Stability preservation in Galerkin-type projection-based model order reduction

Roland Pulch

doi:10.3934/naco.2019003

Article Contents

2019, Volume 9, Issue 1: 23-44. Doi: 10.3934/naco.2019003

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Stability preservation in Galerkin-type projection-based model order reduction

Roland Pulch

Institute of Mathematics and Computer Science, University of Greifswald, Walther-Rathenau-Str. 47, 17489 Greifswald, Germany

Received: November 2017

Revised: May 2018

Published: March 2019

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Abstract

We consider linear dynamical systems consisting of ordinary differential equations with high dimensionality. The aim of model order reduction is to construct an approximating system of a much lower dimension. Therein, the reduced system may be unstable, even though the original system is asymptotically stable. We focus on projection-based model order reduction of Galerkin-type. A transformation of the original system guarantees an asymptotically stable reduced system. This transformation requires the numerical solution of a high-dimensional Lyapunov equation. We specify an approximation of the solution, which allows for an efficient iterative treatment of the Lyapunov equation under a certain assumption. Furthermore, we generalize this strategy to preserve the asymptotic stability of stationary solutions in model order reduction of nonlinear dynamical systems. Numerical results for high-dimensional examples confirm the computational feasibility of the stability-preserving approach.

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Mathematics Subject Classification: Primary: 65L05, 65F10; Secondary: 34C20, 34D20, 93D20.

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Figure 1. Mass-spring-damper configuration

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Figure 2. Bode plot of stochastic Galerkin system for massspring-damper configuration

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Figure 3. Spectral abscissa of the matrices in the ROMs from conventional system (left) and stabilized system (right)

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Figure 4. Error bound in ${\mathscr{H}_2}$-norm for the two MOR approaches in mass-spring-damper example

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Figure 5. Schematic of anemometer

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Figure 6. Bode plot of anemometer benchmark

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Figure 7. Output of the anemometer system

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Figure 8. Spectral abscissa of the matrices in the ROMs from conventional system (left) and stabilized system (right)

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Figure 9. Maximum error of ROMs for the output in the time domain concerning anemometer example

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Figure 10. Error bound in ${\mathscr{H}_2}$-norm for the two MOR approaches in anemometer example

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Figure 11. Condition numbers of reduced mass matrices in anemometer example

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Table 1. Properties of stochastic mass-spring-damper system

dimension n	5440
# non-zero entries in A	25120
# non-zero entries in E	6400
spectral abscissa α(E^-1A)	-0.0048

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Table 2. Properties of anemometer example

dimension n	29008
# non-zero entries in A	201622
# non-zero entries in E	29008
spectral abscissa α(E^-1A)	-146.3

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Table 3. Stability of ROMs for all dimensions $r=1, \ldots, \hat{r}$ using transformation with approximation $M \approx ZZ^\top$ in anemometer example

input rank q_F	number of iterations n_it	maximum dimension $\hat r$
40	10	11
40	28	12
60	10	20
60	20	18

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