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Some notes concerning preconditioning of linear parabolic optimal control problems

Luise Blank

TL;DR

This work analyzes conditioning in linear parabolic PDE-constrained optimization with Neumann boundaries and end-time tracking. It proves mesh-independent conditioning for the reduced system under implicit $dG(0)$ time discretization and conforming spatial FE, and develops a Greif–Schötzau-based preconditioner for the all-at-once saddle-point system that yields spectrally clustered, discretization-insensitive behavior. Numerical experiments in multiple space dimensions validate the theoretical bounds and demonstrate robust, low iteration counts for both reduced and all-at-once formulations. The results provide practical preconditioning strategies for fast solvers in PDE-constrained parabolic optimization and clarify a trade-off between reduced and full-system approaches.

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 Schötzau, 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.

Some notes concerning preconditioning of linear parabolic optimal control problems

TL;DR

This work analyzes conditioning in linear parabolic PDE-constrained optimization with Neumann boundaries and end-time tracking. It proves mesh-independent conditioning for the reduced system under implicit time discretization and conforming spatial FE, and develops a Greif–Schötzau-based preconditioner for the all-at-once saddle-point system that yields spectrally clustered, discretization-insensitive behavior. Numerical experiments in multiple space dimensions validate the theoretical bounds and demonstrate robust, low iteration counts for both reduced and all-at-once formulations. The results provide practical preconditioning strategies for fast solvers in PDE-constrained parabolic optimization and clarify a trade-off between reduced and full-system approaches.

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 Schötzau, 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.
Paper Structure (7 sections, 5 theorems, 36 equations, 1 figure, 1 table)

This paper contains 7 sections, 5 theorems, 36 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation and time discretization
  3. Eigenvalue estimates for the reduced problem formulation
  4. Numerical evidence
  5. Preconditioning of the all-at-once approach
  6. Numerical evidence
  7. Conclusions from the numerical results

Key Result

Theorem 3.1

There exist $\gamma , \bar{\tau} > 0$, such that for all eigenvalues $\eta(F_{\tau})$, $\eta(F_{\tau,h})$, $\eta(\bar{F}_{\tau,h})$ it holds in any space dimension Furthermore, at most $rank (e_N e_N^T \otimes M_N )=\dim (S_h^{(N)})$ eigenvalues of $\bar{F}_{\tau,h}$ as well as of $F_{\tau,h}$ do not equal $\lambda$.

Figures (1)

  • 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 $h_{max}=0.2$.

Theorems & Definitions (7)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 4.1
  • proof
  • Corollary 4.2