Eigenstructure of the linearized electrical impedance tomography problem under radial perturbations

We analyze the Fréchet derivative $F$, that maps a perturbation in conductivity to the linearized change in boundary measurements governed by the conductivity equation. The domain is taken to be the unit ball $B \subset \mathbb{R}^d$ with $d \geq 2$, and we choose perturbations $η$ from the Hilbert space $L^2(B)$. Under the condition that the perturbations are rotationally symmetric, we show that the eigenfunctions of the linear approximation $F η$ correspond to the spherical harmonics. Furthermore, we establish an explicit formula for the associated eigenvalues and show that for perturbations from any bounded subset, the decay of these eigenvalues is uniform with respect to the degree of the spherical harmonics. The established structure of $F η$ enables us to show that the Fréchet derivative $F$ can be approximated by finite-rank operators when restricted to rotationally symmetric perturbations. Both the extension to $L^2(B)$ perturbations and the approximability by finite-rank operators are favorable properties for further analysis of $F$ in numerical algorithms.

💡 Research Summary

The paper investigates the linearized forward operator arising in Electrical Impedance Tomography (EIT) when the conductivity perturbation is restricted to radially symmetric functions. The authors consider the unit ball (B\subset\mathbb{R}^d) with (d\ge 2) and study the Fréchet derivative (F = D\Lambda(1;\cdot)) of the Neumann‑to‑Dirichlet (ND) map at the homogeneous conductivity (\gamma\equiv1). By the well‑known identity
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