1 Introduction
Fluid dynamics is a rich and challenging field at the intersection of physics and engineering. There is still a tremendous amount that we do not understand about turbulence, yet working fluids are at the heart of nearly every major industry, including health, defense, transportation, and energy. One cannot overestimate the significance and impact that a better understanding of fluid flows would have in our ability to predict and manipulate their behavior in the real world. The challenge of modeling and controlling fluids stems from the fact that they are nonlinear with complex multiscale interactions over a large range of spatial and temporal scales. Moreover, data from experiments and simulations are generally represented as exceedingly highdimensional measurements that may obscure underlying patterns. With advances in simulation capabilities and measurement techniques, the volume and quality of such data are rapidly increasing.
Despite the complexity of the governing equations and the overwhelming volume of data, it is often observed that flows evolve on a lowdimensional attractor defined by a few dominant coherent structures [1]. There are a wealth of modal decomposition techniques to mine and characterize these structures from experimental data and numerical solvers [2, 3]. The majority of techniques, such as proper orthogonal decomposition (POD)[1, 4] and dynamic mode decomposition [5, 6, 7] are fundamentally linear, as are many of the standard techniques in control theory [8]. Linear control has been widely applied for flow control [9, 10, 11], for example to stabilize boundary layers. However, many flows are fundamentally nonlinear, limiting the broad use of linear control theory. Loworder nonlinear models may be obtained by Galerkin projection of the Navier Stokes equations onto POD modes, and these models have had great success for uncovering underling mechanisms that drive flows [4]. Regardless, there are issues with Galerkin models, such as instability and mode deformation with changing parameters and boundary conditions [12, 13, 14], limiting the success of PODGalerkin modeling for turbulence.
Recently, network theoretic approaches have been increasingly leveraged to analyze complex, fluid flow systems [15, 16, 17, 18, 19, 20, 21]. Network science [22] characterizes the structure and dynamics on a graph consisting of nodes and the edges connecting them. It is possible to cast a fluid flow in this context, where each grid cell of vorticity is a node, and the induced velocity from the BiotSavart interaction between each node establishes the edge connections [15]. In this way, individual packets of vorticity interact and evolve according to physics that is encoded in the graph. Network analysis provides a complementary perspective to classical techniques in fluid dynamics, especially for flow control. There have been many powerful advances in network control theory [22, 23] surrounding multiagent systems [24, 23] in the past two decades. Multiagent control is designed to be local, efficient to operate, and scalable to extremely large systems, such as the internet [25, 26] or the electric grid [27]. The multiagent control and underlying network dynamics may also be strongly nonlinear, and these controllers are built to handle timedelays and communication failures, which are typical limiting factors for robust performance in classical control [8]. In particular, networks are often characterized by a large collection of elements, represented by nodes on a graph, that each execute their own set of local protocols in response to external stimulus. This analogy holds quite well for a number of large networked dynamical systems, including animals flocking [28, 29], multirobotic cooperative control systems [30], sensor networks [31, 32], biological regulatory networks [33, 34], and the internet [25, 26], to name a few. A longterm goal of characterizing vortical networks is the eventual application of multiagent control to school turbulence into a beneficial configuration for an engineering advantage. In contrast, past closedloop flow control efforts have commonly involved applying linear control techniques to a suitable reducedorder model of the fluid [10].
1.1 Motivation
The application of network theory in fluid dynamics faces the challenge of an exceedingly large number of degrees of freedom (nodes) for a fully turbulent flow, leading to an even larger adjacency matrix of edges. Indeed, the adjacency matrix for a vortical flow network scales as the square of the number of fluid grid points, quickly becoming intractable to store and analyze using classical techniques from linear algebra. Recent approaches have been developed to manage this complexity, including building a graph on the
modes of a flow [35], given by dominant fluid coherent structures from POD, rather than the nodes, given by fluid grid elements. However, there is still interest in developing scalable methods to directly analyze the vortical network in the original ambient measurement space, where nodes are grid elements. Randomized methods for linear algebra provide an emerging alternative to efficiently compute an approximate eigendecomposition of largescale matrices. These methods work with a reduced representation, a socalled sketch, of the input matrix that captures the essential spectral information. This sketch is obtained either through random sampling or by random projections. The sketchcan then be used to efficiently compute a desired lowrank approximation, such as the singular value decomposition. For a comprehensive survey and implementation details see
[36, 37, 38, 39, 40].1.2 Contributions
In this work, we use randomized methods to analyze large networks arising in fluid mechanics. Specifically we are interested in using randomized methods to obtain a qualitatively correct view of the dominant graph structures, which can then be used for the downstream tasks of sensor and actuator placement, optimization, and control. This application domain is the subject of the present work, where we explore the use of random sampling and randomized linear algebra to generate qualitatively accurate estimates of dominant flow structures for cases where the adjacency matrix is exceedingly large. We compare both uniform and quasiuniform sampling, and demonstrate this approach on two highdimensional vortical flow networks. A highlevel principle sketch is provided in Figure
1.2 Vortical interaction network
Recently, network analysis has been employed to represent and analyze vortical interactions in flows [15, 17, 19, 35, 41]. In Nair and Taira [15], spectral sparsification was used to identify a lowdimensional skeleton underlying the highdimensional, nonlinear fluid dynamics. Taira et al. [17] later demonstrated scalefree behavior of the vortex interaction network for twodimensional isotropic turbulence. Network analysis has also been used by Meena et al. [19] for community detection and force prediction in an unsteady wake. More recently, interaction networks based on the energy transfer between modes has been used for modeling and control [35, 41]. These studies provide compelling evidence that the emerging network perspective may complement wellestablished flow modeling techniques.
A graph (network) consists of three components: a set of vertices (nodes) , a set of edges that connect these vertices, and the associated edge weights . In the present work, the vortical elements in a flow field represent the nodes of the vortical flow network, following the formulation of Nair and Taira [15]. The interaction among vortical elements can be quantified by the induced velocity. Following the BiotSavart law, the induced velocity from element to in a twodimensional flow is given by
(1) 
where is the strength of vortical element with vorticity and grid area , and is the Euclidean distance between the vortical elements. This induced velocity represents the edge weights, and may be used to construct an adjacency matrix
(2) 
which mathematically represents the interactions in the vortical network. Since a vortex cannot induce motion upon itself, there are no diagonal entries for the adjacency matrix. Here we have an undirected network with a symmetric adjacency matrix to enable the use of the algorithm described later. However, it is possible to define an asymmetric adjacency matrix for a directed network.
Given the vorticity field of a flow, the corresponding vortical interaction network can be formulated as above. This also suggests that the size of the adjacency matrix scales as where is the number of grid points or vortical elements in the flow field. For a highfidelity simulation or experimental measurements of a flow field, can exceed , making the computation and analysis of the vortical network computationally expensive. Below, we discuss a few network measures that are particularly useful for understanding the interactionbased characteristics of the flow. The current work focuses on computing these measures for large, dense vortical networks in a computationally tractable manner.
2.1 Network measures and computations
The adjacency matrix is the basis for a number of useful network measures, such as eigenvector centrality, Katz centrality, and PageRank [22, 42]. The centrality of a network describes the most important or central elements based on measures that quantify the connection strength of the nodes. One of the most basic centrality measures is the node strength, given by , which measures the overall interaction strength of a node. The eigenvector centrality is another fundamental network property, which also considers the strength of the neighboring elements of a node and is used to identify the most influential nodes in the network [43]. It is computed as the eigenvector of corresponding to the largest eigenvalue :
(3) 
Entries with large absolute value in the vector
correspond to influential nodes. The Katz centrality and PageRank are related and also based on computing the dominant eigenvector of the adjacency matrix.Network analysis is also useful for graph partitioning and clustering [22], where the nodes in the network are grouped into clusters based on their close connectivity. Spectral partitioning or clustering is a common approach based on the dominant eigenvector of the adjacency matrix. Nodes can be divided and subdivided into groups based on the sign of the corresponding elements in the dominant eigenvectors of . Spectral partitioning can also be used in community detection algorithms to identify groups of closely interacting nodes, called communities [44]. These algorithms perform a hierarchical grouping of nodes into communities based on maximizing the overall modular measure of the network. This hierarchical grouping can be performed using the dominant eigenvectors of [45]. In vortical flows, these networkbased measures are useful for identifying the most influential vortical elements and clustering them into coherent structures or communities.
Due to the importance of the spectral information of the adjacency matrix, considerable effort has gone into developing accurate and efficient algorithms to compute dominant eigenvalues for large adjacency matrices. Provided that the largest eigenvalue is real and distinct, the dominant eigenvector can be efficiently computed using the power method [46]. A symmetric adjacency matrix with nonnegative entries will have purely real eigenvalues. For instance, the power method is commonly used to compute the dominant eigenvector of an adjacency matrix [47], and this method is particularly efficient for large sparse adjacency matrices. The computational costs of the power method for computing the dominant eigenvector scales as .
3 Random samplingbased matrix approximation
As stated above, we are interested in computing the eigendecomposition of a real symmetric matrix that takes the form
(4) 
where the columns of the orthogonal matrix
are eigenvectors and the entries of the diagonal matrix are the corresponding eigenvalues . More concretely, we are interested in computing only the dominant eigenvectors and eigenvalues (where ) that yield the rank approximation(5) 
where contains only the columns of that correspond to the largest eigenvalues. Here, we use tools from the field of randomized numerical linear algebra [48, 49, 36, 50, 51, 52, 40, 53, 38, 39, 54] to compute such a lowrank approximation efficiently. Indeed, randomized methods are widely used for computing lowrank approximations [55, 56, 57, 58, 59, 60, 61].
Scalability is achieved by forming a sketched representation of the input (adjacency) matrix which extracts the inherent spectral information. A sketch can be constructed by postmultiplying the adjacency matrix with a an arbitrary ‘sketching’ matrix
. The random matrix
represents either a random projection or a random sampling process.
Projectionbased methods construct a ‘sketch’ by forming a set of randomly weighted linear combinations of the columns of the input (adjacency) matrix [62, 39, 48, 49]. This approach can substantially reduce the computational demands when computing a lowrank approximation. However, projectionbased methods generally require at least a single pass over the entire data matrix.

Samplingbased methods aim to approximate the lowrank structure of the input (adjacency) matrix from a small random subsample of actual columns or rows of the matrix [63, 64, 65]. The sampling process is described in more detail below. In practice, samplingbased methods may bypass the construction of the full adjacency matrix, while sacrificing some accuracy for improved scalability.
In what follows, we generate qualitatively accurate estimates of dominant flow structures for cases where the adjacency matrix is exceedingly large. In this problem context, we require a qualitatively correct view of the dominant graph structures, which can be used for the downstream tasks of sensor and actuator placement, optimization, and control. Therefore, we investigate the use of column sampling to compute a sketched singular value decomposition and the Nyström Method to approximate the leading eigenvalues and eigenvectors. Both techniques have been successfully used to approximate largescale matrices in other applications [66]. It is important to note that samplingbased methods provide tunable error bounds that are based on the number of sampled columns/rows, making them viable as an alternative to wellestablished deterministic algorithms.
3.1 Sketched singular value decomposition
Column sampling for lowrank matrix approximation dates back to the pioneering work of Frieze et al. [62]. Instead of computing the full eigendecomposition of , it is possible to form a rank approximation by first sampling columns from the input matrix
(6) 
where the matrix consists of a subset of columns of and describes the corresponding random sampling process. In practice, it is only necessary to form a list comprised of column indices
(7) 
There are several sampling strategies to choose good columns, and the approximation error depends on the both the sampling process and the number of columns sampled.
In our problem setup, each column of corresponds to the network interaction of a single grid element in the original fluid flow field with every other grid element. Here, we compare uniform random sampling against (quasirandom) Halton sampling [67, 68]. Halton sampling provides better coverage of the domain, as illustrated in Figure 2, and our empirical results show that this translates into faster convergence of the approximation error with respect to the number of columns sampled.
If the columns of are carefully chosen, then provides a basis for the column space of the adjacency matrix . Note that here we assume that is a symmetric matrix. Thus, the dominant left singular vectors of , provide an approximation for the dominant eigenvectors of , i.e., we have . The corresponding eigenvalues are approximated by scaling the singular values of :
(8) 
The cost of constructing the sketch matrix using uniform or Halton sampling is . Then, the cost of computing the SVD on is . For parallel computations, it is possible to compute and by taking the SVD of , which requires to be generated and for the corresponding SVD.
3.2 The Nyström method
The Nyström method provides an efficient approach for lowrank matrix approximations in largescale learning applications. It was initially introduced as a quadrature method for numerical integration to approximate eigenfunction solutions. Recent work has streamlined these computations and provided theoretical foundations for the use of various sampling schemes
[69, 70, 71, 72].In addition to column sampling, the Nyström method further constructs a square matrix by selecting the rows and columns of :
(9) 
When is symmetric and positivesemidefinite (SPSD), is also SPSD. The small matrix can be used to efficiently compute the dominant eigenvectors and eigenvalues of . Following [73], we first compute the eigendecomposition:
(10) 
Then, we reconstruct the dominant eigenvalues as
(11) 
and the corresponding eigenvectors
(12) 
The Nyström method has a computational complexity of .
4 Results
We now demonstrate the use of samplingbased randomized decomposition techniques to characterize the vortical interaction networks for two example flows: the laminar wake flow past a NACA 0012 airfoil with a Gurney flap attached to the trailing edge [74], and a twodimensional homogeneous decaying isotropic turbulence [17].
4.1 Numerical simulations
The first example is the flow past a NACA 0012 airfoil with a chordbased Reynolds number of , at an angle of attack of , and with a Gurney flap of chord length (). This setup generates an unsteady 2P wake [75, 74], with periodic shedding of two pairs of positive and negative vortices. This flow is solved using direct numerical simulations (DNS) via the immersed boundary projection method [76, 77], following the computational setup of Gopalakrishnan Meena et al. [74]. The computational domain consists of five nested levels of multidomains with the finest level of size and the largest being in size. All the domains have a grid resolution of . The time step is limited to a maximum CourantFriedrichsLewy (CFL) number of . Figure 3 (a) shows the vorticity field in a subdomian with a grid resolution of and size , nondimensionalized by the chord length of the airfoil.
The second example is twodimensional decaying homogeneous isotropic turbulence, which is a canonical highdimensional, mutiscale turbulent flow that exhibits complex nonlinear interaction of vortices over a wide range of length scales [78]. The vorticity field is obtained from a twodimensional incompressible DNS, solving the vorticity transport equation without forcing [17]. The simulation, based on the Fourier spectral method and the fourthorder RungeKutta time integration scheme, is performed on a square biperiodic computational domain with grid resolution of . The vorticity field is initialized with vortices with random strengths, core sizes, and centers such that the kinetic energy spectra satisfies with [79]. For this study, the flow field is initialized by an integral lengthscale based Reynolds number of , shown in Figure 3 (b).
The vorticity snapshots of both flows are used to construct the adjacency matrices of the corresponding vortical networks using Eq. 2. The size of the adjacency matrices are of the order and for the airfoil and turbulent flow, respectively, emphasizing the need for dimensionality reduction and computationally tractable analysis tools.
4.2 Computational performance
A naive application of network theory in fluid dynamics can be challenging, because the adjacency matrix is often too large to be stored or analyzed using classical techniques from linear algebra. There are two primary challenges associated with large vortical networks:

The size of the adjacency matrix, which describes the network edges, scales as the square of the number of fluid grid points;

Unlike social graphs, the vortical adjacency matrix is dense.


Here, we use randomized methods to accelerate computations based on the vortical adjacency matrix. We sample a small number of columns from the adjacency matrix to compute a sketched SVD or Nyström approximation. Thus, the approximation quality can be controlled via the number of samples used to compute the lowrank approximation. In the context of networks arising in fluid mechanics, it turns out that sampling a small number of columns is sufficient for computing an approximation that is qualitatively useful. This leads to tremendous memory and computational savings, since the costs are proportional to the number of columns that we sample. We run all of our experiments using Amazon AWS, working on a memory optimized instance ‘x1.16xlarge’ that is powered by two Intel Xeon E7 8880 v3 (Haswell) processors with virtual cores and GB fast memory. Our algorithms are implemented in Python powered by MKL accelerated linear algebra routines, and C extensions for constructing the elements of the adjacency matrix.
We show in Figure 4 the approximation error as a function of the percentage of columns sampled and the corresponding computational time required.
Here, we consider two cropped spatial grids of different dimensions from which we construct the adjacency matrix. The error is reported as the acute angle between the true leading eigenvector and the approximate eigenvector , which provides a useful summary of how closely the two highdimensional eigenvectors align. Since the algorithms are fundamentally based on random sampling, we average the results over initializations and report the distribution of errors.
Halton sampling results in considerably lower error than uniform sampling at the same percentage of columns sampled, which is consistent with the observation that it is sampling the flow domain more efficiently. The results based on Halton sampling also have considerably lower variance, indicating improved numerical robustness. Figures
3(a) and 3(b) illustrate how substantially the computational time increases when the spatial resolution of the flow field is increased only slightly. In the case, only about of the columns need to be sampled for error compared between the approximate and true eigenvectors. In other words, we use only about of the information to compute the lowrank approximation, and this reduces the memory requirements for the example in Figure 3(b) from about GB to roughly GB. At the same time, the computational time is reduced since it is not necessary to construct the full adjacency matrix. The computational time is reduced further by using the Nyström method, while only sacrificing a small amount of accuracy. For comparison, Table 1 summarizes the computational demands and performance for the deterministic power iteration method. Figure 5 shows similar results for the turbulent flow. Again, we see that Halton sampling is favorable and that only a small fraction of columns is required to yield a good approximation of the leading eigenvector.







airfoil  (sec)  74.29 (GB)  (sec)  
airfoil  (sec)  214.67 (GB)  (sec)  
turbulent  (sec)  32.00 (GB)  (sec)  
turbulent  (sec)  101.92 (GB)  (sec) 
4.3 Approximate eigenvectors and spectral clustering
Figure 6 shows the leading eigenvectors of the adjacency matrix for the two flow examples, computed using deterministic and randomized methods. By visual inspection, the Nyström method with quasirandom Halton sampling provides an excellent approximation of the leading eigenvector using only of the columns of the adjacency matrix. The resulting eigenvector centrality of the matrix highlights strong vortex cores in both flows, as shown in Figure 6 (b). Vortices with lower strength have less centrality, emphasizing their decreased role in influencing the flow field. The 2P wake structure is clearly visible in the flow past an airfoil, and some shear layer structures are present in the 2D turbulent flow. The irrotational regions in both flows have the lowest centrality measure. These observations complement the traditional fluid dynamics literature regarding highly influential structures in a vortical flow. The approximate eigenvector centrality measures capture both the highly influential vortex cores as well as the smaller, less influential vortices, as seen in both flows. Moreover, the networkbased measures take into account the spatial arrangements of the vortices to assess their influence.
In the computational comparison above, we only considered relatively low resolution flow fields because it was computationally prohibitive to compute the ground truth eigenvectors using deterministic methods. However, with randomized techniques it is possible to compute the leading eigenvector of the adjacency matrix for a resolution grid and a resolution grid, respectively, as shown in Figure 7. The adjacency matrix for the grid would require about TB of storage, if explicitly constructed. Instead, we sample only about of the columns to compute the Nyström approximation. While we cannot quantify the approximation error, we can see by visual inspection that the eigenvector captures dominant flow structures.
Finally, we investigate spectral clustering, which is one of the common uses of the leading eigenvectors of the adjacency matrix, providing a principled approach to cluster nodes into common communities. Figure
8 shows the results of spectral clustering based on the three leading eigenvectors obtained from the adjacency matrix. In both flow fields, the approximate clusters obtained using randomized methods closely resemble the clusters computed via deterministic eigendecomposition. The right panels in Figure 8 further shows the data plotted in these first three eigenvector coordinates to visualize the clusters and subspaces. Typically, these leading eigenvectors are quite useful for determining relevant community structures within a network. These communities have an important physical interpretation in vortical flow networks, hierarchically identifying and organizing vortex cores within the flow. Because these communities are often determined from the few dominant eigenvectors, the randomized approach here can provide considerable computational advantages. Further, since highvorticity nodes have a large influence on the network interactions, it may be possible to improve convergence by preferential sampling using these community structures.5 Discussion
In this work, we have demonstrated the effective use of scalable algorithms from randomized linear algebra to accelerate network computations for largescale fluid flows, as demonstrated on the wake behind an airfoil and twodimensional isotropic turbulent flow. In particular, we have used samplingbased methods to approximate the leading eigenvectors of the adjacency matrix of the vortical interaction network for cases where the data is so large that even singlepass algorithms are prohibitively expensive. Combining importance sampling, based on the probability distribution of detected communities, and the Nyström method, the leading eigenvector can be accurately computed at a fraction of the computational cost and memory requirement, bypassing the need to construct or query the full
matrix. We also showed that quasirandom Halton sampling techniques outperform uniform sampling in both examples as they provide more comprehensive sampling coverage of the spatial domain.There are a number of interesting future directions based on this work. Further study is required to apply these randomized network based analysis techniques to more complex threedimensional turbulent flows. In addition, there are similar scaling issues in the related fields of almost invariant sets and setoriented methods [80, 81, 82, 83, 84]
, where the statetransition operator may be viewed as a graph on the highdimensional statespace. In the transfer operator case, machine learning has been used to identify a datadriven discretization of the statespace into clusters, which serve as nodes that scale with the intrinsic rank of the attractor, rather than the highdimensional measurement dimension
[85]; similar clustering has been used to identify subspaces for POD–Galerkin models [86]. These methods have been used to determine eigenvectors of the Perron–Frobenius operator, the dual of the Koopman operator, establishing a strong connection between sensitivity and coherence in fluid flows. It will be interesting to mathematically connect those approaches with the randomized methods considered in the present study.Acknowledgments
We acknowledge support from the Army Research Office (W911NF1710118) and the Air Force Office of Scientific Research (FA95501610650). ZB also acknowledges support by the U.S. Department of Energy, Office of Science, under Award Number DEAC0205CH11231. NBE would also like to acknowledge Amazon Web Services for supporting the project with EC2 credits. We would like to thank Bing Brunton, Nathan Kutz, JeanChristophe Loiseau, Krithika Manohar, Aditya Nair, and Bernd Noack for valuable discussions.
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