  This paper concerns the reconstruction of a scalar coefficient of a
second-order elliptic equation in divergence form posed on a bounded domain
from internal data. This theory finds applications in multi-wave imaging,
greedy methods to approximate parameter-dependent elliptic problems, and image
treatment with partial differential equations. We first show that the inverse
problem for smooth coefficients can be rewritten as a linear transport
equation. Assuming that the coefficient is known near the boundary, we study
the well-posedness of associated transport equation as well as its numerical
resolution using discontinuous Galerkin method. We propose a regularized
transport equation that allow us to derive rigorous convergence rates of the
numerical method in terms of the order of the polynomial approximation as well
as the regularization parameter. We finally provide numerical examples for the
inversion assuming a lower regularity of the coefficient, and using synthetic
data.
