Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data.