# Shape reconstruction of a conductivity inclusion using the Faber polynomials

## TL;DR

This paper presents an exact shape reconstruction method for a conductivity inclusion in 2D using Faber polynomials and Polarization Tensors, aiding initial guesses in shape optimization.

## Contribution

It introduces a novel shape recovery formula based on Faber polynomials for inclusions with extreme conductivity, validated through numerical examples.

## Key findings

- Exact shape recovery formula derived for extreme conductivity inclusions.
- Numerical validation confirms the effectiveness of the shape reconstruction method.
- Method provides a good initial guess for further shape optimization.

## 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.