Projection-free approximate balanced truncation of large unstable systems

In this article, we show that the projection-free, snapshot-based, balanced truncation method can be applied directly to unstable systems. We prove that even for unstable systems, the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the Gramians (including the unstable subspace). We then apply the method to a spatially developing unstable system and show that it results in reduced-order models of similar quality to the ones obtained with existing methods. Due to the unbounded growth of unstable modes, a practical restriction on the final impulse response simulation time appears, which can be adjusted depending on the desired order of the reduced-order model. Recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form. Finally, the method is applied to the linearized flow around a cylinder at Re = 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large-scale systems in practice. The well-established approximate balanced truncation numerical framework therefore can be safely applied to unstable systems without any modifications. Additionally, balanced reduced-order models can readily be obtained even for large systems, where the computational cost of existing methods is prohibitive.

Figure 1

First 10 columns of $|{\tilde{U}}_c^{†}{U}_c|$ for $t_{\infty }=20$ (a) and $t_{\infty }=80$ (b). The dashed line separates the stable and antistable parts of the matrix. The color scale is from 0 (black) to 1 (white).

Figure 2

Top left $(10\times 10)$ elements of $|V_c^{†}V|$ for $t_{\infty }=20$ (a) and $t_{\infty }=80$ (b). The dashed lines separate the stable and antistable parts of the matrix. The color scale is from 0 (black) to 1 (white).

Figure 3

Top left $(10\times 10)$ elements of $|U_c^{†}\widehat{T}|$ for $t_{\infty }=20$ (a) and $t_{\infty }=80$ (b). The dashed lines separate the stable and antistable parts of the matrix. The color scale is from 0 (black) to 1 (white).

Figure 4

Convergence of the balancing transformations $T$ and $S$. (a) Evolution of the energy of the forward impulse response (dashed line) and adjoint impulse response (solid line) states with respect to time. (b) Rate of change of the transformation matrices with respect to the final simulation time $t_{\infty }$ used to compute them for the first $n_x$ columns of $T$ and rows of $S$ where $\mathrm{\Delta }t=0.2$.

Figure 5

Convergence of the balanced mode shapes, shown by plotting $1-|{\widehat{T}}_i{\left(t\right)}^{†}{\widehat{T}}_i(t-\mathrm{\Delta }t)|$ with respect to the final simulation time $t_{\infty }$ used to compute $T$ for the first five normalized balanced modes ${\widehat{T}}_i=T_i{\parallel T_i\parallel }^{-1}$. Here $\mathrm{\Delta }t=0.2$ and unstable modes are shown with thick lines.

Figure 6

Balancing error of transformed Gramians computed from a simulation with final simulation time $t_{\infty }=t_2$ and balanced using transformation matrices calculated from simulations with $t_{\infty }=t_1$. The error is quantified by the top left $(4\times 4)$ (thick), $(8\times 8)$ (thin), and $(12\times 12)$ (dashed) elements of $\parallel I-|V_{\left(11\right)}^{†}{V}_{\left(21\right)}{|\parallel }_F$ for $t_{\infty }=t_1=20$, 40, and 80. The value of $t_1$ from each curve can be identified by the zero error values at $t_2=t_1$.

Figure 7

ROM error as a function of the final simulation time $t_{\infty }$. Solid lines: Snapshot projection method [20, 26]. Lines with circles: Projection-free method. For both methods, the different lines correspond to ROMs with 2, 4, 6, 8, 10, and 12 states from top to bottom.

Figure 8

Estimated optimal final simulation time $t_{\mathrm{opt}}\left(r\right)$ for different ROM orders $r$ (symbols) and linear trend line plotted for $r>2$.

Figure 9

Hankel singular values of ROMs obtained with the exact projection method [34] (crosses), the snapshot-based projection method [20, 26] (filled circles), and the projection-free method (open symbols) obtained with $t_{\infty }=t_{\mathrm{opt}}\left(r\right)$ for ROM orders of 4 (diamonds), 8 (squares), and 12 (open circles) from top to bottom. Note that the unstable singular values are not plotted for the first two methods as they are infinite by definition.

Figure 10

ROM error as a function of the ROM order, obtained with $t_{\infty }=t_{\mathrm{opt}}\left(r\right)$ for the projection-free method. Crosses: Exact projection method [34]. Filled circles: Snapshot-based projection method [20, 26]. Open circles: Projection-free method.

Figure 11

Impulse response (a) and transfer function gain (b) of the full system (thick light-grey dashed line, green online) and the ROMs obtained with the exact projection method [34] (thick dark-grey dashed line, blue online), the snapshot-based projection method [20, 26] (thin dash-dotted line), and the projection-free method (thin solid line, red online). ROM orders: 4, 8, and 12, top to bottom for both the impulse responses and the transfer functions.

Figure 12

Unstable base flow over a cylinder at $\mathrm{Re}=100$, showing streamlines and contours of vorticity. The white circle represents the disk of diameter 1 where the input (a vertical body force) is applied and the black triangle shows the location of the output, a sensor measuring the vertical velocity at (2,0).

Figure 13

Hankel singular values of ROMs obtained with the projection-free method with $t_{\infty }=13$ (open circles), $t_{\infty }=25$ (squares), and $t_{\infty }=50$ (diamonds).

Figure 14

Impulse response of the full cylinder flow (thick dashed line), compared to the 12th-order ROM obtained with the projection-free method (thin solid line). The agreement between the two impulse responses is excellent both for the initial transients shown in (a) and long term unstable response shown in (b).