Some notes concerning preconditioning of linear parabolic optimal control problems

Luise Blank

doi:10.3934/naco.2026014

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2026, Volume 19: 121-132. Doi: 10.3934/naco.2026014

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Some notes concerning preconditioning of linear parabolic optimal control problems

Luise Blank

Department of Mathematics, University of Regensburg, Germany

Received: March 21, 2025

Revised: March 2, 2026

Early access: April 9, 2026

Published online: April 9, 2026

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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 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 the all-at-once approach, i.e., for the first-order conditions of the unreduced system a preconditioner based on work by Greif and Schötzau, which provides bounds on the eigenvalue distribution independently of the discretization level. Numerical experiments demonstrate the obtained results and the efficiency of the suggested preconditioners.

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Mathematics Subject Classification: Primary: 49M05, 49M41, 65F08, 65F50.

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Figure 1. The maximal eigenvalues of $ \bar F_{\tau,h} $ for various $ \alpha $ and $ d = 2 $ with $ h = 0.01 $ as well as for $ d = 3 $ and $ h_{max} = 0.2 $

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Table 1. Maximal eigenvalues of $ \bar F_{\tau,h} $ for decreasing values of $ c $

$ c $	$ 100 $	$ 10 $	$ 1 $	$ -1 $	$ -10 $	$ -15 $
$ \mu_{max} $	$ 1.00476 $	$ 1.04975 $	$ 1.43204 $	$ 4.24815 $	$ 2.69623e+7 $	$ 4.70925e+12 $

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