On second order shape optimization methods for electrical impedance tomography

This paper is devoted to the analysis of a second order method for recovering the \emph{a priori} unknown shape of an inclusion $\omega$ inside a body $\Omega$ from boundary measurement. This inverse problem - known as electrical impedance tomography - has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\partial\omega$, then we choose a cost function in order to recover the geometry of $\partial \omega$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.

💡 Research Summary

The paper addresses the inverse problem of Electrical Impedance Tomography (EIT), namely the reconstruction of an unknown inclusion ω inside a bounded domain Ω from boundary measurements. While many works have employed first‑order shape‑gradient methods for this severely ill‑posed problem, none have explored a genuine second‑order (Newton‑type) approach. The authors fill this gap by developing a rigorous shape‑calculus framework that yields the second‑order shape derivative (the Hessian) of the forward state u with respect to perturbations of the interface ∂ω.

The forward model is a transmission problem: the conductivity σ takes two constant values σ₁ in Ω\ω and σ₂ in ω, and the electric potential u satisfies ∇·(σ∇u)=0 in Ω together with Dirichlet data on the outer boundary ∂Ω. The inclusion boundary ∂ω is assumed to be of class C^{2,α}, which guarantees enough regularity for the state to belong to H² locally on each subdomain. A smooth vector field V defines a shape perturbation via the flow Tₜ(x)=x+tV(x). The authors distinguish material and shape derivatives, proving that the first material derivative \dot u exists in H² and satisfies a linearized transmission problem. By differentiating once more they establish the existence of the second material derivative \ddot u, which leads to the second‑order shape derivative of the state.

With the state derivatives in hand, the paper defines a cost functional J(ω)=½‖u|{∂Ω}−u_meas‖²{L²(∂Ω)} that measures the mismatch between simulated and measured boundary data. The first shape derivative ∇J