A reaction coefficient identification problem for fractional diffusion
This work provides a theoretical and numerical framework for identifying reaction coefficients in fractional diffusion models, which is relevant for applications in anomalous transport and porous media, but the results are incremental as they extend existing techniques to a specific fractional operator formulation.
The paper analyzes a reaction coefficient identification problem for fractional diffusion, proving existence of local solutions, optimality conditions, regularity estimates, and rapid decay. It proposes a fully discrete scheme with piecewise constant coefficient discretization and tensorized FEM, deriving convergence and a priori error estimates under a coercivity assumption.
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 $Ω$. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $Ω\times (0,\infty)$. 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 $Ω$ 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.