Adaptive Regularisation for PDE-Constrained Optimal Control

TL;DR

This work tackles PDE-constrained optimal control with a small regularisation parameter, which yields a singularly perturbed problem. It develops a finite element framework that couples adaptive mesh refinement with spatially varying regularisation $\alpha_h$, guided by rigorous $a$ posteriori$ error estimators that bound both discretisation and regularisation inconsistencies. The authors derive global upper and local lower bounds for the energy error, and implement an adaptive loop (solve-estimate-mark-refine) with a maximum-marking strategy to balance $\eta_h$ and $\eta_α$, demonstrating reliability on smooth targets, boundary layers, and discontinuous data. Numerical experiments in 1D and 2D, including condition-number studies, show improved accuracy and stability, with the regularisation adapting to local solution features and the mesh focusing refinement where needed. The approach provides a robust, efficient pathway to solve nonlinear PDE-constrained optimisation problems by dynamically balancing regularisation and discretisation errors and suggests several avenues for future extensions to alternative regularisation norms and spatially varying parameters.

Abstract

PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both regularisation and discretisation adaptivity, varying both the regularisation parameter and mesh-size locally based on rigorous a posteriori error estimates aiming to dynamically balance induced regularisation and discretisation errors, offering a robust and efficient method for solving these problems. We demonstrate the efficacy of our analysis with several numerical experiments.

Figures (14)

• Figure 1: Example \ref{['gaussian']}. Plots of the target, state, and adjoint variables with $\alpha = 10^{-8}$.
• Figure 2: Example \ref{['bl']}. Plots of state, and adjoint variables, $\alpha = 10^{-6}$.
• Figure 3: Convergence plots for \ref{['sec:conv']} for examples \ref{['gaussian']} and \ref{['bl']}. The left column showcases the convergence of the mesh estimator $\eta_T$ as the mesh is uniformly refined with no adaptive regularisation. The results showcase that this estimator is robust in $h$. The middle column plots the effectivity index $\eta_h/\|\boldsymbol{e}\|_X$ for a fixed uniform mesh for different choices of $\alpha$. The effectivity is constant and therefore $\eta_h$ is robust in $\alpha$. The right column showcases the behaviour of the error and estimators on a fixed uniform mesh for the adaptive regularisation scheme as $\alpha_h$ was decreased uniformly to $\alpha$.
• Figure 4: Example \ref{['gaussian']} (smooth target), \ref{['sec:regadpt']}. Convergence results for the regularisation refinement only scheme. $\text{dim}V_h = 10^4$, $\rho = 0.5$ and $\alpha = 10^{-8}$, $\texttt{tol}_{\alpha} = 10^{-8}$
• Figure 5: Example \ref{['gaussian']} (smooth target): \ref{['sec:regadpt']}, regularisation refinement only. Plots of $\alpha_h$ (left), $\eta_\alpha$ (middle) and the approximate solution $u_h$ compared to $u$ and $d$ (right) over the domain at three different iterations. $\text{dim}V_h = 10^4$, $\rho = 0.5$ and $\alpha = 10^{-8}$, $\texttt{tol}_\alpha = 10^{-8}$. Videos of these stills for every iteration can be found in the supplementary material.

Theorems & Definitions (21)

• Remark : Equivalence of OCP to a singularly perturbed equation
• Theorem 2.1: Well-posedness
• proof
• Lemma 3.1: Well-posedness of the FE approximation
• proof
• Theorem 3.2: Galerkin Orthogonality
• proof
• Lemma 3.3: Strang's Lemma
• proof
• Remark : A priori coupling