Simultaneous Reconstruction of Coefficients and Source Parameters in Elliptic Systems Modelled with Many Boundary Value Problems

In this work the inverse problem for determination of unknown parameters related to both intensities and support of sources and materials coefficients in second-order elliptic equations models is posed with over specification of data on the boundary. A discrepancy function based on difference of two mixed problems formulated by splitting the Cauchy data is introduced. This function controls the measured difference between the two solutions for the same set of Cauchy data. Parameters can be determined by minimization of this function under guess values. The concept of Calderón projector gap is introduced as a tool for checking the consistency of Cauchy data. Numerical implementations based on quadratic finite elements are presented in a two dimensional square (−1, +1) × (−1, +1) model with unknown source, conductivity, and absorption supported by an also unknown characteristic square shape interior domain. Since this minimization involves the iterative solution of a huge number of direct boundary value problems, the adoption of a non-differentiable minimization algorithm is recommended and the Nelder-Mead simplex method is used to search for optimal parameters.