An Iterative and Adaptive Lie-Group Method for Solving the Calderón Inverse Problem

Abstract

We solve the Calderon inverse conductivity problem (Calderon (1980, 2006)), for an elliptic type equation in a rectangular plane domain, to recover an unknown conductivity function inside the domain, from the over-specified Cauchy data on the bottom of the rectangle. The Calderon inverse problem exhibits three- fold simultaneous difficulties: ill-posedness of the inverse Cauchy problem, ill- posedness of the parameter identification, and no information inside the domain being available on the impedance function. In order to solve this problem, we discretize the whole domain into many sub-domains of finite strips, each with a small height. Thus the Calderon inverse problem is reduced to an inverse Cauchy problem and a parameter identification problem in each finite strip. An effective combination of the Lie-group adaptive method (LGAM), together with a finite- strip method is developed, where the Lie-group equation can adaptively solve the semi-discretized ODEs to find the unknown conductivity coefficients through it- erations. The success of the present method hinges on a rationale that the local ODEs and the global Lie-group equation have to be self-adaptive during the itera- tion process. Thus, we have a computationally inexpensive mathematical algorithm to solve the Calderon inverse problem. The feasibility, accuracy and efficiency of present method are evaluated by comparing the estimated results for the unknown impedance function in the domain, in the Calderon inverse problem, with some postulated exact solutions. It may be concluded that the iterative and adaptive Lie- group method presented in this paper, may provide a simple and effective means of solving the Calderon inverse problem in general domains.