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
Artikel, Bibliographie
Kurzbeschreibung (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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2022 |
| Autor(en): | Manns, Paul ; Ulbrich, Stefan |
| Art des Eintrags: | Bibliographie |
| Titel: | A Simplified Newton Method to Generate Snapshots for POD Models of Semilinear Optimal Control Problems |
| Sprache: | Englisch |
| Publikationsjahr: | 2022 |
| Ort: | Philadelphia |
| Verlag: | SIAM |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | SIAM Journal on Numerical Analysis |
| Jahrgang/Volume einer Zeitschrift: | 60 |
| (Heft-)Nummer: | 5 |
| Kollation: | 27 Seiten |
| DOI: | 10.1137/21M1439821 |
| URL / URN: | https://epubs.siam.org/doi/10.1137/21M1439821 |
| Kurzbeschreibung (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. |
| Fachbereich(e)/-gebiet(e): | DFG-Sonderforschungsbereiche (inkl. Transregio) DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen > Projektbereich B: Modellierung und Simulation DFG-Sonderforschungsbereiche (inkl. Transregio) > Sonderforschungsbereiche > SFB 1194: Wechselseitige Beeinflussung von Transport- und Benetzungsvorgängen > Projektbereich B: Modellierung und Simulation > B04: Simulationsbasierte Optimierung und Optimales Design von Experimenten für Benetzungsvorgänge 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Optimierung 04 Fachbereich Mathematik > Optimierung > Nonlinear Optimization |
| Hinterlegungsdatum: | 07 Dez 2023 12:54 |
| Letzte Änderung: | 07 Dez 2023 12:54 |
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