Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST)
Reconstruction of a conductivity inclusion using the Faber polynomials
In this talk, I present analytical shape recovery methods for the conductivity inclusion in two dimensions, obtained in collaboration with Doosung Choi and Junbeom Kim. For a simply connected planar domain, there exists a function that conformally maps a region outside a circular disk to the region outside the domain. The conformal mapping then defines the so-called Faber polynomials, which form a basis for analytic functions. An electrical inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a far-field expansion, whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from the exterior measurements. As a modification of GPTs, we recently introduced the Faber polynomial polarization tensors (FPTs). We design two analytical approaches for the shape recovery of a conductivity inclusion by employing the concept of FPTs. Numerical experiments demonstrate the validity of the proposed analytical methods.