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From the previous chapter it is clear that a single POD-based ROM is generally reliable in a restricted zone around the point it is constructed (root-point), and it needs to be updated as the optimization proceeds from the root-point to other points in the decision space. To converge to the optimum we can a) solve the ROM-based optimization problem within tight bounds, b) take a step and construct a new ROM by generating new snapshots at this new point, and c) repeat the process until the optimum is achieved. Construction and updation of ROM should be coupled with the progress in the optimization process. Here, the computational advantage comes from the fact that the ROM is used for optimization instead of the detailed rigorous model. In order to guarantee convergence of the above process, we additionally embed it in a trust-region framework. Trust-region methods [54, 61] are suitable and quite appropriate for ROM-based optimiza- tion since they not only ensure that the step computed by the optimizer using ROM stays reasonably close to the root-point (as demonstrated in section 6.5.4.1), but also allow ROM updation decisions based on the information obtained during the optimization procedure, and provide a robust and globally convergent framework with ROMs. With such an adaptive framework we can regulate the amount of optimization done with a ROM before we return to the detailed model to update it. By comparing the improvement predicted by the ROM to the improvement realized for the true system being optimized, we not only deduce how well ROM is predicting the behavior of the system, but also decide if it should be updated or re-used during the course of the optimization. The idea of using a trust-region framework to manage approximate models in optimization was first introduced by Alexandrov et al. [8]. However, they developed their generic framework only for unconstrained optimization problems. They use a basic trust-region algorithm [54] which involves solving the ROM-based problem in a trust-region, taking a step if the reduction in the original objective function is reasonable compared to the one predicted by ROM-based problem, and repeating this until convergence. In order to ensure convergence to the correct 7.1 Introduction 7.1 Introduction Chapter 7. Trust-region Framework for ROM-based Optimization 135PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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