POD-based reduced order methods for optimal control problems governed by parametric partial differential equation with varying boundary control

TL;DR

This work tackles reduced-order modeling for varying boundary optimal control problems (vbOCPs) governed by parametric PDEs, where the active boundary segment $\Gamma_C^{\mu_u}$ shifts with the geometric parameter $\mu_u$. Standard POD struggles due to transport-like state features and the non-affine dependence introduced by the boundary control; to address this, the authors develop tailored ROM strategies—Geometric Recasting (Geo-R) and Local POD (L-POD)—and augment POD with DEIM where needed. Numerical experiments on a simple geometry and a complex one with a dolphin-hole domain demonstrate that Geo-R delivers large online speed-ups with modest offline costs when applicable, while L-POD provides robust, high-accuracy reduced models in challenging geometries where Geo-R is not feasible. The results offer practical guidelines for selecting ROM strategies in vbOCPs and point to future work on error certification and 3D extensions.

Abstract

In this work we propose tailored model order reduction for varying boundary optimal control problems governed by parametric partial differential equations. With varying boundary control, we mean that a specific parameter changes where the boundary control acts on the system. This peculiar formulation might benefit from model order reduction. Indeed, fast and reliable simulations of this model can be of utmost usefulness in many applied fields, such as geophysics and energy engineering. However, varying boundary control features very complicated and diversified parametric behaviour for the state and adjoint variables. The state solution, for example, changing the boundary control parameter, might feature transport phenomena. Moreover, the problem loses its affine structure. It is well known that classical model order reduction techniques fail in this setting, both in accuracy and in efficiency. Thus, we propose reduced approaches inspired by the ones used when dealing with wave-like phenomena. Indeed, we compare standard proper orthogonal decomposition with two tailored strategies: geometric recasting and local proper orthogonal decomposition. Geometric recasting solves the optimization system in a reference domain simplifying the problem at hand avoiding hyper-reduction, while local proper orthogonal decomposition builds local bases to increase the accuracy of the reduced solution in very general settings (where geometric recasting is unfeasible). We compare the various approaches on two different numerical experiments based on geometries of increasing complexity.

Figures (12)

• Figure 1: Schematic representation of the computational domain and of the control action of a vbOCP($\boldsymbol \mu$) for different values of $\mu_u$, say $\mu_u = \mu_u^{*}$ (left) and $\mu_u = \overline{\mu_u}$ (right).
• Figure 2: Schematic representation of the computational domain and of the control action of a classical geometrical OCP($\boldsymbol \mu$) under affine transformation for different values of $\mu$, say $\mu = \mu^{*}$ (left) and $\mu = \overline{\mu}$ (right).
• Figure 3: (Test 1). Domain $\Omega$. Observation domain:$\Omega_{\text{obs}} = \Omega_3\cup \Omega_4$, Control domain:$\Gamma_C^{\mu_u}$ (red dashed line). Blue solid line: Dirichlet boundary conditions. Black dotted line: Neumann boundary conditions. We represent the domain for $\mu_u = 0.3$ (left) and $\mu_u = 0.7$ (right).
• Figure 4: (Test 1: POD). Left. Averaged relative log-error for the two variables. Right. Decay of the eigenvalues for the two variables.
• Figure 5: (Test 1: POD). Some high-fidelity solutions for fixed physical parameters ($\mu_1 = 12, \mu_2 = 2.5$) and varying $\mu_u = 0.1, 0.5, 0.99$, from left to right. The state solutions and the adjoint ones are represented on the top and on the bottom of the Figure, respectively.

Theorems & Definitions (2)

• Remark 1
• Remark 2