Some notes concerning preconditioning of linear parabolic optimal control problems

TL;DR

This paper investigates the conditioning of linear parabolic optimal control problems and proposes preconditioners that ensure bounded condition numbers regardless of discretization level, supported by numerical experiments.

Contribution

It introduces a preconditioning strategy for all-at-once systems in parabolic optimal control problems that maintains stable eigenvalue bounds across discretizations.

Findings

Condition numbers are bounded independently of discretization.

Proposed preconditioners improve computational efficiency.

Numerical experiments confirm theoretical results.

Abstract

In this paper we study the conditioning of optimal control problems constrained by linear parabolic equations with Neumann boundary conditions. While we concentrate on a given end-time target function the results hold also when the target function is given over the whole time horizon. When implicit time discretization and conforming finite elements in space are employed we show that the reduced problem formulation has condition numbers which are bounded independently of the discretization level in arbitrary space dimension. In addition we propose for the all-at-once approach, i.e. for the first-order conditions of the unreduced system a preconditioner based on work by Greif and Sch\"otzau, which provides also bounds on the eigenvalue distribution independently of the discretization level. Numerical experiments demonstrate the obtained results and the efficiency of the suggested preconditioners.