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

TL;DR

This paper investigates an optimization problem involving the order of a fractional elliptic operator, establishing existence, uniqueness, and developing algorithms with convergence analysis, supported by numerical experiments.

Contribution

It introduces a new framework for identifying the fractional order in elliptic equations, including theoretical analysis and practical algorithms.

Findings

Existence of an optimal fractional order and associated state.

Conditions for local uniqueness of the solution.

Algorithms with proven convergence and numerical validation.

Abstract

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.