Optimization with respect to order in a fractional diffusion model: analysis, approximation and algorithmic aspects
It provides theoretical and numerical foundations for estimating the fractional order in elliptic PDEs, a problem relevant to inverse problems and parameter identification.
This paper studies the identification of the fractional order parameter in an elliptic PDE, proving existence and local uniqueness of optimal solutions, and developing convergent numerical algorithms with supporting experiments.
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.