A reaction coefficient identification problem for fractional diffusion

TL;DR

This paper addresses the inverse problem of identifying a reaction coefficient in fractional diffusion models, proposing a numerical scheme with proven convergence and error estimates based on spectral fractional operators.

Contribution

It introduces a novel approach to the reaction coefficient identification problem for fractional diffusion, including regularity analysis, a truncation strategy, and a fully discrete FEM scheme with convergence guarantees.

Findings

Existence of local solutions and optimality conditions.

Regularity estimates and solution decay properties.

Convergence and error estimates for the proposed numerical scheme.

Abstract

We analyze a reaction coefficient identification problem for the spectral fractional powers of a symmetric, coercive, linear, elliptic, second-order operator in a bounded domain $\Omega$. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $\Omega \times (0,\infty)$. We thus consider an equivalent coefficient identification problem, where the coefficient to be identified appears explicitly. We derive existence of local solutions, optimality conditions, regularity estimates, and a rapid decay of solutions on the extended domain $(0,\infty)$. The latter property suggests a truncation that is suitable for numerical approximation. We thus propose and analyze a fully discrete scheme that discretizes the set of admissible coefficients with piecewise constant functions. The discretization of the state equation relies on the tensorization of a first-degree FEM in $\Omega$ with a suitable $hp$-FEM in the extended dimension. We derive convergence results and obtain, under the assumption that in neighborhood of a local solution the second derivative of the reduced cost functional is coercive, a priori error estimates.