Assessing the information content of complex flows

Complex dynamical systems can potentially contain a vast amount of information. Accurately assessing how much of this information must be captured to retain the essential physics is a key step for determining appropriate discretization for numerical simulation or measurement resolution for experiments. Using recent mathematical advances, we define spatiotemporally compact objects that we term dynamical linear neighborhoods (DLNs) that reduce the amount of information needed to capture the local dynamics in a well-defined way. By solving a set-cover problem, we show that we can compress the information in a full dynamical system into a smaller set of optimally influential DLNs. We demonstrate our techniques on experimental data from a laboratory quasi-two-dimensional turbulent flow. Our results have implications both for assessments of the fidelity of simulations or experiments and for the compression of large dynamical data sets.

Figure 1

(a) Most influential DLNs as determined by our greedy algorithm for a flow with $\mathrm{Re}=77$. Here, $T=2$ s, corresponding to one eddy turnover time for the largest Reynolds number in our data set ($\mathrm{Re}=288$). (b) Corresponding downsampled velocity field (only 85% of the velocity vectors are shown, for clarity) at the initial time $t_0$. The locations of the center trajectories of the most influential DLNs shown in (a) are marked. (c), (d) The same quantities as in (a) and (b) but for a flow with $\mathrm{Re}=234$.

Figure 2

Probability density functions of the area of individual DLNs for our six different Reynolds numbers. As $\mathrm{Re}$ grows, the typical area of a DLN shrinks.

Figure 3

The number of DLNs required to cover the full domain as a function of the flow Reynolds number. The dashed line is a ${\mathrm{Re}}^{3/2}$ power law.

Figure 4

Quality of a reconstruction of the flow field using a finite set of DLNs containing only 5.7% of the measured trajectories for flow with $\mathrm{Re}=216$. (a) Probability density functions of the velocity magnitude for both the original measurement (solid line) and the reconstruction (dashed line). (b) Energy spectra of the original measurement (solid line) and the reconstruction (dashed line). The vertical line shows the energy injection scale. (c) A section of the original velocity field (downsampled by 50% for clarity). (d) The same section of the reconstructed velocity field with vectors plotted at the same points as in (c).

Figure 5

Comparison of the quality of our DLN-based reconstruction scheme (solid line) with one based on a random sampling of the original velocity field (dashed line; see text for details). The two methods are compared by measuring the mean of the magnitude of the difference between the measured velocity and the reconstructed velocity at every point in the flow field, scaled by the root-mean-square velocity. This error is plotted against the number of trajectories used for the reconstruction, scaled by the total number of measured trajectories. Regardless of the number of trajectories used, the DLN-based scheme always outperforms the random scheme.