Reconstruction of the shape of an inclusion from Generalized Polarization Tensors and Elastic Moment Tensors

다국어 초록 (Multilingual Abstract)

The concept of the Generalized Polarization Tensors(GPTs) and the Elastic Moment Tensors(EMTs) have been studied particularly in the context of imaging small inclusions. The GPTs and EMTs contain significant information on the shape of the domain and its material parameter. It is known that given an arbitrary shape, one can find an equivalent ellipse or ellipsoid with the same first order GPT or EMT. In this paper we consider the problem of recovering finer details of the shape of a given domain using higher-order GPTs and EMTs. We design an optimization approach which solves the problem by simply 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 and EMTs that are due to a small deformation of the boundary of the domain. We perform some numerical experiments to demonstrate the validity and the limitations of the proposed method. The results clearly show that our approach is very promising in recovering fine shape details.