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A Simplified Newton Method to Generate Snapshots for POD Models of Semilinear Optimal Control Problems

Manns, Paul ; Ulbrich, Stefan (2022)
A Simplified Newton Method to Generate Snapshots for POD Models of Semilinear Optimal Control Problems.
In: SIAM Journal on Numerical Analysis, 60 (5)
doi: 10.1137/21M1439821
Article, Bibliographie

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.

Item Type: Article
Erschienen: 2022
Creators: Manns, Paul ; Ulbrich, Stefan
Type of entry: Bibliographie
Title: A Simplified Newton Method to Generate Snapshots for POD Models of Semilinear Optimal Control Problems
Language: English
Date: 2022
Place of Publication: Philadelphia
Publisher: SIAM
Journal or Publication Title: SIAM Journal on Numerical Analysis
Volume of the journal: 60
Issue Number: 5
Collation: 27 Seiten
DOI: 10.1137/21M1439821
URL / URN: https://epubs.siam.org/doi/10.1137/21M1439821
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.

Divisions: DFG-Collaborative Research Centres (incl. Transregio)
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting Processes
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting Processes > Research Area B: Modeling and Simulation
DFG-Collaborative Research Centres (incl. Transregio) > Collaborative Research Centres > CRC 1194: Interaction between Transport and Wetting Processes > Research Area B: Modeling and Simulation > B04: Simulation based Optimization and Optimal Design of Experiments for Wetting Phenomena
04 Department of Mathematics
04 Department of Mathematics > Optimization
04 Department of Mathematics > Optimization > Nonlinear Optimization
Date Deposited: 07 Dec 2023 12:54
Last Modified: 07 Dec 2023 12:54
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