Inverse problem for a planar conductivity inclusion

TL;DR

This paper develops an analytical method to recover a planar homogeneous conductivity inclusion from exterior measurements using generalized polarization tensors, providing an inversion formula linked to conformal mapping coefficients.

Contribution

It introduces a new inversion formula for planar conductivity inclusions based on GPTs and establishes matrix factorizations to prove it, advancing inverse problem techniques.

Findings

Derived an explicit inversion formula for the inclusion

Established matrix factorizations for GPTs

Demonstrated the method's theoretical validity

Abstract

This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.