Analytical shape recovery of a conductivity inclusion based on Faber polynomials

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A conductivity inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from multistatic measurements. As a modification of GPTs, the Faber polynomial polarization tensors (FPTs) were recently introduced in two dimensions. In this study, we design two novel analytical non-iterative methods for recovering the shape of a simply connected inclusion from GPTs by employing the concept of FPTs. First, we derive an explicit expression for the coefficients of the exterior conformal mapping associated with an inclusion in a simple form in terms of GPTs, which allows us to accurately reconstruct the shape of an inclusion with extreme or near-extreme conductivity. Secondly, we provide an explicit asymptotic formula in terms of GPTs for the shape of an inclusion with arbitrary conductivity by considering the inclusion as a perturbation of its equivalent ellipse. With this formula, one can non-iteratively approximate an inclusion of general shape with arbitrary conductivity, including a straight or asymmetric shape. Numerical experiments demonstrate the validity of the proposed analytical approaches.
Publisher
SPRINGER HEIDELBERG
Issue Date
2021-12
Language
English
Article Type
Article
Citation

MATHEMATISCHE ANNALEN, v.381, no.3-4, pp.1837 - 1867

ISSN
0025-5831
DOI
10.1007/s00208-020-02041-1
URI
http://hdl.handle.net/10203/288744
Appears in Collection
MA-Journal Papers(저널논문)
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