A Note on Multigrid Preconditioning for Fractional PDE-Constrained Optimization Problems

TL;DR

This paper introduces a multigrid preconditioning method for fractional PDE-constrained optimization problems, effectively reducing iterative solver complexity despite dense fractional operator matrices.

Contribution

It develops a novel multigrid preconditioning strategy based on a reduced approach, addressing the challenge of dense fractional operator matrices in PDE-constrained optimization.

Findings

Significant reduction in CG iterations with the new preconditioner

Preconditioner quality assessed via spectral distance

Numerical experiments support the theoretical conjecture

Abstract

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-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.