A note on multigrid preconditioning for fractional PDE-constrained optimization problems

In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order optimality Karush–Kuhn–Tucker (KKT) systems are widely popular, but their development have relied on the underlying systems being sparse. On the other hand, for most discretizations, the matrix representation of fractional operators is expected to be dense. We develop a preconditioning strategy for our problem based on a reduced approach, namely we eliminate the state constraint using the control-to-state map. Our multigrid preconditioning approach shows a dramatic reduction in the number of CG iterations. We assess the quality of preconditioner in terms of the spectral distance. Finally, we provide a partial theoretical analysis for this preconditioner, and we formulate a conjecture which is clearly supported by our numerical experiments.