M. S. Vogelius, Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume, ESAIM Control Optim. Calc. Var. 9 (2003), 49-66. MR1957090 (2004j:78025)
10. H. Ammari, M. S. 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. 80 (2001), 769-814. MR1860816 (2003b:78002)
11. M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal. 32 (2001), 1327-1341. MR1856252 (2002j:35303)
12. M. Brühl, M. Hanke, and M. S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities, Numer. Math. 93 (2003), 635-654. MR1961882 (2004b:65169)
13. Y. Capdeboscq and M. S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements, Math. Model. Numer. Anal. 37 (2003), no. 2, 227-240. MR1991198 (2005c:35067)
14. _, A general representation formula for boundary voltage perturbations caused by internal conductivity in homogeneities of low volume fraction, Math. Model. Numer. Anal. 37 (2003), no. 1, 159-173. MR1972656 (2004b:35334)
15. D. Cedio-Fengya, S. Moskow, and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998), no. 3, 553-595. MR1629995 (99d:78011)
16. M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Problems 17 (2001), no. 4, 591-595. MR1861470
17. D. Colton, J. Coyle, and P. Monk, Recent developments in inverse acoustic scattering theory, SIAM Rev. 42 (2000), no. 3, 369-414. MR1786932 (2001f:76066)
18. D. Colton, K. Giebermann, and P. Monk, A regularized sampling method for solving three-dimensional inverse scattering problems, SIAM J. Sci. Comput. 21 (2000), no. 6, 2316-2330. MR1762044 (2000m:65165)
19. D. Colton, H. Haddar, and P. Monk, The linear sampling method for solving the electromagnetic inverse scattering problem, SIAM J. Sci. Comput. 24 (2002), no. 3, 719-731. MR1950509 (2003m:78020)
20. D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems 12 (1996), no. 4, 383-393. MR1402098 (97d:35032)
21. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613-626. MR0937473 (89h:35090)
22. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 2. Functional and Variational Methods, Springer-Verlag, Berlin, 1988. MR0969367 (89m:0000l)
23. A. J. Devaney, Super-resolution processing of multi-static data using time reversal and MUSIC, preprint.
24. A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rational Mech. Anal. 105 (1989), no. 4, 299-326. MR0973245 (90c:35198)
25. B. Gebauer, The factorization method for real elliptic problems, Z. Anal. Anwend. 25 (2006), 81-102. MR2216883
26. B. Gebauer, M. Hanke, A. Kirsch, W. Muniz, and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems 21 (2005), 2035-2050. MR2183666 (2006k:78009)
27. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. MR0851383 (88b:65l29)
28. M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems 19 (2003), S65-S9O. MR2036522
29. A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems 14 (1998), no. 6, 1489-1512. MR1662460 (99k:35193)
30. _, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems 18 (2000), no. 4, 1025-1040. MR1929280 (2003f:35286)
31. _, The factorization method for Maxwell's equations, Inverse Problems 20 (2004), no. 6, S117-S134. MR2107232 (2005i:35285)
32. _, The factorization method for a class of inverse elliptic problems, Math. Nachr. 278 (2005), no. 3, 258-277. MR2110531 (2006h:35281)
33. R. Kress, Linear Integral Equations, Applied Mathematical Sciences, vol. 82, Springer-Verlag, Berlin, 1989. MR1007594 (90j:4500l)
34. _, A factorization method for an inverse Neumann problem for harmonic vector fields, Georgian Math. J. 10 (2003), no. 3, 549-560. MR2023274 (2004m:35276)
35. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. MR1742312 (2001a:35051)
36. J.-C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, vol. 144, Springer-Verlag, New York, 2001. MR1822275 (2002c:35003)
37. M. S. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, Math. Model. Numer. Anal. 34 (2000), 723-748. MR1784483 (2001f:78024)