State-based nested iteration solution of optimal control problems with PDE constraints

TL;DR

The paper develops an abstract, operator-based framework for PDE-constrained OCPs with tracking objectives, encompassing elliptic, parabolic, and hyperbolic PDEs and both $L^2$ and energy regularizations. It introduces space-time finite element discretizations, derives regularization- and discretization-error estimates, and establishes an optimal balance between the regularization parameter $\rho$ and mesh size $h$, e.g., $\rho=h^{2}$ for $L^2$-regularization and $\rho=h^{4}$ for energy regularization in appropriate settings. A key contribution is the state-based Schur-complement formulation solved by robust pcg preconditioned with lumped mass matrices, enabling asymptotically optimal complexity within a nested iteration framework over adaptively refined meshes; this is complemented by variational-inequality handling for state/control constraints via semi-smooth Newton methods. The framework is then specialized to distributed Poisson control and demonstrated through extensive 1D, 2D, and 3D numerical experiments, validating theoretical error bounds and showing substantial computational speedups from nesting while preserving accuracy. The authors also discuss extensions to Dirichlet boundary control, parabolic and hyperbolic state equations, and potential future work on nonlinear PDEs and more general constraints, highlighting the practical impact for scalable PDE-constrained optimization in engineering and physics.

Abstract

We consider an abstract framework for the numerical solution of optimal control problems (OCPs) subject to partial differential equations (PDEs). Examples include not only the distributed control of elliptic PDEs such as the Poisson equation discussed in this paper in detail but also parabolic and hyperbolic equations. The approach covers the standard $L^2$ setting as well as the more recent energy regularization, also including state and control constraints. We discretize OCPs subject to parabolic or hyperbolic PDEs by means of space-time finite elements similar as in the elliptic case. We discuss regularization and finite element error estimates, and derive an optimal relation between the regularization parameter and the finite element mesh size in order to balance the accuracy, and the energy costs for the corresponding control. Finally, we also discuss the efficient solution of the resulting systems of algebraic equations, and their use in a state-based nested iteration procedure that allows us to compute finite element approximations to the state and the control in asymptotically optimal complexity. The numerical results illustrate the theoretical findings quantitatively.

Figures (8)

• Figure 1: Targets $\overline{y}_i$, state solutions $y_{i,\varrho_{L^2}}$ and $y_{i,\varrho_{H^{-1}}}$, and errors $\| y_{i,\varrho} - \overline{y}_i \|_{L^2(\Omega)}$ for different choices of regularization parameters and regularization norms.
• Figure 2: Error and different types of regularization with $\varrho = \varrho_{H^{-1}}=\varrho_{L^2}$.
• Figure 3: Meshes and dual meshes in 1D (left) and 2D (right).
• Figure 4: Targets $\overline{y}_1$, $\overline{y}_3$, (exact) reconstructed states using the $H^{-1}$ and $L^2$ regularization $y_{i,\varrho}$ and $y_{i,\varrho h}$, respectively. And reconstruction of the controls $\widetilde{u}_{i,\varrho h}$ on the primal and dual mesh.
• Figure 5: Errors $\|\overline{y}-y_\varrho\|_{L^2(\Omega)}$ and $\|\overline{y}-y_{\varrho h}\|_{L^2(\Omega)}$ for the different targets $\overline{y}_1$ and $\overline{y}_3$ and for the $H^{-1}$ and the $L^2$ regularization.

Theorems & Definitions (23)

• Lemma 1
• proof
• Corollary 1
• Lemma 2
• proof
• Theorem 1
• Lemma 3
• proof
• Theorem 2
• Lemma 4