Recent Advances in Stability for Inverse Inclusion Problems in Electrical Impedance Tomography

Inverse inclusion problems arise in several applications such as electrical impedance tomography, nondestructive testing, and geophysical imaging. The purpose of this review is to present some recent advances concerning stability estimates for the determination of inclusions from boundary measurements. We focus on elliptic conductivity equations with discontinuous coefficients and discuss different settings including isotropic and anisotropic conductivities, variable coefficients, local boundary measurements, and layered media. Particular attention is devoted to the analytical tools underlying logarithmic stability estimates, namely singular solutions, asymptotic analysis of fundamental solutions, quantitative unique continuation, and propagation of smallness arguments. We also compare different stability regimes, emphasizing how finite-dimensional geometric priors, such as polygonal or polyhedral inclusions, may lead to Hölder or Lipschitz stability estimates in contrast with the logarithmic stability typical of broader infinite-dimensional classes. We highlight how a common methodological framework can be adapted to increasingly complex geometrical and physical configurations. Finally, we briefly discuss some open problems and possible future developments in the study of inverse inclusion problems.