|
Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements
Author(s):
Habib
Ammari;
Elena
Beretta;
Elisa
Francini;
Hyeonbae
Kang;
Mikyoung
Lim.
Journal:
Math. Comp.
79
(2010),
1757-1777.
MSC (2000):
Primary 35R30, 35B34
Posted:
January 15, 2010
MathSciNet review:
2630011
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this paper, we propose an original and promising optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenfunctions associated with the transmission problem for the Laplacian. Based on a rigorous asymptotic analysis, we derive an asymptotic formula for the perturbations in the modal measurements that are due to small changes in the interface of the inclusion. Using fine gradient estimates, we carefully estimate the error term in this asymptotic formula. We then provide a key dual identity which naturally yields to the formulation of the proposed optimization problem. The viability of our reconstruction approach is documented by a variety of numerical results. The resolution limit of our algorithm is also highlighted.
References:
-
- 1.
- H. Ammari, E. Beretta, E. Francini, H. Kang, and M. Lim, Reconstruction of small interface changes of an inclusion from modal measurements II: The elastic case, J. Math. Pures Appl., to appear.
- 2.
- H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math., 68 (2008), 1557-1573. MR 2424952 (2009h:35439)
- 3.
- H. Ammari, P. Garapon, F. Jouve, and H. Kang, A new optimal control approach toward reconstruction of extended inclusions, preprint.
- 4.
- H. Ammari, H. Kang, E. Kim, and H. Lee, Vibration testing for anomaly detection, Math. Meth. Appl. Sci., 32 (2009), 863-874. MR 2507936
- 5.
- H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs, Vol. 153, American Math. Soc., Providence, 2009. MR 2488135
- 6.
- H. Ammari, H. Kang, M. Lim, and H. Zribi, Layer potential techniques in spectral analysis. Part I: Complete asymptotic expansions for eigenvalues of the Laplacian in domains with small inclusions, Trans. Amer. Math. Soc., to appear.
- 7.
- H. Ammari, H. Kang, M. Lim, and H. Zribi, Conductivity interface problems. Part I: small perturbations of an interface, Trans. Amer. Math. Soc., to appear.
- 8.
- H. Ammari and S. Moskow, Asymptotic expansions for eigenvalues in the presence of small inhomogeneities, Math. Meth. Appl. Sci., 26 (2003), 67-75. MR 1943110 (2003j:35035)
- 9.
- Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN, 37 (2003), 159-173. MR 1972656 (2004b:35334)
- 10.
- P.R. Garabedian and M. Schiffer, Convexity of domain functionals, J. Anal. Math., 2 (1953), 281--368. MR 0060117 (15:627a)
- 11.
- D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Springer-Verlag, 1983. MR 737190 (86c:35035)
- 12.
- J. Hadamard, Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées, OEuvres de Jacques Hadamard, 515-631, Vol. 2, Ed. CNRS, Paris, 1968.
- 13.
- T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976. MR 0407617 (53:11389)
- 14.
- V. Kozlov, On the Hadamard formula for nonsmooth domains, J. Differ. Equat., 230 (2006), 532-555. MR 2269932 (2007h:35244)
- 15.
- Y.Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153 (2000), 91-151. MR 1770682 (2001m:35083)
- 16.
- Y.Y. Li and L. Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math., 56 (2003), 892-925. MR 1990481 (2004k:35097)
- 17.
- J. Osborn, Spectral approximation for compact operators, Math. of Comp., 29 (1975), 712-725. MR 0383117 (52:3998)
- 18.
- S. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum, J. Comput. Phys., 171 (2001), 272-288. MR 1843648 (2002f:65088)
- 19.
- O.S. Salawu, Detection of structural damage through changes in frequency: a review, Engineering Structures, 19 (1997), 718-723.
- 20.
- J. Sanchez Hubert and E. Sanchez Palencia, Vibration and Coupling of Continuous Systems, Asymptotic Methods, Springer-Verlag, Berlin, 1989. MR 996423 (91c:00018)
- 21.
- B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal., 28 (1978), 377-385. MR 0500266 (58:17937)
- 22.
- P. Stollmann, A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains, Math. Z., 219 (1995), 275-287. MR 1337221 (96d:60119)
- 23.
- J. Weidmann, Continuous dependence of eigenvalues and eigenfunctions of elliptic differential operators on the domain (in German), Math. Scand., 54 (1984), 51-69. MR 753063 (86f:35133)
Similar Articles:
Retrieve articles in Mathematics of Computation
with
MSC (2000):
35R30, 35B34
Retrieve articles in all Journals with
MSC (2000):
35R30, 35B34
Additional Information:
Habib
Ammari
Affiliation:
Centre de Mathématiques Appliquées, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France
Email:
ammari@cmapx.polytechnique.fr
Elena
Beretta
Affiliation:
Dipartimento di Matematica ``G. Castelnuovo'' Università di Roma ``La Sapienza'', Piazzale Aldo Moro 5, 00185 Roma, Italy
Email:
beretta@mat.uniroma1.it
Elisa
Francini
Affiliation:
Dipartimento di Matematica, Università degli Studi di Firenze ``Ulisse Dini'', Viale Morgagni 67/A, 50134 Firenze, Italy
Email:
francini@math.unifi.it
Hyeonbae
Kang
Affiliation:
Department of Mathematics, Inha University, Incheon 402-751, Korea
Email:
hbkang@inha.ac.kr
Mikyoung
Lim
Affiliation:
Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, 335 Gwahangno (373-1 Guseong-dong), Yuseong-gu, Daejeon 305-701, Korea
Email:
mklim@kaist.ac.kr
DOI:
10.1090/S0025-5718-10-02344-6
PII:
S 0025-5718(10)02344-6
Keywords:
Shape reconstruction,
vibration analysis,
asymptotic expansion,
reconstruction algorithm,
optimization problem
Received by editor(s):
June 5, 2008
Received by editor(s) in revised form:
March 25, 2009 and August 9, 2009
Posted:
January 15, 2010
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
