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The Generalized Polarization Tensors for Resolved Imaging. Part I: Shape Reconstruction of a Conductivity
 

Summary: The Generalized Polarization Tensors for Resolved
Imaging. Part I: Shape Reconstruction of a Conductivity
Inclusion
Habib Ammari
Hyeonbae Kang
Mikyoung Lim
Habib Zribi
Abstract
With each Lipschitz domain and material parameter, an infinite number of tensors,
called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain
significant information on the shape of the domain and its material parameter. They
generalize the concept of Polarization Tensor (PT), which can be seen as the first-order
GPT. It is known that given an arbitrary shape, one can find an equivalent ellipse or
ellipsoid with the same PT. In this paper we consider the problem of recovering finer
details of the shape of a given domain using higher-order polarization tensors. We
design an optimization approach which solves the problem by minimizing a weighted
discrepancy functional. In order to compute the shape derivative of this functional, we
rigorously derive an asymptotic expansion of the perturbations of the GPTs that are
due to a small deformation of the boundary of the domain. Our derivations are based on
the theory of layer potentials. We perform some numerical experiments to demonstrate

  

Source: Ammari, Habib - Centre de Mathématique Appliquées, École Polytechnique

 

Collections: Mathematics