On the simultaneous recovery of boundary impedance and internal conductivity

Consider an inverse problem of the simultaneous recovery of boundary impedance and internal conductivity in the electrical impedance tomography (EIT) model using local internal measurement data, which is governed by a boundary value problem for an elliptic equation in divergence form with Robin boundary condition. We firstly express the solution to the forward problem by volume and surface potentials in terms of the Levi function. Then, for the inverse problem, we prove the uniqueness of the solution in an admissible set by unique extension of the solution under some {a-prior} assumption. Finally we establish the regularizing reconstruction schemes for boundary impedance and internal conductivity using noisy measurement data with rigorous error estimates. The mollification method is proposed to recover the boundary impedance from the boundary condition, and the internal conductivity with known boundary value is recovered from an integral system, where the Tikhonov regularization is applied to seek the stable solution, considering that the error involved in the boundary impedance coefficient reconstruction will propagate to the recovering process for internal conductivity. Numerical implementations are presented to illustrate the validity of the proposed method.