  The problem of identifying the diffusion parameter appearing in a nonlocal
steady diffusion equation is considered. The identification problem is
formulated as an optimal control problem having a matching functional as the
objective of the control and the parameter function as the control variable.
The analysis makes use of a nonlocal vector calculus that allows one to define
a variational formulation of the nonlocal problem. In a manner analogous to the
local partial differential equations counterpart, we demonstrate, for certain
kernel functions, the existence of at least one optimal solution in the space
of admissible parameters. We introduce a Galerkin finite element discretization
of the optimal control problem and derive a priori error estimates for the
approximate state and control variables. Using one-dimensional numerical
experiments, we illustrate the theoretical results and show that by using
nonlocal models it is possible to estimate non-smooth and discontinuous
diffusion parameters.
