classification of spatiotemporal data via asynchronous ...lin491/pub/glin-16-siam-mms.pdf · 824...

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MULTISCALE MODEL. SIMUL. c© 2016 Society for Industrial and Applied MathematicsVol. 14, No. 2, pp. 823–838

CLASSIFICATION OF SPATIOTEMPORAL DATA VIAASYNCHRONOUS SPARSE SAMPLING: APPLICATION TO FLOW

AROUND A CYLINDER∗

IDO BRIGHT† , GUANG LIN‡ , AND J. NATHAN KUTZ†

Abstract. We present a novel method for the classification and reconstruction of time-depen-dent, high-dimensional data using sparse measurements, and we apply it to the flow around a cylin-der. Assuming the data lies near a low-dimensional manifold (low-rank dynamics) in space and hasperiodic time dependency with a sparse number of Fourier modes, we employ compressive sensingfor accurately classifying the dynamical regime. We further show that we can reconstruct the fullspatiotemporal behavior with these limited measurements, extending previous results of compressivesensing that apply for only a single snapshot of data. The method can be used for building improvedreduced-order models and designing sampling/measurement strategies that leverage time asynchrony.

Key words. dynamical systems, bifurcations, classification, compressive sensing, sparse repre-sentation, proper orthogonal decomposition

AMS subject classifications. 37E99, 37G99, 37L65

DOI. 10.1137/15M1023609

1. Introduction. Data-driven methods, often rooted in new innovations of ma-chine learning [27, 13, 46], are becoming transformative tools in the study of complexdynamical systems [41]. Such methods aim to take advantage of the ubiquitous ob-servation across the physical, engineering, and biological sciences that meaningfulinput/output of macroscopic variables are encoded in low-dimensional patterns of dy-namic activity [21, 36, 56, 55, 42, 33, 61]. Reduced-order models (ROMs) capitalizeon this observation with the goal of producing real-time computational schemes formodeling complex systems. (See the recent edited monograph addressing the state ofthe art in ROMs for computational modeling of complex parametrized systems [54].)Complementing these efforts are new theoretical efforts using sparsity promoting tech-niques [16, 17, 18, 25, 32, 9, 10, 8] which exploit this fact in order to deal with thepractical constraints of a limited number of sensors for data acquisition in the complexsystem [14, 15, 58, 64, 30, 49]. Such findings lead us to conjecture that a principledapproach to sensing in such complex systems will allow us to probe the latent, low-dimensional structure of dynamic activity relating to the fundamental quantities ofinterest. A direct implication of such a data-driven modeling strategy is that we cangain traction on prediction, state estimation, and control of the complex system. Tothis end, we demonstrate that asynchronous sparse measurements in both time andspace allow for a robust method, i.e., a spatiotemporal compressive sensing method,for classifying the dynamical behavior of a given system. Once classified, it can be usedfor full state reconstruction and/or future state prediction of the full spatiotemporalsystem, extending previous works [14, 15, 58] which only reconstructed single timesnapshots of the data. Moreover, the method presented provides a spatiotemporal

∗Received by the editors May 29, 2015; accepted for publication (in revised form) March 3, 2016;published electronically May 12, 2016.

http://www.siam.org/journals/mms/14-2/M102360.html†Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925

([email protected], [email protected]). The work of the third author was supported by the Air ForceOffice of Scientific Research (grant FA9550-15-1-0385).‡Department of Mathematics and School of Mechanical Engineering, Purdue University, West

Lafayette, IN 47907 ([email protected]).

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http://www.siam.org/journals/mms/14-2/M102360.html

mailto:[email protected]




824 IDO BRIGHT, GUANG LIN, AND J. NATHAN KUTZ

classification, using data coming from multiple time points, rather than the spatialclassification, with data coming from a single snapshot, presented in previous results.Thus the classification is more robust as it uses multiple time points and significantlymore data for the classification task. Specifically, for time-varying nonlinear prob-lems, classification using only one snapshot at different times may provide differentclassification results. Here, we provide a spatiotemporal classification using data frommultiple time points, which will provide a more robust and consistent classificationresult.

To demonstrate our methodology, we consider a practical implementation of ourtechnique on time-varying fluid flows that are ubiquitous in modern engineering and inthe life sciences. Specifically, we use the asynchronous sparse measurements in timeand space on the pressure field around a cylinder to classify the Reynolds numberand reconstruct the full pressure field. This problem was previously and successfullyconsidered [14, 15, 58] with the view of only considering spatial compressive sensing.The addition of asynchronous time measurements is the new innovation consideredhere allowing for a more robust architecture for performing the classification task.

Our method is motivated by bio-inspired engineering design principles [3, 20,66, 47] whereby birds, bats, insects, and fish routinely harness limited measurementsof unsteady fluid phenomena to improve their propulsive efficiency, maximize thrustand lift, and increase maneuverability [12, 22, 57, 60, 65, 67]. Indeed, despite theirlimited number of computational resources and spatially distributed, noisy sensors,they achieve robust flight dynamics and control despite potentially large perturbationsin flight conditions (e.g., wind gusts, bodily harm). Such observations suggest theexistence of low-dimensional structures in the flow-field that are sparsely sampledand which are exploited for robust control purposes.

The primary innovation here revolves around the compressive sensing ideology.Compressed sensing allows the reconstruction of a signal using a small number ofmeasurements, based on the fact that the signal has a sparse representation in an ap-propriate basis [16, 17, 18, 25, 32, 9, 10]. This allows the use of fewer measurementsthan the Nyquist–Shannon criterion requires [48, 59]. Although typical applicationsare found in areas spanning signal processing to computer vision, our objective is touse the methodology for the numerical solution of partial differential equations (PDE)and/or direct observations of complex systems. We present a specific example: theincompressible Navier–Stokes flow around a cylinder. However, the framework pre-sented can be applied to a much broader set of equations. At the method’s core isthe fact that most complex systems, including the flow around a cylinder, exhibitan underlying low-dimensional structure in their dynamics; such an assumption is atthe heart of the ROM methodology. Unlike the standard application of compressivesensing for image processing, for example, where the sparse (wavelet) basis is alreadyknown, the optimal sparse basis considered here is computed directly from PDE sim-ulations (e.g., through a proper orthogonal decomposition, or POD). Previously, themethod was developed to classify the Reynolds number for flow around a cylindergiven a single snapshot measurement [14]. However, by using asynchronous temporalmeasurements as well, the method can be greatly improved. Thus a spatiotemporalversion of the compressive sensing architecture is developed and advocated here asthe primary result. Note that asynchronous sampling is required for the compressivesensing architecture in order to avoid aliasing errors [41].

The paper is outlined as follows: In section 2 we outline the underlying meth-ods used for the spatiotemporal dimensionality reductions, highlighting how low-rankstructures in both space and time are equally exploited. Section 3 outlines the method

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ASYNCHRONOUS SPARSE SAMPLING 825

used for reconstruction of the solution given a sparse set of measurements. Classifica-tion of the dynamical regime of the system, which is based upon libraries of dynamicalmodes, is demonstrated in section 4. Sections 2–4 in combination form the core in-novation of this article, leading to the example problem of flow around a cylinder ofsection 5. A brief conclusion and outlook are given in section 6.

2. Dimensionality reduction for data snapshots. Dimensionality reductionhas a long history in regard to theoretical methods applied to complex dynami-cal systems. Before the advent of computers, dimensionality reduction often wasachieved through asymptotic reductions of the governing equations. Indeed, manywell-characterized theoretical examples of fluid dynamics are due to perturbative re-ductions of the full system in some parameter regime, e.g., the low Reynolds numberlimit. Such reductions allow for a simplified model where known analytic methodscan be applied to great effect. These asymptotic techniques also serve as the back-bone for normal form reductions for characterizing the low-dimensional embeddingsubspaces where bifurcations in complex systems occur. With the advent of the com-puter and modern high-performance computing, these same kinds of reductions canbe achieved through principled approaches such as the proper orthogonal decompo-sition (POD) [11], which is also known with minor variation as principal componentanalysis (PCA) [50], the Karhunen–Loeve (KL) decomposition, the Hotelling trans-form [34, 35], and/or empirical orthogonal functions (EOFs) [44]. These methodsidentify the low-dimensional subspaces and modal basis that optimally represent, inan `2 sense, the dynamics of the system, thus forming the basis of the ROM architec-ture [54].

For our purposes, consider data that comes from a dynamical system of a singlespatial variable x of the form

(2.1)∂u

∂t= N (ν, x, t, u, ux, uxx, . . . ) ,

where N(·) determines the evolution of the complex system which can be a function ofspace (x ∈ [−L,L]), time (t ∈ [0, T ]), and linear and nonlinear terms involving u(x, t)and its derivatives. Additionally, the parameter ν is a bifurcation parameter thatdetermines the dynamical regime of the system. Such systems are typically solvedusing a numerical discretization scheme where the kth time point (k = 1, 2, . . . ,m) isgiven by

(2.2) u(x, tk)→ uk = [u(x1, tk) u(x2, tk) · · · u(xn, tk)]T

with the discretization grid ∆x = xj+1−xj � 1, x1 = −L, xn = L, and n� 1. Thusin a simulation, one has access to all the data in a discretized form in space and time.Moreover the data is high-dimensional since a fine grid (of size n) is typically requiredfor accurate solutions of the PDE.

For various values of the bifurcation parameter ν we assume we have access to thedata, either from computation or from dense measurements in time and space. Wecan arrange the full data into a matrix whose columns are the full discretized state ofthe system, or snapshots, sampled evenly in time,

(2.3) AνD = [u1 u2 · · · um] ,

where AνD ∈ Rn×m. In the application we wish to consider in this article, we assume

that we have access only to a subset of the data, based on the position in the spatial

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grid xj ; i.e., we have a limited number of spatial sensors available that are fixed toa specific spatial location. To be more precise, consider the projection matrix Pk,whose k nonzero entries determine the spatial locations for data sensors (let k = 3 forillustrative purposes):

(2.4) P3 =

1 0 · · · · · · 00 · · · 0 1 0 · · · · · · 00 · · · · · · 0 1 0 · · · 0

.This example maps out the full state of the system onto 3 measurement locations:

(2.5) u = P3u.

More generally, for k measurement locations, we can construct the data matrix

(2.6) AνS = [u1 u2 · · · um] ,

where AνS ∈ Rk×m. Thus Aν

S is a subset of data generated from a small number of sen-sors k where k � n. Our hypothesis is that the data satisfies the following conditions:

H1 The data in AνS has a low-dimensional representation in the phase space, namely,it can be approximated by a low-dimensional vector space while maintainingmost of the energy (variance) in the `2 sense.

To extract the low-rank representation of the data, we apply a singular valuedecomposition (SVD) to the data matrix to obtain

(2.7) AνS = Vν

SΣνSUν∗

S ,

where the ∗ denotes the complex-conjugate transpose of a matrix. The SVD de-composition can be used for producing a principled low-dimensional representationof the data. Truncation is typically achieved by inspection of the diagonal singularvalue matrix Σν

S . Specifically, dν modes are selected that, for instance, represent 99%of the total variance in the data. For data with noise, recent work by Gavish andDonoho [31] provides a principled truncation algorithm to account for the effects ofnoise.

In our analysis, we reduce the data to dν dimensions for a variety of dynamicalregimes ν. This allows us to obtain the approximation to the data matrix

(2.8) AνS = Vν

SΣνSUν∗

S ,

where the barred matrices are dν-rank approximations to their unbarred counterparts,ultimately providing the approximation to the full data,

(2.9) AνS ≈ Aν

S =

dν∑i=1

σivνi u

ν∗i ,

where VνS = [vν1 , . . . ,v

νdν ], Uν

S = [uν1 uν2 . . . uνdν ], and σi are the corresponding diag-onal elements Σν

S .In addition to our assumption that (2.1) exhibits low-dimensional dynamics, there

is a large subclass of dynamical systems that also exhibit time dynamics that are nearlyperiodic. More precisely, in the application considered here, the time dynamics canbe approximated by a small number of Fourier modes, thus allowing for a secondsparsification step. This leads to the second hypothesis of the current work:

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H2 Each row of the matrix UνS can be approximated, in the `2 sense, by a small

number of Fourier modes in time, namely by cosines and sines.

Given this second hypothesis, our goal now is to further approximate the full datamatrix Aν

S by a sparse number of Fourier modes. Thus in the modal decomposition(2.7), the matrix containing the time dynamic modes uνi is approximated by

(2.10) uνi ≈nνi∑j=1

(ac,νi,j cνi,j + as,νi,j sνi,j

),

where nνi is the small (sparse) number of Fourier modes retained with

cνi,j =[cos(ωνi,jt1

), cos

(ωνi,jt2

), . . . , cos

(ωνi,jtm

)],(2.11a)

sνi,j =[sin(ωνi,jt1

), sin

(ωνi,jt2

), . . . , sin

(ωνi,jtm

)],(2.11b)

and where ωνi,j is the angular frequency obtained by the Fourier transform.The low-rank approximation by the SVD (2.9) and the sparse Fourier mode ap-

proximation (2.10) can be combined to give the time-space reduction

(2.12) AνS ≈

dν∑i=1

σivi

nνi∑j=1

(ac,νi,j cνi,j + as,νi,j sνi,j

)=

dν∑i=1

nνi∑j=1

(αc,νi,j vic

νi,j + αs,νi,j vis

νi,j

),

where αc,νi,j = σiac,νi,j and αs,νi,j = σia

s,νi,j .

We now perform the final dimensionality-reduction step in our analysis. Specifi-

cally, we define(ανi,j)2

=(αs,νi,j

)2+(αc,νi,j

)2and choose mν of the largest ανi,j so that

their sum of squares approximates the energy up to a given threshold for trunca-tion. This allows us to obtain the final low-rank approximation in the l2 sense (seesection 5.2 for the specifics) of the data matrix

(2.13) AνS ≈

mν∑k=1

(αc,νk vνikc

νik,jk

+ αs,νk vνiksνik,jk

).

The difference between the approximations (2.9) and (2.13) is that we assume in thelatter that the time representation can be well approximated by the Fourier modescνik,jk and sνik,jk . So although there is a more dimensionally compact, and optimal,representation for uνi in (2.9), such a time representation is generally in terms ofunknown functions; i.e., the SVD gives a time representation not in terms of well-known functions, but simply as the best least-squares fit to the data. By specifyingthe sinusoidal functional form, a linear problem can be constructed capable of bothclassifying and reconstructing the spatiotemporal data in a sparse way.

Note that we do not need to store the complete vectors cνik,jk and sνik,jk , but onlythe vectors vνik , and the frequencies ωνik,jk obtained by the fast Fourier transform.Approximation (2.13) allows us to use a small number of measurements in time andspace to reconstruct an approximation to the entire data matrix. This is the primaryinnovation of this work.

3. Reconstruction from sparse measurements. The primary objective inthis work is to use sparse measurements in space and time to reconstruct the fullstate u of (2.1) and (2.2). To this end, consider that we are given N measurements

(3.1) p = [p1 p2 · · · pN ]

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at the space-time points

(3.2) (xi, ti) = (χi, τi) for i = 1, 2, . . . , N.

Recall that we assumed that the measurements can be represented sparsely by theparameter ν in (2.13), namely,

(3.3) pi ≈mν∑k=1

[vνik]χi

(cos(ωνik,jk (τi + φ)

)+ sin

(ωνik,jk (τi + φ)

)),

where [v]l denotes the lth element of the vector, and φ is an unknown phase.We relax this representation by folding in the unknown phase into additional

amplitude parameters. The new representation takes the form

(3.4) pi ≈mν∑k=1

[vνik]χi

(Aνik,jk cos

(ωνik,jkτi

)+Bνik,jk sin

(ωνik,jkτi

)).

Note that for each ωνik,jk there are distinct Aνik,jk and Bνik,jk that do not depend on i.Thus there are a total of 2mν amplitude parameters that need to be determined.

The representation (3.4) is for a single measurement. The total number of mea-surements mν , as given by (3.1), can be represented in matrix form by

(3.5) p = Φνb,

where the odd and even columns of Φν are [Φν ]i,2j−1 =[vνik]χi

cos(ωνij ,jjτi) and

[Φν ]i,2j=[vij]χi

sin(ωνij ,jjτi) so that

(3.6)

Φν=

[vi1 ]χ1

cos(ωνi1,j1τ1

)[vi1 ]χ1

sin(ωνi1,j1τ1

)· · · [vimν ]χ1

sin(ωνimν ,jmν τ1

)[vi1 ]χ2

cos(ωνi1,j1τ2

)[vi1 ]χ2

sin(ωνi1,j1τ2

)· · · [vimν ]χ2

sin(ωνimν ,jmν τ2

)...

.... . .

...[vi1 ]χN cos

(ωνi1,j1τN

)[vi1 ]χN sin

(ωνi1,j1τN

)· · · [vimν ]χN sin

(ωνimν ,jmν τN

)

with

(3.7) b =

Aνi1,j1Bνi1,j1Aνi2,j2Bνi2,j2

...Bνimν ,jmν

.

Note that the matrix Φν depends on the time-space locations where measurementsare made. The reconstruction based upon (2.13) can then be accomplished using astandard Moore–Penrose pseudoinverse [63] given the measurements p. This allowsfor computing the unknowns b and performing the full state reconstruction withlimited measurements. Note that the vector b is thus not anticipated to be sparsedue to the pseudoinverse operation.

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4. Dynamical libraries and classification. The previous section consideredour ability to perform a reconstruction or approximation of the full state of the systemusing a limited number of measurements. The reconstruction was for a specific value ofthe bifurcation parameter ν. However, there may be many different dynamical regimesof (2.1) that are of interest and have low-dimensional dynamics [14, 15, 58]. For eachdynamical regime νj , we can then construct the measurement matrices Φν1 , . . . ,ΦνM

where we assume there are M dynamical regimes of interest (ν1, ν2, . . . , νM ). We cancollect these various measurement matrices into a single library matrix:

(4.1) Ψ = [Φν1 Φν2 . . . ΦνM ] .

Thus Ψ contains all the low-rank approximations for space-time measurements fromthe M dynamical regimes. Note that the library matrix is constructed from measure-ments made on the cylinder itself. In previous work [14], we demonstrated that iflibraries were built for the flow in the domain and on the cylinder surface, measure-ments on the surface alone could be used for reconstruction both on the cylinder andin the wake.

Such a database building strategy is also at the forefront of ROM architecturessince ROMs are neither robust with respect to parameter changes nor cheap to gen-erate. Thus methods based on a database of ROMs coupled with a suitable interpo-lation scheme greatly reduce the computational cost for aeroelastic predictions whileretaining good accuracy [54, 5]. Indeed, a variety of recent works, for instance, fromAmsallem and coworkers [6, 19, 2], Haasdonk and coworkers [39, 24], and Peherstorferand Willcox [51, 52, 53], have produced libraries of ROM models that can be selectedand/or interpolated through measurement and classification (typically clustered withk-means-type algorithms). Alternatively, cluster-based ROMs use a k-means cluster-ing to build a Markov transition model between dynamical states [40]. Before thesemore formal techniques based upon machine learning were developed, it was alreadyrealized that parameter domains could be decomposed into subdomains and a localROM/POD computed in each subdomain. Eftang, Patera, and Rønquist [28] used apartitioning based on a binary tree, whereas Amsallem, Cortial, and Farhat [4] useda Voronoi tessellation of the domain. Such methods were closely related to the workof Du and Gunzburger [26], where the data snapshots were partitioned into subsetsand multiple reduced bases computed. The multiple bases were then recombined intoa single basis, so it does not lead to a library per se. For a review of these domainpartitioning strategies, please see [7].

Our objective ultimately is to use a small number of measurements to reconstructthe full state of the system. However, in order to do so efficiently and effectively, oneneeds to classify which dynamical regime νj the system is in so that the correct Φνj

can be used for the reconstruction. The classification is predicated on the idea thatour measurements (3.1) have the sparsest reconstruction (of the form (3.4)), given thecorrect parameter νj [14, 15, 58]. To employ this fact in a classification algorithm, weshall represent the point pi using all the possible parameters νj , generate an under-determined system, and use a compressive sensing algorithm to obtain the sparsestrepresentation. Our hypothesis is that the nonzero coefficients Aνik,jk and Bνik,jk willbe concentrated in the correct ν. Specifically we shall construct an underdeterminedlinear system and solve it using a sparsity promoting algorithm in order to identify thecorrect µj . Thus classification is based upon `1 optimization, whereas reconstructionrelies on an `2 projection unto the correctly identified basis.

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To be more precise, our classification is based on the minimization problem

(4.2) min ‖a‖1 s.t. p = Ψa,

where the unknown vector a, corresponding to the Aνi,j and the Bνi,j , determines theweighting of the library elements Ψ unto the measurement p. The `1 minimization isa sparsity promoting algorithm that attempts to give a solution a that has as manyzeros as possible. The nonzero components are used as an indicator for the dynamicalregime and modes, νj and Φνj , respectively. Here, however, the classification exploitsboth time and space measurements, whereas previous work [14, 15, 58] only consideredmeasurements at a single fixed time for classification.

The compressed-sensing formulation in (4.2) assumes there is no noise in themeasurements and that equality holds. Dealing with noise classification can be ac-complished by considering a relaxed form of the minimization by a lasso algorithm [62]

(4.3) min ‖a‖1 s.t. ‖p−Ψa‖2 < δ

for some parameter δ. This minimization problem minimizes the error, in an `2 sense,between the measurements and the projection onto the library with the sparsest vectora. The sparse solution a can then be used to determine the dynamical regime νj thesystem is in. Once determined, reconstruction can be achieved via (3.4). Specifically,the classification of the dynamical regime is done by summing the absolute value ofthe coefficients of a that corresponds to each dynamical regime νj . To account forregimes with a larger number of coefficients allocated for their dynamical regime, wenormalize by dividing by the square root of the number of POD modes allocated in afor each µj . The classified regime µj is the one that has the largest magnitude afterexecution of this algorithm. The normalization proposed is important to include, asthe regimes with a larger number of POD modes tend to spread their energy over theirbasis set, and thus the normalization accounts for this phenomenon and the correctenergy in each library grouping.

It should be noted that the proposed method for classification is one of only manythat might be envisioned. Indeed, the computer science literature has a vast numberof innovations around clustering and classification that might be suitably adapted tothe task outlined here. The advantages of the method proposed is that it takes clearadvantage of many aspects of the problem in hand, specifically that a very sparserepresentation of the dynamics for each dynamical regime can be achieved. Thus thelasso minimization is performed on a small problem which makes it computationallyefficient. Depending upon the specific application or nature and strength of the noiseapplied to the data, modifications to the basic algorithm might be appropriate.

5. Application: Flow around a cylinder. We demonstrate the sparse time-space sampling algorithm developed on a canonical problem of applied mathematics:the flow around a cylinder. This problem is well understood and has already beenthe subject of studies concerning sparse spatial measurements [14, 64, 38, 37]. Specif-ically, it is known that for low to moderate Reynolds numbers, the dynamics is spa-tially low-dimensional and POD approaches have been successful in quantifying thedynamics [64, 23, 45, 29, 43]. Additionally, the time dynamics is nearly periodic, thussuggesting that the asynchronous sampling in time advocated here can be exploitedfor this problem. The Reynolds number plays the role of the bifurcation parameter νin (2.1).

The data we consider comes from numerical simulations of the incompressible

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ASYNCHRONOUS SPARSE SAMPLING 831Pressure

0.5

-0.5

0.0

-1.0

ν = 40

ν = 300

ν = 150

ν = 1000

-3

3

0

-0.5 0 4 8x

y

Fig. 1. Streamlines and pressure field near the cylinder for a typical snapshot taken for Reynoldsnumbers ν = 40, 150, 300, 1000.

Navier–Stokes equation:

∂u

∂t+ u · ∇u+∇p− ν∇2u = 0,(5.1a)

∇ · u = 0,(5.1b)

where u (x, y, t) ∈ R2 represents the two-dimensional velocity, and p (x, y, t) ∈ R2 thecorresponding pressure field. The boundary condition are as follows: (i) constant flow

of u = (1, 0)T

at x = −15, i.e., the entry of the channel; (ii) constant pressure of p = 0at x = 25, i.e., the end of the channel; and (iii) Neumann boundary conditions, i.e.,∂u∂n = 0 on the boundary of the channel and the cylinder (centered at (x, y) = (0, 0)and of radius unity).

5.1. The data: Snapshots of fluid pressure. Our data comes from numer-ical simulation of the incompressible Navier–Stokes equation (5.1), computed usingthe Nektar++ package [1], which produces accurate results using a high-order finiteelement method. The algorithm uses a nonuniform mesh consisting of 228 grid points,with the high-order method producing 66,000 Gaussian quadrature points, i.e., thenumerical discretization yields an n = 66,000-dimensional system. Figure 1 showstypical snapshots of the streamlines and pressure profiles taken once the system tran-sients have died away and the flow has reached equilibrium or a time-periodic statefor various Reynolds numbers.

The different dynamical regimes of (5.1) correspond to different Reynolds numberflows. Thus the classification and reconstruction problem we consider involves libraryelements extracted from the flow at Reynolds numbers ν = 40, 150, 300, 800, 1000.Specifically, our measurements are based on the pressure field at the perimeter ofthe cylinder where sensors could be easily placed. In particular, a sparse number of

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Pressure

0.5

-0.5

0.0

-1.0

ν = 40

ν = 300

ν = 150

ν = 1000

0

π/2

π

3π/2

time

0T−−−−−−−−→

Fig. 2. Time evolution of the pressure field around the cylinder for increasing Reynolds numbersν = 40 (T = 8), ν = 150 (T = 8), ν = 300 (T = 20), and ν = 1000 (T = 20).

such sensors are used to classify (identify) the Reynolds number and reconstruct thepressure field around the cylinder and in the fluid flow. Note that placing sensorsin the fluid flow itself may be valid for lab experiments and computer simulations.However, no realistic system of interest, such as an airfoil or insect wing, places sensorsdirectly in the flow-field behind the structure. Thus there is a clear reliance on limitedmeasurements on the cylinder itself.

The time evolution of the pressure field on the perimeter of the cylinder is shownin Figure 2 for several Reynolds numbers. As can be seen, increasing the Reynoldsnumber drives the dynamics from a steady-state configuration to a time-periodic evo-lution. The waterfall plots of the pressure dynamics on the cylinder are essentially thesnapshots used for projecting the dynamics onto a low-dimensional manifold throughthe SVD; i.e., the snapshots provide the data necessary for the low-rank, POD method-ology [11].

For each relevant value of the parameter ν we perform an SVD on the data matrixin order to apply the dimensionality reduction method described in section 2. It is wellknown that for relatively low Reynolds numbers, a fast decay of the singular valuesis observed. This can be seen in Figure 3 along with the three most dominant PODmodes. Thus for every Reynolds number ν = 40, 150, 300, 800, 1000, we apply theprocedures described in sections 2 and 3 on the data matrix Aν in order to computethe quantities vνik and ωνik,jk .

5.2. Sparse representation. For every Reynolds number we keep 99% of thetotal energy (variance), which gives for the Reynolds numbers ν = 40, 150, 300, 800,1000 considered a total of 1, 3, 3, 9, 9 POD modes to represent the dynamics. Moreprecisely, it determines the number of nonzero coefficients (1, 3, 3, 9, 9) in the sparse

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30

210

60

240

90

270

120

300

150

330

180 0

30

210

60

240

90

270

120

300

150

330

180 0

30

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60

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90

270

120

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150

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180 0

30

210

60

240

90

270

120

300

150

330

180 0

ν = 40

ν = 300

ν = 150

ν = 1000

ν= 40

150

300

1000

0

-5

-10

-15Sin

gu

lar

valu

es(σν j/σν 1)

log( ∑ j 1

σν k/∑ N 1

σν k

)

0

1

1 5 10 15 1 5 10 15

-0.2 0.2 Pressure

u uw w

w w

wDominant Modes

mode 1mode 2mode 3cylinder

Fig. 3. Normalized decay of singular values (top left) of the data matrix for various Reynoldsnumbers ν= 40, 150, 300, 1000 (top right on logarithmic scale). As the Reynolds number increases,a higher dimensional space is needed to preserve the same amount of energy. The dominant threepressure modes are shown in polar coordinates. The pressure scale is shown by the thick line (bottomleft).

approximation (2.13) (i.e., nonzero αc,νk ’s and αs,νk ’s). The specific modes chosen byour representation, based on the singular value analysis, namely, on the magnitude of(ανi,j)2

=(αc,νi,j

)2+(αs,νi,j

)2, is found in Table 1. Representative spatiotemporal modes

for ν = 40, 150, and 800 are depicted in Figure 4.

5.3. Results. We applied the sparse classification algorithm in section 4 to thetraining data collected in the previous subsection. Based on previous results [14],we have chosen the sparse measurements at positions corresponding to maxima andminima points of the dominant spatial modes (the vνi ). Specifically 20 such positionswere chosen. The classification algorithm was applied to 100 measurements randomlyselected over the time interval of training. Namely, classification was based on 2.5%of the data. The results of classification subject to noise with a random distributionare presented in Table 2.

It should be noted once again that the method demonstrated here extends pre-

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Table 1Sparse representation: The table contains the top 3 coefficients in the expansion (2.13), where

we specify (ανi,j)2 = (αc,νi,j )2 + (αs,νi,j )2.

Reynolds number # Modes Top 3 coefficients

ν = 40 1 α401,0 = 1

ν = 150 3 α1501,0 = 0.97 α150

2,T=170 = 0.03

ν = 300 3 α3001,0 = 0.92 α300

2,T=375 = 0.07

ν = 800 9 α8001,0 = 0.87 α800

2,T=343 = 0.11 α8003,T=172 = 0.04

ν = 1000 9 α10001,0 = 0.85 α1000

2,T=333 = 0.12 α10003,T=166 = 0.06

ν = 800

ν = 40 ν = 150

mod

e1

mod

e2

mod

e3

mod

e4

mod

e1

mod

e1

mod

e2

0

T−−−−−→

0

π/2

π

3π/2

Fig. 4. The figure above shows the spatiotemporal modes. These modes correspond to thespecified Reynolds number, and the mode number corresponds to the magnitude of the energy of thespecific mode.

Table 2The error rate of our algorithm with noise that is normally distributed with the standard devia-

tion increasing across the columns. It can be seen that for low to no additive error, choosing a smallparameter δ for the lasso (4.3) provides a good performance; however, once the error is increased,the algorithm fails, providing result that are not any better than guessing. Note that increasing thelasso parameter δ provides improved performance for higher error, while decreasing the performancefor low or no error.

Error (std) 0 2−7 2−5 2−4

δ = 0.1 0.89 0.88 0.2 0.2

δ = 0.7 0.8 0.82 0.83 0.82

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SIAM MULTISCALE MODELING SIMULATION c© xxxx Society for Industrial and Applied MathematicsVol. xx, pp. x x–x

P (x, t)

P (x, t)

E(x, t)

π

0

−π0 T/2 T

Figure 5.5. (a) Full pressure field P (x, t) and its (b) reconstruction P (x, t) for flow around a cylinderusing the proposed asynchronous sampling scheme with Reynolds number ν = 300. The absolute value of thedifference between the two E(x, t) = |P (x, t) − P (x, t)| is shown in (c) with T = 20. The error initially isdominated by the fact that there are transients in the flow while errors later in time are from a phase-slipbetween the periodic solutions.

structures.The combination of dimensionality reduction and sparsity promoting techniques advocated

here can be applied in an exceptionally broad context. Indeed, such algorithmic strategies canbe used to enhance computation and efficiently identify measurement locations in a givensystem by remembering the system’s key characteristics in both time and space. The low-rankspatio-temporal nature of the approximations used allows construction of control algorithmsthat wrap around the dimensionality reduction and sparsity infrastructure—another appealingaspect of the methodology. In conclusion, we advocate a general theoretical framework forcomplex systems whereby low-rank, spatio-temporal libraries representing the optimal modalbasis in time and space are constructed, or learned, from snapshot sampling of the dynamics.

Acknowledgements. J. N. Kutz acknowledges support for this work from the Air ForceOffice of Scientific Research (GRANT 11832892).

14

Fig. 5. Full pressure field P (x, t) (top) and its reconstruction P (x, t) (middle) for flow arounda cylinder using the proposed asynchronous sampling scheme with Reynolds number ν = 300. Theabsolute value of the difference between the two E(x, t) = |P (x, t)− P (x, t)| is shown in the bottompanel with T = 20. The error initially is dominated by the fact that there are transients in the flow,while errors later in time are from a phase-slip between the periodic solutions.

vious work [14, 15, 58] by considering the full spatiotemporal dynamics. Specifically,previous findings were associated with successfully classifying and reconstructing asingle snapshot in time. Here, the much more difficult task is considered of recon-struction of the entire spatiotemporal behavior with limited measurements both intime and space. The results presented show the method is quite effective at thistask, thus providing a potentially transformational paradigm for taking advantage oflimited measurements of a complex, spatiotemporal system. To exemplify the abilityof the method to reconstruct the spatiotemporal pressure field, Figure 5 shows thepressure field as a function of time around the cylinder for Reynolds number ν = 300.In addition, the reconstruction of the field is shown along with the residual betweenthe full field and its spatiotemporal approximation.

6. Conclusion and outlook. Modern data methods and analysis techniqueslook to exploit underlying low-dimensional structures in complex data sets. In thecase of time-dependent data generated from complex systems, effective methods mustcapitalize simultaneously on the underlying low-rank structures in both time andspace. In doing so, we have demonstrated that one can use a sparse number of spatialsensors that sample randomly (asynchronously) in time to accurately classify thedynamical state of the system even in the presence of noise. The method advocatedintegrates two transformative tools of analysis: (i) model reduction (machine learning)methods for complex systems, and (ii) compressive sensing/sparse representation.

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The current mathematical architecture and innovation extend previous work onsparse sampling of complex systems by exploiting the nearly periodic time behaviorknown to be exhibited by many complex systems, including the flow around a cylinderexample considered here. Indeed, taking advantage of the sparse signal representationin time allows for better classification of the dynamical state. Moreover it equally, andmaximally, exploits the low-dimensional behavior in both time and space. Previousanalysis relied on a single measurement in time for performing such classificationtasks, thus only exploiting the low-dimensional spatial structures.

The combination of dimensionality reduction and sparsity promoting techniquesadvocated here can be applied in an exceptionally broad context. Indeed, such algo-rithmic strategies can be used to enhance computation and efficiently identify mea-surement locations in a given system by remembering the system’s key characteristicsin both time and space. The low-rank spatiotemporal nature of the approximationsused allows construction of control algorithms that wrap around the dimensionality re-duction and sparsity infrastructure—another appealing aspect of the methodology. Inconclusion, we advocate a general theoretical framework for complex systems wherebylow-rank, spatiotemporal libraries representing the optimal modal basis in time andspace are constructed, or learned, from snapshot sampling of the dynamics.

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