A~posteriori error analysis for optimization with PDE constraints

TL;DR

This work develops an a posteriori error analysis for PDE-constrained optimization with a linear state equation and box constraints on the control, using a reduced and rescaled optimality system with the adjoint scaled as $z=p/\sqrt{\alpha}$. A computable estimator $\eta$ (with $\eta^2=\sum_{z\in\mathcal V}\eta_z^2$) bounds the total error $\mathsf{err}$ via $c_\Delta\eta \le \mathsf{err} \le C_\Delta \min\{1+O(1/\sqrt{\alpha}),\,1+O(1/\alpha)\frac{(\sum h_z^2\eta_z^2)^{1/2}}{\eta}\}\eta$, with constants independent of $\alpha$; as $\alpha\to0$ the gap between bounds grows like $O(1/\sqrt{\alpha})$, improved to $O(1/\alpha)$ in prior work. The analysis exploits the compactness of the control and observation operators $C$ and $I$ to decompose the residual into a dominant invertible part and a compact perturbation, enabling an auxiliary reconstruction $R\tilde{x}$ to bound the error more sharply when compactness is present. In the unconstrained case ($K=Q$) the bounds simplify and depend on a symmetric, linear residual norm, offering α-independent estimates. The framework is illustrated on model problems with distributed and boundary control, including finite element discretizations and adaptive mesh refinement guided by the residual-based estimators, and demonstrates how compactness and regularity influence estimator performance.

Abstract

We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a posteriori bounds capturing the approximation of the state, the adjoint state, the control and the observation. The upper and lower bounds show a gap, which grows with decreasing cost or Tikhonov regularization parameter. This growth is mitigated compared to previous results and can be countered by refinement if control and observation involve compact operators. Numerical results illustrate these properties for model problems with distributed and boundary control.

Figures (5)

• Figure 1: Domain $\Omega$ and subdomains $\Omega_\textsf{W}, \Omega_\textsf{Q}$ for Example \ref{['distributed-control-example']}.
• Figure 2: $d_\alpha$-error (+) and associated general upper ($\bullet$) and lower ($\circ$) bound, as well as upper bound with compactness ($$) versus DOFs for Example \ref{['distributed-control-example']}.
• Figure 3: Adaptive mesh refinement history for Example \ref{['distributed-control-example']} with $\alpha = 10^{-4}$.
• Figure 4: $\left\|{\cdot}\right\|$-error (+), general upper ($\bullet$) and lower ($\circ$) bound, as well as upper ($$) and lower bound with compactness versus DOFs for Example \ref{['neumann-control-example']}.
• Figure 5: Adaptive mesh refinement history for Example \ref{['neumann-control-example']} with $\alpha = 10^{-1}$.

Theorems & Definitions (31)

• Theorem 2.1: Continuity and inf-sup stability of form $b_\alpha$
• proof
• Theorem 3.1: Bounding the $d_\alpha$-error -- general case
• proof
• Remark 3.2: Gap in bounding $d_\alpha$-error -- general case
• Corollary 3.3: Bounding the augmented $d_\alpha$-error -- general case
• proof
• Remark 3.4: Comparison with KohlsRoeschSiebert:2014KohlsRoeschSiebert:2012
• Remark 3.5: Connection with KohlsRoeschSiebert:2014, elliptic reconstruction, Wheeler's trick, and Schatz's argument
• Theorem 3.6: Bounding the $d_\alpha$-error -- compact case