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

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