Optimization with Respect to Order in a Fractional Diffusion Model: Analysis, Approximation and Algorithmic Aspects

We consider an identification (inverse) problem, where the state u is governed by a fractional elliptic equation and the unknown variable corresponds to the order s(0,1) of the underlying operator. We study the existence of an optimal pair (s) 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.