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Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements


Authors: Habib Ammari, Elena Beretta, Elisa Francini, Hyeonbae Kang and Mikyoung Lim
Journal: Math. Comp. 79 (2010), 1757-1777
MSC (2000): Primary 35R30, 35B34
DOI: https://doi.org/10.1090/S0025-5718-10-02344-6
Published electronically: January 15, 2010
MathSciNet review: 2630011
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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.


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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: https://doi.org/10.1090/S0025-5718-10-02344-6
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
Published electronically: January 15, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.