  We consider an identification problem, where the state $u$ is governed by a
fractional elliptic equation and the unknown variable corresponds to the order
$s \in (0,1)$ of the underlying operator. We study the existence of an optimal
pair $(\bar s, \bar u)$ and provide sufficient conditions for its local
uniqueness. We develop semi-discrete and fully discrete algorithms to
approximate the solutions to our identification problem and provide a
convergence analysis. We present numerical illustrations that confirm and
extend our theory.
