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