On the reconstruction of medium conductivity by integral equation method based on the Levi function

Consider an inverse problem of recovering the medium conductivity governed by an elliptic system, with partial information of the solution specified in some internal domain as inversion input. We firstly establish the uniqueness of this inverse problem and the conditional stability of H$ \ddot{\text{o}} $lder type in internal domain in terms of the analytic extension of the solution. Then by representing the solution of the direct problem with variable coefficient under the Levi function framework, this nonlinear inverse problem is reformulated as solving a linear integral system provided that the boundary value of the conductivity be known. Then this linear system is regularized to deal with the ill-posedness of the function extension, with an efficient numerical realization scheme for seeking the regularizing solution firstly for the density pair and then for the conductivity to be recovered. Numerical implementations are presented to show the validity of the proposed scheme.