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Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

Monika Wolfmayr

TL;DR

This work addresses computable, guaranteed lower bounds for two cost functionals in time-periodic parabolic optimal control, complementing existing upper bounds to yield two-sided estimates. It leverages the multiharmonic finite element method to decouple the problem across Fourier modes, enabling mode-wise analysis, discretization, and adaptive refinement. The main contributions include explicit minorants for the first functional and both majorants and minorants for the gradient-target second functional, together with rigorous a posteriori error bounds and robust MinRes-AMLI preconditioning. Numerical experiments across smooth and non-smooth data validate the tightness of the bounds and demonstrate the practical effectiveness of the adaptive time-space framework AMhFEM for time-periodic optimization problems.

Abstract

In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimization problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work provides the basis for an adaptive scheme for time-periodic optimization problems.

Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

TL;DR

This work addresses computable, guaranteed lower bounds for two cost functionals in time-periodic parabolic optimal control, complementing existing upper bounds to yield two-sided estimates. It leverages the multiharmonic finite element method to decouple the problem across Fourier modes, enabling mode-wise analysis, discretization, and adaptive refinement. The main contributions include explicit minorants for the first functional and both majorants and minorants for the gradient-target second functional, together with rigorous a posteriori error bounds and robust MinRes-AMLI preconditioning. Numerical experiments across smooth and non-smooth data validate the tightness of the bounds and demonstrate the practical effectiveness of the adaptive time-space framework AMhFEM for time-periodic optimization problems.

Abstract

In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to large systems of linear equations having a saddle point structure. The derivation of preconditioners for the minimal residual method for the new optimization problem is discussed in more detail. Finally, several numerical experiments for both optimal control problems are presented confirming the theoretical results obtained. This work provides the basis for an adaptive scheme for time-periodic optimization problems.

Paper Structure

This paper contains 27 sections, 9 theorems, 134 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Time-periodic parabolic PDE, the two cost functionals and preliminary results
  3. Preliminaries
  4. Optimization problem I
  5. Majorant for cost functional \ref{['equation:minfunc1']}
  6. Optimization problem II
  7. Guaranteed lower bounds leading to two-sided bounds for optimization problem I
  8. Minorant for cost functional \ref{['equation:minfunc1']}
  9. Guaranteed upper bounds for the discretization errors of the control and the state
  10. Two-sided bounds for optimization problem II
  11. Majorant for cost functional \ref{['equation:minfunc2']}
  12. Minorant for cost functional \ref{['equation:minfunc2']}
  13. Guaranteed upper bounds for the discretization errors of the control and the state
  14. Multiharmonic finite element (MhFE) discretization
  15. Optimization problem I
  16. ...and 12 more sections

Key Result

Theorem 1

Let $y_d \in L^2(Q)$ be given. Let $p_\eta \in H^{1,\frac{1}{2}}_{0,per}(Q)$ solve equation:NOCforPeta1 for an arbitrary $\eta \in H^{1,1}_{0,per}(Q)$. The following estimate holds: for any $\zeta \in H^{1,1}_{0,per}(Q)$, where $\underline{\mu_{1}} = \frac{1}{\sqrt{2}} \min\{\underline{\nu},\underline{\sigma}\}$, $\mathcal{R}_3(\zeta,\boldsymbol{\rho},\eta) = \eta - y_d + \nabla \cdot \boldsymbol

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 4
  • Theorem 3
  • proof
  • ...and 11 more