  In this paper, we propose and analyze a reconstruction algorithm for imaging
an anisotropic conductivity tensor in a second-order elliptic PDE with a
nonzero Dirichlet boundary condition from internal current densities. It is
based on a regularized output least-squares formulation with the standard
$L^2(\Omega)^{d,d}$ penalty, which is then discretized by the standard Galerkin
finite element method. We establish the continuity and differentiability of the
forward map with respect to the conductivity tensor in the
$L^p(\Omega)^{d,d}$-norms, the existence of minimizers and optimality systems
of the regularized formulation using the concept of H-convergence. Further, we
provide a detailed analysis of the discretized problem, especially the
convergence of the discrete approximations with respect to the mesh size, using
the discrete counterpart of H-convergence. In addition, we develop a projected
Newton algorithm for solving the first-order optimality system. We present
extensive two-dimensional numerical examples to show the efficiency of the
proposed method.
