Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Calderón cavities inverse problem as a shape-from-moments problem


Authors: Alexandre Munnier and Karim Ramdani
Journal: Quart. Appl. Math. 76 (2018), 407-435
MSC (2010): Primary 31A25, 45Q05, 65N21, 30E05
DOI: https://doi.org/10.1090/qam/1505
Published electronically: April 11, 2018
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Abstract: In this paper, we address a particular case of Calderón's (or conductivity) inverse problem in dimension two, namely the case of a homogeneous background containing a finite number of cavities (i.e., heterogeneities of infinitely high conductivities). We aim to recover the location and the shape of the cavities from the knowledge of the Dirichlet-to-Neumann (DtN) map of the problem. The proposed reconstruction method is non-iterative and uses two main ingredients. First, we show how to compute the so-called Generalized Pólia-Szegö tensors (GPST) of the cavities from the DtN of the cavities. Secondly, we show that the obtained shape from the GPST inverse problem can be transformed into a shape-from-moments problem, for some particular configurations. However, numerical results suggest that the reconstruction method is efficient for arbitrary geometries.


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Additional Information

Alexandre Munnier
Affiliation: Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
Email: alexandre.munnier@univ-lorraine.fr

Karim Ramdani
Affiliation: Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
Email: karim.ramdani@inria.fr

DOI: https://doi.org/10.1090/qam/1505
Received by editor(s): April 5, 2017
Published electronically: April 11, 2018
Article copyright: © Copyright 2018 Brown University

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