A numerical investigation of the inverse potential conductivity problem in a circular inclusion

Abstract

In this paper we consider the inverse conductivity problem which requires the determination of the location, size and/or non-dimensional conductivity, k, of a circular inclusion D contained in a domain Ω from measured electric voltage, ϕ, and electric current flux, (∂ϕ/∂n), on the boundary ∂Ω. Concrete practical applications concern electrical impedance tomography experiments which are performed in order to monitor patients' internal diseases. This difficult inverse problem, yet completely unsolved, is approached using the boundary element method in conjunction with a least-squares minimization procedure. The influence of each variable (such as the location and size of the inclusion, the non-dimensional conductivity k and the ‘noisy’ measured boundary values) on the stability and convergence of the numerical solution is thoroughly investigated. It was found that the retrieval of the location and size of the circular inclusion is convergent and stable no matter whether k is known or unknown. The retrieval of k is stable with respect to noise in the boundary measurements, but unstable with respect to noise in the location and/or size of the inclusion.