Shape reconstruction of a conductivity inclusion using the Faber polynomials

Abstract: We consider the shape reconstruction of a conductivity inclusion in two dimensions. We use the concept of Faber polynomials Polarization Tensors (FPTs) introduced in \cite{choi:2018:GME} to derive an exact shape recovery formula for an inclusion with the extreme conductivity. This shape can be a good initial guess in the shape recovery optimization for an inclusion with either small or large conductivity values. We illustrate and validate our results with numerical examples.