Continuity of the linearized forward map of electrical impedance tomography from square-integrable perturbations to Hilbert-Schmidt operators

TL;DR

This paper proves that the linearized forward map in 2D electrical impedance tomography is continuous from square-integrable conductivity perturbations to Hilbert-Schmidt operators, supporting advanced reconstruction algorithms.

Contribution

It establishes the boundedness of the Fréchet derivative as a Hilbert-Schmidt operator under specific conditions, providing a rigorous foundation for linearization-based EIT reconstructions.

Findings

Fréchet derivative is bounded from L^2 to Hilbert-Schmidt operators.

Results hold for constant background conductivity and C^{1,\alpha} boundary domains.

Supports the theoretical analysis of linearization methods in EIT.

Abstract

This work considers the Fr\'echet derivative of the idealized forward map of two-dimensional electrical impedance tomography, i.e., the linear operator that maps a perturbation of the coefficient in the conductivity equation over a bounded two-dimensional domain to the linear approximation of the corresponding change in the Neumann-to-Dirichlet boundary map. It is proved that the Fr\'echet derivative is bounded from the space of square-integrable conductivity perturbations to the space of Hilbert--Schmidt operators on the mean-free $L^2$ functions on the domain boundary, if the background conductivity coefficient is constant and the considered simply-connected domain has a $C^{1,\alpha}$ boundary. This result provides a theoretical framework for analyzing linearization-based one-step reconstruction algorithms of electrical impedance tomography in an infinite-dimensional setting.