POD Suboptimal Control of Evolution Problems: Theory and Applications

TL;DR

The work addresses efficiently solving PDE-constrained optimal control problems by deploying Proper Orthogonal Decomposition (POD) to derive reduced-order models. It develops both discrete and continuous POD frameworks, connects POD to the singular value decomposition through the snapshot operator, and analyzes perturbations, projection operators, and error bounds within embedded Hilbert spaces (Gelfand triples). A POD-Galerkin scheme for linear evolution problems is presented, with comprehensive a-priori and a-posteriori error analyses, and the methodology is extended to quadratic programming with basis-update and to nonlinear/state-constrained/multiobjective control in later sections. The results demonstrate that a small number of POD modes can capture the essential dynamics of PDEs, enabling fast, accurate PDE-constrained optimization with rigorous error guarantees and practical guidance for snapshot design and basis selection.

Abstract

The work is organized as follows. First an introduction is given in Chapter 1. In Chapter 2 we introduce the POD method in finite and infinite-dimensional Hilbert spaces and discuss various applications. Chapter 3 is devoted to to POD-based Galerkin schemes for evolution problems. Mainly, we study linear problems taking different discretization methods into account. We provide a certified a-priori and a-posteriori error analysis. Furthermore, the numerical realizations are explained and illustrated by test examples. Quadratic programming problems governed by liner evolution problems are investigated in Chapter 4. As in Chapter 3 we present a certified a-priori and a-posteriori error analysis. Moreover, we discuss basis update strategies. In Chapter 5 we give an outlook to further directions in reduced-order modeling in optimal control and optimization. More precisely, a nonlinear optimal control problem is studied. Moreover, state-constrained optimization problems are solved by a tailored combination of primal-dual active set methods and POD-aesed reduced-order modeling. Furthermore, POD Galerkin methods for multiobjective optimal control problems are investigated. Finally, some required theoretical foundations are summarized in the appendix.

Figures (29)

• Figure 1.3.1: Numerical setup in Example \ref{['ex:generalNumericSetup']}.
• Figure 1.3.2: Run \ref{['ex:generalNumericSetup']}. Time-average temperature evolution of the solutions to \ref{['SIAM:EqMotPDE1']} for the fixed control inputs $u_1$, $u_2$, and $u_3$.
• Figure 1.3.3: Run \ref{['ex:generalNumericSetup']}. Solution snapshots of the solutions to \ref{['SIAM:EqMotPDE1']} for the fixed control inputs $u_1$, $u_2$, and $u_3$.
• Figure 2.1.4: Example \ref{['Example:cosExample']}. Function $y_1(t,{\boldsymbol x})=\cos(t)\cos({\boldsymbol x})$ (left plot); projection of $y_1$ onto the POD space $V^\ell=\mathop{\mathrm{span}}\limits\{\hat{\psi}_1^n,\ldots,\hat{\psi}_\ell^n\}$ for $\ell=1$ (right plot).
• Figure 2.1.5: Example \ref{['Example:cosExample']}. Function $y_2(t,{\boldsymbol x})=\cos(t+{\boldsymbol x})$ (left plot); projection of $y_2$ onto the POD space $V^\ell=\mathop{\mathrm{span}}\limits\{\hat{\psi}_1^n,\ldots,\hat{\psi}_\ell^n\}$ for $\ell=1$ (middle plot) and $\ell=2$ (right plot).

Theorems & Definitions (254)

• Remark 1.2.1
• Definition 2.1.3
• Remark 2.1.4
• Remark 2.1.5
• Example 2.1.6: Three simple functions
• Example 2.1.7: Evolution problem
• Example 2.1.8: Elliptic problem
• Remark 2.1.9
• Lemma 2.1.10
• Remark 2.1.11