An adaptive finite element method for the sparse optimal control of fractional diffusion
Provides a rigorous error analysis and adaptive method for a challenging class of sparse optimal control problems with fractional diffusion, benefiting researchers in computational PDE-constrained optimization.
This paper develops an a posteriori error estimator for PDE-constrained optimization with nondifferentiable cost, fractional diffusion, and control constraints, and proves its local efficiency and reliability. The adaptive scheme achieves optimal experimental convergence rates.
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.