Optimization with respect to order in a fractional diffusion model: analysis, approximation and algorithmic aspects

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.