Fig 1.
Identification of dominant eigenvalues and eigenvectors of the adjacency matrix obtained from a fluid flow field using randomized methods.
Path I randomly samples columns of the adjacency matrix to approximate the leading eigenvalues and eigenvectors; path II uses both column and row sampling to form a sketch. Note, it is not required to explicitly construct the full adjacency matrix.
Fig 2.
Examples of two different random sampling approaches based on uniform sampling and Halton random sampling.
Halton sampling provides a better coverage of the domain than uniform sampling does.
Fig 3.
(a) Vorticity field of two-dimensional DNS of the flow over a NACA 0012 airfoil with Gurney flap at an angle of attack of 20° and flap height of 0.1 chord length at Re = 1000; the full illustration can be found in [74]. A subdomain of the original vorticity field is used in this example. (b) Vorticity field of two-dimensional decaying homogeneous isotropic turbulence, initialized by an integral length-scale based Reynolds number of Re(t0) = 814.
Fig 4.
Computational performance for the flow over an airfoil using two different spatial resolutions.
The results are averaged over 20 runs using different random seeds. The shaded region indicates the variance of approximation error in multiple runs.
Fig 5.
Computational performance for the isotropic turbulent flow using two different spatial resolutions.
The results are averaged over 20 runs using different random seeds. The shaded region indicates the variance of approximation error in multiple runs.
Table 1.
Summary of computational results for constructing the full adjacency matrix and computing the dominant eigenvector using the deterministic power method.
Here the computational bottleneck is the memory required to construct the adjacency matrix. Note that the power method does not generally require that the full adjacency matrix is constructed explicitly; however, the power method is not efficient because the adjacency matrices we consider are dense.
Fig 6.
Qualitative comparison of the deterministic (b) and approximate (c) dominant eigenvector of the adjacency matrix generated from the flow field in (a).
The top row shows the results for the airfoil wake and the bottom shows the results for the two-dimensional isotropic turbulence. The approximation in (c) uses the Nyström method with Halton sampling with 10% of the columns of the adjacency matrix.
Fig 7.
Relative approximation error between the deterministic and approximate dominant eigenvector as in Fig 6, distributed in the spatial domain: (a) airfoil wake flow, (b) two-dimensional isotropic turbulent flow.
Fig 8.
Approximated leading eigenvectors of the adjacency matrices for higher-resolution isotropic flow fields.
Here, we are using the Nyström method with Halton sampling. By visual inspection we can see that sampling only about 10% of the columns of the full adjacency matrix is sufficient for computing an approximated eigenvector that captures the dominant graph structure.
Fig 9.
Spectral clustering using the top three eigenvectors of the adjacency matrix for the airfoil wake flow.
In (a) we show the clusters that we found using the exact eigenvectors. It can be seen that the approximate eigenvectors allow us to reveal the same seven clusters, as shown in (b). In (c) and (d) we show the first three approximated eigenvector coordinates to visualize the clusters and subspaces.