Reconstruction of Inhomogeneous Conductivities via the Concept of Generalized Polarization Tensors

TL;DR

This paper generalizes the concept of generalized polarization tensors (GPTs) to inhomogeneous conductivities, establishing their properties, relation to the NtD map, and proposing a reconstruction algorithm for conductivity distributions.

Contribution

It extends GPTs to inhomogeneous cases, relates them to the NtD map, and introduces a new algorithm for reconstructing conductivity distributions.

Findings

GPTs uniquely determine conductivity distributions

Properties like symmetry and positivity of GPTs are established

Numerical examples demonstrate the reconstruction algorithm's viability

Abstract

This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.