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

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  • 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.

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


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  • Figure 1.  Mass-spring-damper configuration

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

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

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

    Figure 5.  Schematic of anemometer

    Figure 6.  Bode plot of anemometer benchmark

    Figure 7.  Output of the anemometer system

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

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

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

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

    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
     | Show Table
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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
     | Show Table
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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 qF number of iterations nit maximum dimension $\hat r$
    40 10 11
    40 28 12
    60 10 20
    60 20 18
     | Show Table
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