# An adaptive finite element method for the sparse optimal control of fractional diffusion

## TL;DR

This paper develops an adaptive finite element method with an a posteriori error estimator for fractional diffusion optimal control problems, achieving optimal convergence rates through local error assessment.

## Contribution

It introduces a novel a posteriori error estimator for fractional PDE-constrained optimization, combining state, adjoint, and control discretization errors, with proven efficiency and reliability.

## Key findings

- The estimator is locally efficient and reliable.
- The adaptive scheme attains optimal convergence rates.
- Numerical experiments confirm the effectiveness of the approach.

## Abstract

We propose and analyze an a posteriori error estimator for a PDE-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence.