A simplified Newton method to generate snapshots for POD models of semilinear optimal control problems

TL;DR

This paper introduces a simplified Newton method to efficiently generate high-quality POD models for semilinear PDE control problems, improving approximation accuracy with minimal additional computational effort.

Contribution

It proposes a novel simplified Newton approach to enhance POD model quality for PDE-constrained optimization, especially in MPC contexts, with an additional Newton step extending impulse response snapshots.

Findings

Adding a second Newton step significantly improves POD model accuracy.

The method introduces moderate extra computational costs during optimization.

Illustrative example confirms the effectiveness of the approach.

Abstract

In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however, and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context. In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.