Comparison of optimized Dynamic Mode Decomposition vs POD …inavon/pubs/WNMF4.pdf · 2014. 7. 31. · Comparison of optimized Dynamic Mode Decomposition vs POD for the shallow water

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2014; 00:1–25Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld

Comparison of optimized Dynamic Mode Decomposition vs PODfor the shallow water equations model reduction with

large-time-step observations

D.A. Bistrian1∗ and I.M. Navon2

1Department of Electrical Engineering and Industrial Informatics, University ”Politehnica” of Timisoara, 331128Hunedoara, Romania,

2Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306-4120, USA

SUMMARY

We propose a framework for dynamic mode decomposition of 2D flows, when numerical or experimentaldata snapshots are captured with large time steps. Such problems originate for instance from meteorology,when a large time step acts like a filter in obtaining the significant Koopman modes, therefore the classicdynamic mode decomposition method is not effective. This study is motivated by the need to further clarifythe connection between Koopman modes and POD dynamic modes. We apply dynamic mode decomposition(DMD) and proper orthogonal decomposition (POD) to derive reduced-order models of Shallow WaterEquations (SWE). A new algorithm for extracting the dominant Koopman modes of the flow field and anew criterion of selecting the optimal Koopman modes are proposed. A quantitative comparison of thespatial modes computed from the two decompositions is performed and a rigorous error analysis for theROM models obtained by the classic POD and the optimized DMD is presented.Copyright © 2014 John Wiley & Sons, Ltd.

Received . . .

KEY WORDS: dynamic mode decomposition; proper orthogonal decomposition; model order reduction;shallow water equations.

1. INTRODUCTION

While until the last decade, techniques like Direct Numerical Simulations (DNS), Laser DopplerVelocimetry (LDV) or Particle Image Velocimetry (PIV) have provided more accurate spatiallyand temporally resolved turbulent flows, there arose a need to understand the physics of fluid flowbesides flow visualization, as a preliminary step in the process of developing efficient control andmodel reduction techniques.

The modal decomposition of fluid dynamics is a frequently employed technique, capable ofproviding tools for studying dominant and coherent structures in turbulent flows. The coherentstructures [1] represent spatially or temporally evolving vortical motions, either growing with onerate, oscillating with one frequency or containing the largest possible kinetic energy. A complexturbulent flow often consists of a superposition of such coherent structures, whose development isresponsible for the bulk mass, energy transfer or hydrodynamic instability.

Among several snapshot-based model order reduction (MOR) modal decomposition methods,Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) have beenwidely applied to study the physics of the dynamics of the flows in different applications. Themodel order reduction using the method of POD has been illustrated on a variety of examples

∗Correspondence to: Revolutiei Str. Nr.5, 331128 Hunedoara, Romania. E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.Prepared using fldauth.cls [Version: 2010/05/13 v2.00]

2 D.A. BISTRIAN AND I.M. NAVON

ranging from fluid mechanics (Luchtenburg and Rowley [2], Liberge and Hamdouni [3]), turbulentflows and oceanography (Wang et al. [4], Abramov and Majda [5], Osth et al. [6]) or engineeringstructures (Mariani and Dessi [7], Buljak and Maier [8]). More recently, the POD approach hasbeen incorporated for reduced order modeling purposes, within an unstructured mesh finite elementocean model by Du et al. [9], Fang et al. [10], Stefanescu and Navon [11]. POD proved to be aneffective technique also in inverse problems, as demonstrated the work of Winton et al. [12], Chenet al. [13, 14] and Cao et al. [15].

In particular, certain global modes types are considered. Linear global eigenmodes are small-amplitude perturbations that grow or decay exponentially and pulsate with one frequency. They areused in hydrodynamic stability analysis [16]. Balanced modes are used to construct low-dimensionalmodels of large-scale flow systems in order to capture the relation between input disturbances andthe output sensors used for flow measurements. Projecting the original linear system onto thesemodes results in a high-fidelity model that accurately reproduces the input-output dynamics of thereduced model [17, 18].

Koopman modes represent spatial flow structures with time-periodic motion used to characterizeoscillatory nonlinear flow dynamics. They have been increasingly used because they provide apowerful way of analysing nonlinear flow dynamics using linear techniques (see e.g. the workof Bagheri [19], Mezic [20], Rowley et al. [21]). The Dynamic Mode Decomposition generalizesthe global stability modes and approximates the eigenvalues of the Koopman operator [22]. TheKoopman modes are extracted from the data snapshots and a unique frequency is associated toeach mode. This is of major interest for fluid dynamics applications where phenomena occurring atdifferent frequencies must be individualized.

The application of proper orthogonal decomposition is primarily limited to flows whose coherentstructures can be hierarchically ranked in terms of their energy content. But there are situations whenthe energy content is not a sufficient criterion to accurately describe the dynamical behavior of aboveflows. Instead, dynamic mode decomposition links the dominant flow features by a representationin the amplitudes-temporal dominant frequencies space.

A comparative analysis of proper orthogonal decomposition and dynamic mode decompositionhas been performed in the literature, to identify which of these decomposition techniques is moreefficient. Recent studies performed in various fields have demonstrated that these are complementarymethods contributing to the identification of systems in different ways. Simultaneous applicationof the two methods provides an a priori knowledge of the dynamics of the system. For example,Semeraro et al. [23] present a comparative analysis of POD-DMD computed from experimentaldata of a turbulent jet. The extracted DMD modes exhibit many similarities with the POD modesand the flapping mode was easily identified using both methods. The transition to unsteadinessand the dynamics of weakly turbulent natural convection in a differentially heated 3D cavity wassuccessfully investigated by Soucasse et al. [24] by modal decomposition. In a novel approachmanner, Frederich and Luchtenburg [25] consider the Proper Orthogonal Decomposition andDynamic Mode Decomposition. They show how the correlation matrix which is needed for POD canbe re-used in the computation of the DMD modes. In the field of aerodynamics, Muld [26] and hiscoworkers applied Proper Orthogonal Decomposition and Dynamic Mode Decomposition to extractthe most dominant flow structures of a simulated flow in the wake of a high-speed train model. Theyperform a comparison between the modes from the two different decomposition methods.

The present study is motivated by the need to further clarify the connection between Koopmanmodes and POD dynamic modes, as well as address their physical significance, in modaldecomposition of flows with large time span. So far, POD and DMD were applied to problems witha rapidly developing dynamic, considering very small time steps. In general, for problems occurringin meteorology [27, 28] or oceanography [29, 30], the use of large time-step for observables isjustified. It was realized that large time steps act like a filter in obtaining the significant Koopmanmodes, therefore the classic dynamic mode decomposition method is not effective when it is appliedto problems with large time steps. In this work, a new approach is proposed to derive an improvedDMD based procedure that is able to extract dynamically relevant flow features from time-resolvedexperimental or numerical data. Our objective is to employ the improved DMD technique in parallel

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2014)Prepared using fldauth.cls DOI: 10.1002/fld

COMPARISON OF OPTIMIZED DYNAMIC MODE DECOMPOSITION VS POD MODEL REDUCTION 3

with the classic POD method in order to analyze which of these procedures better highlight thecoherent structures of the flow dynamics. The novelty introduced in this paper resides in applicationof the optimized DMD technique to problems originating from meteorology, when numerical orexperimental data snapshots are captured with large time steps. The modes selection, which iscentral in model reduction, represents the subject that we aim to investigate in this paper. We proposea new criterion of selecting the optimal Koopman modes. Additionally, we present a rigorous erroranalysis for the ROM models obtained by the classic POD and the optimized DMD and we alsocompare the relative computational efficiency of above-mentioned ROM methods.

The remainder of this article is organized as follows. The procedure of numerical data acquisitionis presented in Section 2. In Section 3 we recall the principles governing the Dynamic ModeDecomposition and we give the description of the optimized DMD for large time step observables.In particular, we discuss the implementation of the dynamic mode decomposition for twodimensional flows and the criterion for optimal selection of the Koopman modes. The principlesgoverning the Proper Orthogonal Decomposition are discussed in detail in Section 4 that includesalso the algorithm of computing the 2D proper orthogonal modes. This strategies are applied tothe SWE model in Section 5 along with a qualitative analysis of Koopman and POD modes, whilethe reduced order models obtained by involving the DMD and POD expansion of the variables arediscussed in detail in Section 6. Summary and conclusions are drawn in the final section.

2. NUMERICAL DATA ACQUISITION

We consider a bounded open domain Ω ⊂ R3 and let L2 (Ω) be the Hilbert space of square integrablevector functions over Ω, associated with the energy norm ∥w∥L2 = (w,w)

1/2L2 and the standard inner

product (v, w)L2 =∫Ω

v · w dz. Let H∇ be the Hilbert space of divergence free functions given by

H∇ =w ∈ L2 (Ω)

∣∣∣∇ · w = 0 in Ω, w · →n = 0 on ∂Ω, (1)

where→n is the outward normal to the boundary. We define Hd (Ω) ⊂ L2 (Ω) to be the Hilbert space

of functions with d distributional derivatives that are all square integrable. Let V be the Hilbert space

V =

w ∈ H∇

∣∣∣∣w ∈ H1 (Ω) , w = 0,∂w

∂→n

= 0, on ∂Ω

, (2)

with norm ∥w∥V = (w,w)1/2V and the inner product (v, w)V =

d∑i=1

(∇vi,∇wi).

In the Cartesian coordinates formulation, we suppose there exists a time dependent flow w =(u, v, h) (x, y, t) ∈ V and a given initial flow w (x, y, 0) = (u0, v0, h0) (x, y), that are are solutionsof the Saint Venant equations, also called the Shallow Water Equations (SWE) [31],

ut + uux + vuy + ηx − fv = 0, (3)

vt + uvx + vvy + ηy + fu = 0, (4)

ηt + (ηu)x + (ηv)y = 0, (5)

where u (x, y, t) and v (x, y, t) are the velocity components in the x and y axis respectively,η (x, y, t) = gh (x, y, t) is the geopotential height, h (x, y, t) represents the depth of the fluid, fis the Coriolis factor and g is the acceleration of gravity. Subscripts represent the derivatives withrespect to time and the streamwise and spanwise coordinates.

We consider that the reference computational configuration is the rectangular 2D domain Ω =[0, Lmax]× [0, Dmax]. The model (3)-(5) is considered here in a β-plane assumption [32], in whichthe effect of the earth’s sphericity is modeled by a linear variation in the Coriolis factor

f = f +β

2(2y −Dmax) , (6)



where f and β are constants, Lmax and Dmax are the dimensions of the rectangular domain ofintegration Ω.

The shallow-water equations have been used for a wide variety of hydrological and geophysicalfluid dynamics phenomena such as tide-currents [33], pollutant dispersion [34], storm-surges ortsunami wave propagation [35].

The test problem used in this paper is consisting of the nonlinear shallow-water equations (3)-(5)in a channel on the rotating earth, associated with periodic boundary conditions in the x-directionand solid wall boundary condition in the y-direction:

w (0, y, t) = w (Lmax, y, t) , v (x, 0, t) = v (x,Dmax, t) . (7)

The initial condition I1 introduced by Grammeltvedt [36] was adopted as the initial height field,which has been tested by different researchers (Cullen and Morton [37], Navon [38], Stefanescu andNavon [11], Fang et al. [39]), i.e.

h0 (x, y) = H0 +H1 tanh

(9(Dmax/2− y)

2Dmax

)+H2 sin

(2πx

Lmax

)cosh−2

(9(Dmax/2− y)

Dmax

),

(8)which propagates the energy in wave number one, in the streamwise direction. Using the geostrophicrelationship, u = −hy (g/f), v = hx (g/f), the initial velocity fields are derived as:

u0 (x, y) = − g

f

9H1

2Dmax

(tanh2

(9Dmax/2− 9y

2Dmax

)− 1

)−

− 18g

fH2 sinh

(9Dmax/2− 9y

Dmax

) sin(

2πxLmax

)Dmaxcosh

3(

9Dmax/2−9yDmax

) , (9)

v0 (x, y) = 2πH2g

fLmaxcos

(2πx

Lmax

)cosh−2

(9(Dmax/2− y)

Dmax

). (10)

In developing a higher order scheme for approximating the quadratically nonlinear terms thatappear in the equations of hydrological dynamics, we have followed the approach used by Navon[32], which implements a two-stage finite-element Numerov-Galerkin method for integratingthe nonlinear shallow-water equations on a β-plane limited-area domain. The use of numericalintegration methods to study the behavior of theoretical models in oceanography or to predictthe evolution of an actual state is subject to several predicaments, one of the major difficultiesbeing nonlinear computational instability of the finite difference analogues of the governing partialdifferential equations. The approach adopted in the work noted above involves the use of a weightedselective lumping scheme in the finite-element method, combined with a successive overrelaxation(S.O.R.) iterative method for solving the resulting systems of linear equations. Determination ateach time-step of the values of the three integral invariants [40, 41] of the shallow-water equations,i.e. the total mass, the total energy and the potential enstrophy, proved that the two stage Numerov-Galerkin is attaining a consistently higher accuracy than the single-stage finite-element method (seeNavon (1983) [42]).

In several seminal papers, Arakawa [43, 44] indicated that the integral constraints on quadraticquantities of physical importance, such as conservation of mean kinetic energy and mean squarevorticity, will not be maintained in finite difference analogues of the equation of motion for two-dimensional incompressible flow, unless the finite difference Jacobian expression for the advectionterm is restricted to a form which properly represents the interaction between grid points (i.e. use ofstaggered C or D grids).

Thus the effect of conservation of integral invariants by finite-element discretization scheme of theshallow-water equations (3)-(5) as a measure of the correct discretization of long-term integrationshas a pivotal importance. The numerical integration scheme is detailed in [32]. Using this program,we have captured the shallow-water dynamics over long-term numerical integrations (10-20 days).We will further detail the results in the section dedicated to numerical experiments.



3. DYNAMIC MODAL DECOMPOSITION OF FLOW FIELDS BASED ON KOOPMANOPERATOR

So far we have noticed two directions in dynamic mode decomposition technique. First one isseeking a companion matrix that helps to construct in a least squares sense the final data vectoras a linear combination of all previous data vectors [21, 45, 46]. Because this version may be ill-conditioned in practice, Schmid [47] recommends an alternate algorithm, upon which the workwithin this article is based.

The aim of this section is twofold: first we describe the classical analytical method for dynamicmode decomposition. The optimized DMD algorithm is then introduced as an improvement of theoriginal algorithm, together with a criterion of optimal selection of the Koopman modes.

3.1. The Koopman Operator and the General Description of DMD

Employing numerical simulations or experimental measurements techniques, different quantitiesassociated with the flow are measured and collected as observations at one or more time signals,called observables. It turns out (see the survey of Bagheri [48]) that monitoring an observable overa very long time interval allows the reconstruction of the phase space.

Considering a dynamical system evolving on a manifold M such that, for all wi ∈ M

wi+1 = f(wi), (11)

the Koopman operator, defined by Koopman [22] in 1931 is a linear operator that maps any scalar-valued function g : M → R into a new function Ug given by

Ug (w) = g (f (w)) . (12)

The Koopman operator is infinite-dimensional and it steps forward in time an observable. Relatedwith spectral properties of the Koopman linear operator, the reader is invited to refer to Rowley et al.[21] for rigorous treatment on the subject. There is a unique expansion that expands each snapshotin terms of vector coefficients vj which are called Koopman modes and Koopman eigenfunctionsaj (w) associated with mode amplitudes, such that

g (w) =

∞∑j=1

aj (w) vj . (13)

Iterates of w0 are then given by

g (wk) =

∞∑j=1

λkj aj (w0) vj , λj = eσj+iωj , (14)

where λj are the called the Ritz eigenvalues of the modal decomposition, that are complex-valuedflow structures associated with the growth rate σj and the frequency ωj .

Assuming that w0, w1, ...wN is a data sequence collected at a constant sampling time ∆t, wedefine the following matrices

V N−10 =

(w0 w1 ... wN−1

), V N

1 =(w1 w2 ... wN

). (15)

The DMD algorithm is based on the hypothesis that a Koopman linear operator A exists, thatsteps forward in time the snapshots, such that

wi+1 = Awi, i = 0...N − 1. (16)

It follows that the snapshots data set

V N−10 =

(w0 Aw0 A2w0 ... AN−1w0

)(17)



corresponds to the N th Krylov subspace generated by the Koopman operator from w0.Since the eigenvalues of the unknown matrix operator A must be obtained, a Galerkin projection

of A onto the subspace spanned by the snapshots is performed. For a sufficiently long sequenceof the snapshots, we suppose that the last snapshot wN can be written as a linear combination ofprevious N − 1, vectors, such that

wN = c0w0 + c1w1 + ...+ cN−1wN−1 +R, (18)

which can be written in matrix notation as

wN = V N−10 c +ReTN−1, (19)

in which cT =(c0 c1 ... cN−1

)is a complex column vector and R is the residual vector. We

assemble the following relations

Aw0, w1, ...wN−1 = w1, w2, ...wN =w1, w2, ...V

N−10 c

+ReTN−1 (20)

in the matrix notation form,

AV N−10 = V N

1 = V N−10 C +ReTN−1, C =

0 ... 0 c01 0 c1...

......

...0 . . . 1 cN−1

, (21)

where C is the companion matrix and eTj represents the jth Euclidean unitary vector of length N − 1.A direct consequence of (21) is that decreasing the residual increases the overall convergence and

therefore the eigenvalues of the companion matrix C will converge toward some eigenvalues of theKoopman operator A. Therefore, the way that we will monitor this convergence is by evaluating thesize of the residual during the modal decomposition and plotting its L2 -norm.

Several methods have been employed so far to compute the companion matrix. The last columnof the companion matrix may be found using the Moore-Penrose pseudo-inverse [49] of V N−1

0 , as

c =(V N−10

)+wN =

((V N−10

)∗V N−10

)−1(V N−10

)∗wN .

A solution for the linear least-square problem obtained from (21) is given by the economy sizeQR-decomposition of V N−1

0 , as it is discussed in [50].Instead of using QR-decomposition, SVD was also applied on V N−1

0 in order to find the singulareigenvalues and vectors, this approach being helpful when the matrix V N−1

0 is rank deficient [47].The eigen-elements of the companion matrix span the original data and the decomposition (14) isachieved.

The procedure described above turns out to be ill-conditioned in practice. This may especially bethe case of snapshots collected at large time steps, that behave like using noisy experimental data.In this situation, the modal decomposition is unable to find proper eigen-elements of the Koopmanoperator and the flow reconstruction is inaccurate. For these cases, we propose in the following animproved dynamic mode decomposition algorithm.

3.2. Description of the Optimized DMD for Large Time Step Observables

In this section, we consider that dynamical system (11) represents an approximation of the SWEmodel (3)-(5), discretized in both time and space, and the observables wi = w (ti), ti = i∆t,i = 0..Nt, consist of the time dependent variables w = u, v, h (x, y, t) in the spatial domain Ω.The main objective is to find a representation of the flow field in the form

wDMD (x, y, t) = Wb +

rDMD∑j=1

aje(σj+iωj)tϕj (x, y), σj =

log (|λj |)∆t

, ωj =arg (|λj |)

∆t, (22)

where ϕj ∈ C are the DMD modes, aj ∈ C are the amplitudes of the modes, λj ∈ C are the Ritzeigenvalues of the modal decomposition associated with the growth rate σj and the frequency ωj and



Wb is a constant offset that represents the data mean, usually called the base flow in hydrodynamicstability analysis,

Wb =1

1 +Nt

Nt∑i=0

wi. (23)

Theoretically, we apply the dynamic mode decomposition on the mean-subtracted data w′

i =wi −Wb, i = 0..Nt. As noticed by Noack et al. [51], the use of the mean of a data set as thebase flow represents a common practice in application of modal decomposition like POD. Recently,Chen, Tu and Rowley [52] pointed out that constructing the Koopman modes from base-flow-subtracted data offers the advantage that the reduced order model will satisfy the same boundaryconditions employed for the full model, while computing DMD modes without first subtracting abase flow results in the boundary conditions not being satisfied. On the other hand, they indicatedthat DMD and discrete Fourier transform (DFT) constitute formally equivalent methods in a periodicregime case and when the data set under consideration is centered. As a result, in this situation, allRitz values have a unit magnitude and the temporal DFT is wholly incapable of determining themodal growth rates, therefore a mean-subtracted approach is not recommended. A different opinionwas expressed by Cammilleri et al. [53]. They mention that the equivalence of DMD and DFT onlyapplies when the number of snapshots is lower than the dimension of the snapshot vector.

We extend in this paper the application of the DMD to a mean-subtracted flow field. The mainobjective of this work is to prove that for large time step observables, dynamic mode decompositionof mean-flow-subtracted data and temporal DFT analysis are not equivalent and we will emphasizethe excellent behavior of the optimized method in computing the modal decomposition for largetime step data.

The method of snapshots formulation is well-suited for large data, because the eigenvalue problemdoes not depend on the dimension of the snapshot vector (see Holmes et al. [54]). Here we applythe method of snapshots introduced by Sirovich [55] in 1987 and we solve the resulting eigenvalueproblem by a matrix multiplication method.

A few modal decomposition methods, along with Petrov-Galerkin projection of linear systemsonto modes, are implemented by Belson et al. [56], but we pursue a different approach yieldinga supplementary subroutine for extracting the optimal Koopman modes. We summarize below thesteps of the algorithm.

Algorithm 1: DMD for 2D flows with large time step observations

(i) Collect data wi (x, y) = w (x, y, ti), ti = i∆t, i = 0..Nt from the flow field, equallydistributed in time.

(ii) Transform snapshots into columns wi (x, y) of the matrix

V =

| |

... |

w0 (x, y) w1 (x, y)... wNt (x, y)

| |... |

. (24)

(iii) Compute the mean flow W b (x, y) =1

1+Nt

Nt∑i=0

wi (x, y) and the mean-subtracted snapshot

matrix V ′ = V −W b.(iv) A matrix V Nt−1

0 is formed with the first Nt columns and the matrix V Nt1 contains the last Nt

columns of V ′,

V Nt−10 =

| |

... |

w0 (x, y) w1 (x, y)... wNt−1 (x, y)

| |... |

, (25)



V Nt1 =

| |

... |

w1 (x, y) w2 (x, y)... wNt (x, y)

| |... |

. (26)

Performing a Galerkin projection of the unknown Koopman operator A onto the subspacespanned by the snapshots, we express the vectors of V Nt

1 as a linear combination of theindependent sequence V Nt−1

0 :

V Nt1 = AV Nt−1

0 = V Nt−10 S +R, (27)

where R is the residual matrix and S approximates the eigenvalues of A when the norm∥R∥2 → 0. The objective at this step is to solve the minimization problem

MinimizeS

R =∥∥V Nt

1 − V Nt−10 S

∥∥ . (28)

(v) We identify a singular value decomposition of V Nt−10 :

V Nt−10 = UΣWH , (29)

where U contains the proper orthogonal modes of V Nt−10 , Σ is a square diagonal matrix

containing the singular values of V Nt−10 and WH is the conjugate transpose of W . It follows

from (27), that S can be obtained by multiplying V Nt1 by the Moore-Penrose pseudoinverse

of V Nt−10 :

S =(V Nt−10

)+V Nt1 = WΣ+UHV Nt

1 = XΛX−1, (30)

where X and Λ represent the eigenvectors, respectively the eigenvalues of S, and Σ+ iscomputed according to Moore-Penrose pseudoinverse definition of Golub and van Loan [49]:

Σ+ = diag

(1

σ1, · · · , 1

σr, 0 · · · , 0

), r = rank

(V Nt−10

). (31)

(vi) Calculation of Koopman modes and amplitudes. After solving the eigenvalue problem

SX = XΛ, (32)

the diagonal entries of Λ represent the Ritz eigenvalues λ. The frequency and damping areprovided by these eigenvalues. The projection of V Nt

1 on the modes V Nt−10 X yields:

V Nt1 =

(V Nt−10 X

)ΛX−1 = AV Nt−1

0 . (33)

From this expression, the contribution of each dynamic mode to the data sequence V Nt1 is

obtained. Thus the Koopman modes are the columns of the matrix ϕ = V Nt−10 X . Unflatten

these columns to obtain the dynamic modes ϕi (x, y) = ϕ (x, y, ti), ti = i∆t, i = 0..Nt − 1.The amplitudes are given by the norm of the corresponding column vector of V Nt−1

0 X , as

aj =

∥∥V Nt−10 X (:, j)

∥∥2∥∥V Nt−1

0 X∥∥2

, j = 1..r, (34)

where r represents the number of the Koopman modes stored for modal decomposition.

3.3. Optimal Selection of the Koopman modes

Selection of Koopman modes and amplitudes used for the flow reconstruction constitutes thesource of many discussions among modal decomposition practitioners. For instance, Jovanovicet al. [57] introduced a low-rank DMD algorithm to identify an a priori specified number of



modes that provide optimal approximation of experimental or numerical snapshots at a certain timeinterval. Consequently, the modes and frequencies that have strongest influence on the quality ofapproximation have been selected. Chen et al. [52] introduced an optimized DMD, which tailorsthe decomposition to an optimal number of modes. This method minimizes the total residual overall data vectors and uses simulated annealing and quasi-Newton minimization iterative methods forselecting the optimal frequencies.

The superposition of all Koopman modes, weighted by their amplitudes and complex frequencies,approximates the entire data sequence,

wDMD (x, y, t) = Wb +

r∑j=1

ajλjϕj (x, y), T ∈ 0, . . . Nt − 1 , (35)

but there are also modes that have a weak contribution.We address in his section the problem of identification of an optimal truncated representation of

the flow field in order to capture the most important dynamic structures. To this end, the flow can bereconstructed at a certain moment in time T , using the first optimal Koopman modes and associatedamplitudes and Ritz eigenvalues as:

wDMD (x, y, T ) = Wb +

rDMD∑j=1

ajλjϕj (x, y), T ∈ 0, . . . Nt − 1 , (36)

where rDMD < Nt represents the optimal number of the selected modes that must be identified. Inmatrix formulation, relation (36) yields:

wDMD (x, y, T ) = Wb +(a1 a2 . . . arDMD

) λ1 . . . 0...

......

0 . . . λrDMD

ϕ1 (x, y)

...ϕrDMD

(x, y)

.

(37)In Proper Orthogonal Decomposition or POD Balanced Truncation Method [46], the flow field

is decomposed into orthogonal modes which are by construction ranked by energy level throughthe POD or Hankel eigenvalues [48]. Thereafter, the order of the modes is in decreasing amountof energy and the POD modes are designed to contain the largest amount of energy with anygiven number of modes. In Dynamic Mode Decomposition, the modes are not orthogonal, but oneadvantage of DMD compared to POD is that each DMD mode is associated with a pulsation, agrowth rate and each mode has a single distinct frequency. The modes are arranged in descendingorder of their energy content, but unlike the POD modes, they give the energy of the fluctuations atdistinct frequencies. For this feature, DMD method was originally used in non-linear dynamics, forinstance Muld et al. [26] and was just recently introduced in fluid mechanics [58, 25, 59].

A criterion of selecting the DMD modes can be their amplitude aj , or based on theirfrequency/growth rate ωj/σj . As reported by Noack et al. [60], the amplitude criterion is notsufficient since there exist modes which are very rapidly damped, having very high amplitudes. Themodal selection based on frequency/growth rate is not rigorous since it relies on a priori physicalknowledge of the flow. The frequencies resolved by the DMD are still subjected to the Nyquistsampling theorem [61] and the researcher has to know in advance frequencies that are essential inthe flow physics to adjust the sampling interval ∆t. On the other hand, the non-orthogonality of theKoopman modes may raise the projection error while increasing the order of the DMD basis.

To avoid these difficulties, we introduce in the following a new method to optimize the selectionof the Koopman modes involved in the reconstruction of the flow. The optimized DMD thatwe propose, is based on the conservation of quadratic integral invariants by the finite-elementdiscretization scheme of the shallow-water model (3)-(5). We assume that the reduced orderreconstructed flow (36) also preserves the conservation of the total flow energy. In parallel, weaim to eliminate the amplitudes that contribute weakly to the data sequence. Let

E0 =1

Nt + 1

Nt∑j=0

∫ ∫Ω

h0(x, y, j)u0(x, y, j)2+ v0(x, y, j)

2+ gh0(x, y, j)

2dx dy, (38)



be the total energy of the high fidelity flow and

EDMD =1

Nt + 1

Nt∑j=0

∫ ∫Ω

h(x, y, j)u(x, y, j)2+ v(x, y, j)

2+ gh(x, y, j)

2dx dy, (39)

be the total energy of the reduced order flow.We arrange the Koopman modes in descending order of the energy of the modes, weighted by the

inverse of the Strouhal number St = arg (λj) / (2π∆t):

eDMDj =

1

St·

∥ϕj (x, y)∥F∥wDMD (x, y, j)∥F

, j = 1..Nt. (40)

We denote by ∥∥F the Frobenius matrix norm in the sense that for any matrix A ∈ Cm×n havingsingular values σ1, ..., σn and SVD of the form A = UΣV H , then

∥A∥F =∥∥UHAV

∥∥F= ∥Σ∥F =

√σ1

2 + ...+ σn2. (41)

Determination of the optimal vector of amplitudes and corresponding eigenvalues andKoopman modes (aj , λj , ϕj (x, y)) , j = 1..r then amounts to finding the solution to the followingoptimization problem Minimize

r

1Nt+1

Nt∑j=0

∥w0(x,y,j)−wDMD(x,y,j)∥F

∥w0(x,y,j)∥F,

Subject to∣∣E0 − EDMD

∣∣ < ε,

(42)

where w0 (x, y, j) and wDMD (x, y, j), j = 0..Nt represents the full rank flow, respectively theKoopman decomposed flow at time j and ε = 1.0000e− 15 sets an upper bound on the relativeerror due to rounding in floating point arithmetic.

We are interested in finding the Koopman modes that provide the maximum energy of thefluctuations at distinct frequencies and we continue minimizing the residual under the lineardynamics constraint and the flow energy conservation assumption.

4. DYNAMIC MODAL DECOMPOSITION OF FLOW FIELDS BY PROPER ORTHOGONALDECOMPOSITION SUBJECT TO THE LINEAR DYNAMICS

Apart from Krylov subspace methods, the Proper Orthogonal Decomposition (POD), represents atthe moment state-of-the-art for many model reduction problems. The strong point of POD is that itcan be applied to non-linear partial differential equations, especially for smooth systems in whichthe energetics can be characterized by the first few modes. The applicability of POD to complexsystems is limited mainly due to errors associated with the truncation of POD modes. The POD andits variants are also known as Karhunen-Loeve expansions in feature selection and signal processing,empirical orthogonal functions in atmospheric science or principal component analysis in statistics.In weather and climate modeling [5, 10, 11, 30], as well as in other complex systems such as dataassimilation [13, 14], the development of accurate and reliable low dimensional models representsan extremely important task.

The idea underlying this method is that the time response of a system given a certain input,contains the essential behavior of the system. Therefore, the set of outputs serves as a starting-pointfor POD. We consider that the observables wi = w (ti), ti = i∆t, i = 0..Nt, consist of the timedependent variables w = u, v, h (x, y, t) of the SWE model (3)-(5), discretized in both time andspace in the spatial domain Ω. The main objective in POD is to find a representation of the flow fieldof the form

wPOD (x, y, T ) = Wb +

rPOD∑j=1

bjTΦn (x, y), T ∈ 0, . . . Nt − 1 , (43)



where Wb represents the data mean flow, rPOD represents the optimal number of the POD selectedmodes that must be identified. The time dependent coefficients bj are called Fourier coefficients. Weare looking for an orthonormal basis Φj (x, y), j = 1..rPOD such that the averages of the first fewFourier-coefficients represent 99% of the total energy of the snapshots. This leads to the followingPOD algorithm:

Algorithm 2: POD for 2D flows with large time step observations

(i) Collect data wi (x, y) = w (x, y, ti), ti = i∆t, i = 0..Nt from the flow field, equallydistributed in time.

(ii) We build the matrix V ,

V =

| |

... |

w0 (x, y) w1 (x, y)... wNt (x, y)

| |... |

. (44)

consisting in the columns wi (x, y) of the snapshots.

(iii) Compute the mean flow W b (x, y) =1

1+Nt

Nt∑i=0

wi (x, y) and the mean-subtracted snapshot

matrix V ′ = V −W b.(iv) Calculate the empirical correlation matrix

C =1

NtV ′V ′T , (45)

where Nt represents the number of snapshots and V ′T represents the transpose of the meansubtracted snapshot matrix.

(v) Compute the singular eigenvalue decomposition

Cvj = λjvj , j = 1..m, vj ∈ Rm, (46)

where m represents the number of the total eigenvalues.(vi) We find the number of POD basis vectors capturing 99.99% of the snapshots energy, defined

as

ePOD =

rPOD∑j=1

λj

/m∑j=1

λj . (47)

(vii) We can choose the first orthonormal basis of eigenvectors v1, ...vrPOD and thecorresponding POD basis functions are given by

Φj =1√λj

V ′vj , j = 1, ...rPOD. (48)

(viii) The temporal coefficients are stored in the matrix B, which is obtained by relation

B = ΦTV ′, B ∈ MrPOD×m. (49)

Hence, in each row we find the trajectories of the dynamical system at discrete time events.(ix) Flow reconstruction. The reconstruction of the flow fluctuating part is achieved as V ′

POD =ΦB, and the reconstruction of the flow field is VPOD = V ′

POD +W b.



a. b. c.

Figure 1. Spectrum of the Dynamic Mode Decomposition: a. Geopotential field h; b. Streamwise velocityfield u; c. Spanwise velocity field v.

5. ANALYSIS OF COHERENT STRUCTURES BY KOOPMAN MODES AND POD

We perform the numerical experiments in a rectangular channel whose dimensions are Dmax =4400km, Lmax = 6000km. The integration time window was 60h and we record the unsteadysolution of the two-dimensional shallow water equations model (3)-(5), with time step ∆t = 600s.The dimensional constants used for the model are

f = 10−4s−1, β = 1.5× 10−11s−1m−1, g = 10ms−1,

H0 = 2000m, H1 = 220m, H2 = 133m. (50)

In this section, the application of POD and DMD is illustrated by comparing the evolution ofthe flow field along the integration time window. There are several major differences between thesetwo decomposition methods. The spatial basis functions ϕj (x, y) and Φj (x, y), for DMD and PODrespectively, offer an insight of the coherent structures in the flow field. The differences betweenϕj (x, y) and Φj (x, y) occur due to the principles of the decomposition methods. The time evolutionof a DMD mode is influenced by the multiplication of the complex eigenvalue λj of the Koopmanoperator weighted by the amplitude, while the time evolution of POD modes is described by thefunctions bj (t). The POD modes are orthogonal in space with the energy inner product. In DMD,each mode oscillates at a single frequency, hence the expression that the DMD modes are orthogonalin time.

The DMD spectra for the mean-subtracted fields (u, v, h) (x, y, t) are presented in Figure 1. It iseasy to see that the Ritz values are not the roots of unity, therefore the mean subtracted DMD isnot identical with the discrete Fourier transform, for the case presented here. The optimized DMDtechique presented herein is fully capable of determining the modal growth rates and the associatedfrequencies. The optimization problem described by Equation (42) leads to the number of the DMDmodes rDMD = 4 to be used for the flow reconstruction.

The amplitudes of the DMD decomposition of the geopotential, obtained from Equation (34)are displayed in Figure 2a, in descending order. Figure 2b presents the normalized vector energy(40) versus the Strouhal number. The lighter colored dots indicate the four amplitude values forwhich the corresponding modes and Ritz eigenvalues are used in the flow reconstruction. Figure2 shows that the higher amplitudes are associated to the most energetic Koopman modes. Thisdemonstrates that, in DMD decomposition, the amplitudes are directly proportional to the energyin the coherent structures, as defined by Equation (40), unlike the POD decomposition wherethe eigenvalues capturing most of the snapshots energy indicate the coresponding POD basisfunctions. The Koopman modes associated to these amplitudes are depicted in Figure 3, for modaldecomposition of the geopotential field.

The optimization criterion (42) was applied in this paper to find the optimal number of DMDbasis functions and their associated Ritz values and amplitudes. The absolute error between the flowtotal energy E0 and the total energy of the flow decomposed by DMD technique EDMD, definedrespectively by Equations (38), (39), is presented in Figure 4a, as a function of the number of theselected DMD basis functions.



a. b.

Figure 2. a. The amplitudes of the DMD decomposition of the geopotential, sorted in descending order;b. The normalized vector energy versus the Strouhal number. The lighter colored dots indicate thefour amplitude values for which the corresponding modes and Ritz eigenvalues are used in the flow

reconstruction.

We plot in Figure 5 the singular values obtained from POD decomposition. Most of the energydefined by Equation (47) is contained in the first few modes. Specifically, the number of optimalPOD basis functions is rPOD = 17, because the first 17 eigenvalues yield more than 99.99% of thesnapshots energy (see Figure 4b). The first eight POD basis functions are depicted in Figure 6.

A quantitative comparison of the spatial modes computed from the two decompositions discussedhere can be obtained from the Modal Assurance Criterion (MAC), as recommended by Brown andhis coworkers [62]. The Modal Assurance Criterion is a measure of the degree of linearity betweentwo vectors. The MAC value for a pair of modes is defined as

MACij

(ϕi

DMD,ΦjPOD

)=

(∥∥∥(ϕjDMD

)H · ΦjPOD

∥∥∥F

)2∥∥∥(ϕj

DMD)H · ϕj

DMD∥∥∥F

∥∥∥(ΦjPOD

)H · ΦjPOD

∥∥∥F

, (51)

where (·) represents the Hermitian inner product, H denotes the conjugate transpose and ∥∥F is theFrobenius matrix norm. The computed MACij

(ϕi

DMD,ΦjPOD

)takes values in the interval [0, 1],

where 1 indicates identical modes and 0 indicates the orthogonality of the modes. In practice [63],two vectors are considered correlated when the MAC value is greater than 0.9, which correspondsto an angle lower than 18 degrees. The vectors are considered uncorrelated when the MAC value islower than 0.6, which means that they are separated by an angle greater than 39 degrees.

In the following, we compare the computed DMD modes with the POD modes used as basisfunctions in the two modal decomposition methods. The MAC values computed between the firstrDMD = 4 modes and the first rPOD = 17 orthogonal modes are represented in Figure 7.

As expected, the first mode corresponding to the mean flow is well captured by both methods,with MAC

(ϕ1

DMD,Φ1POD

)= 1. The fourth POD mode Φ4 exhibits a strong similarity with the

DMD modes, having an increased MAC value MAC41 = 0.89, MAC42 = 0.89, MAC43 = 0.96,MAC44 = 0.96. Analyzing the modal assurance matrix, we conclude that only four POD modesare correlated with the DMD modes, namely Φ1, Φ4, Φ6 and Φ16, exhibiting a MAC number graterthan 0.59. The other POD modes differ from the DMD modes, as with all they ensure the caption of99.99% of the snapshots energy in the POD modal decomposition.

Unlike POD, it is evident that the first four optimal DMD modes are sufficient to describe the flowfield, as indicated the higher MAC values between the second, third and fourth DMD modes andthe first POD mode: MAC21 = 1, MAC31 = 0.99, MAC41 = 0.99. Hence the conclusion that theDMD modal decomposition is more efficient than POD decomposition, because the DMD modaldecomposition is achieved with a smaller number of terms.



05000

100000

5000

0

DMD Mode nr.1

Rea

l par

t

05000

100000

5000

0

DMD Mode nr.1

Imag

inar

y pa

rt

05000

100000

5000

0

DMD Mode nr.2

Rea

l par

t

05000

100000

5000

0

DMD Mode nr.2

Imag

inar

y pa

rt

05000

100000

5000

0

DMD Mode nr.3

Rea

l par

t

05000

100000

5000

0

DMD Mode nr.3

Imag

inar

y pa

rt

05000

100000

5000

0

DMD Mode nr.4

Rea

l par

t

05000

100000

5000

0

DMD Mode nr.4

Imag

inar

y pa

rt

Figure 3. The first four Koopman modes used in optimized DMD. Left column: the real parts, right column:the imaginary parts.

6. ANALYSIS OF OPTIMIZED DMD-ROM AND POD-ROM MODELS

The two-stage finite-element Numerov-Galerkin method for integrating the nonlinear shallow-waterequations on a β-plane limited-area domain proposed by Navon [32] was employed in order toobtain the numerical solution of the SWE model (3)-(5). In Figure 8 the initial velocity fields arepresented. The solutions of geopotential height field and (u, v) field at T = 24h are illustrated inFigure 9. The DMD and POD computed geopotential height field and (u, v) field at T = 24h arealso depicted in Figures 10 and 11.

These reconstructions, when plotted on the same length and time scales as the simulations of thefull system, exhibit strikingly similar features, both quantitatively and qualitatively. The validityof the optimized DMD/POD approach has been checked by comparing our results with thoseobtained by Stefanescu and Navon [11], when an alternating direction fully implicit (ADI) finite-difference scheme was used for discretization of 2-D shallow-water equations on a β-plane. The



a. b.

Figure 4. a. The absolute error between the flow total energy and the total energy of the flow decomposedby DMD method, as the number of the DMD modes; b. The energy captured in the POD decoposition as the

number of the POD modes.

a.0 50 100 150 200 250 300

10−25

10−20

10−15

10−10

10−5

Number of POD eigenvalues

Sin

gula

r va

lue

b.0 5 10 15

10−10

10−9

10−8

10−7

10−6

10−5

Eigenvalue number

Firs

t 17

sing

ular

val

ues

Figure 5. a. POD eigenvalues; b. First rPOD = 17 singular values used for POD decomposition.

flow reconstructions presented in Figure 10a and Figure 11a are very close to those computed in[11] (see p. 103, Figure 2(a) indicates the results used in comparison).

The similarity between these characteristics of the geopotential height field and those obtained inthe previous investigation validates the method presented here and certifies that the optimized DMDcan be applied successfully to 2D flows.

We focus in this section on employing tools of DMD, POD and Galerkin projection to provide aconsistent way for producing reduced-order models from data.

By collecting snapshots of the velocity and geopotential field and applying the DMD method, aDMD-ROM of the flow is constructed from the DMD basis by writing

w (x, y, t) ≈ wDMD (x, y, t) = Wb (x, y) +

rDMD∑j=1

aj (t)λjϕj (x, y), (52)

where Wb is the centering trajectory, rDMD is the number of DMD basis functions and λj , ϕj (x, y)represent the Ritz eigenvalues of the Koopman operator and the DMD basis functions, respectively.We now replace the velocity w with wDMD in the SWE model (3)-(5) associated with the initialconditions (8), (9), (10), compactly written

∂w∂t (x, y, t) = f (t, w (x, y, t))w (x, y, t0) = w0 (x, y)

(53)



0 5000 100000

5000

0

POD Mode nr.1

0 5000 100000

5000

0

POD Mode nr.2

0 5000 100000

5000

0

POD Mode nr.3

0 5000 100000

5000

0

POD Mode nr.4

0 5000 100000

5000

0

POD Mode nr.5

0 5000 100000

5000

0

POD Mode nr.6

0 5000 100000

5000

0

POD Mode nr.7

0 5000 100000

5000

0

POD Mode nr.8

Figure 6. First eight POD basis functions.

1 2 3 4

01

35

79

1113

1517

0

0.2

0.4

0.6

0.8

1

DMD modes φj

POD modes Φj

MA

C M

atri

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4123456789

1011121314151617

DMD modes φj

POD

mod

es Φ

j

MAC matrix

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10.289970.590070.328110.236020.436050.314050.246070.487620.369150.174880.233380.650190.207930.894440.468860.19544

0.289970.590070.328110.236020.436050.314050.246070.487620.369150.174880.233380.650190.207930.894440.468860.19544

0.311490.658170.403160.2771

0.494340.379510.336020.548010.461150.254290.241040.755410.352040.965840.588910.313380.99251

0.311490.658170.403160.2771

0.494340.379510.336020.548010.461150.254290.241040.755410.352040.965840.588910.313380.992511.0001 1.0001

Figure 7. Modal Assurance Criterion - MAC Matrix between DMD and POD modes.

and then project the resulting equations onto the subspace XDMD =span ϕ1(·), ϕ2(·), ..., ϕrDMD (·) spanned by the DMD basis:⟨

ϕi (·) ,rDMD∑j=1

λjϕj (·) aj (t)

⟩=

⟨ϕi (·) , f

(t,

rDMD∑j=1

λjϕj (·) aj (t)

)⟩, (54)



1800018500

18500190001950020000205002100021500

2150022000

x [km]

y [k

m]

0 2 4 6

x 106

0

2

4x 10

6

05x 10

60

5

x 106

1.5

2

2.5

x 104

x [km]

h0 field

y [km]

0

0

0

0

5050

5050 −50 −50100100−100 −100150150−150

−150200

x [km]

y [k

m]

0 2 4 6

x 106

0

2

4x 10

6

05x 10

60

5

x 106

−500

0

500

x [km]

u0 field

y [km]

0

0

−50 5050

−100 100100

x [km]

y [k

m]

0 2 4 6

x 106

0

2

4x 10

6

05x 10

60

5

x 106

−200

0

200

x [km]

v0 field

y [km]

Figure 8. Initial velocity fields: Geopotential height field for the Grammeltvedt initial condition (first line),streamwise and spanwise velocity fields calculated from the geopotential field by using the geostrophic

approximation (second and third line).

a.

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19500

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20000

2000020500

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2050021000

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Solution of geopotential height field at T=24h

x [km]

y [k

m]

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

b.0 1000 2000 3000 4000 5000 6000

0

500

1000

1500

2000

2500

3000

3500

4000

4500

x [km]

Solution of (u,v) field at T=24h

y [k

m]

Figure 9. a. Solution of geopotential height field at T = 24h; b. Solution of (u, v) field at T = 24h.

⟨ϕi (·) ,

rDMD∑j=1

λjϕj (·) aj (t0)

⟩= ⟨ϕi (·) , w0⟩ , for i = 1..rDMD. (55)



a.

1800018000

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2100021000

21000

2150021500

21500

2200

0

22000

22000

x [km]

y [k

m]

DMD computed geopotential height field at T=24h

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

b.0 1000 2000 3000 4000 5000 6000

0

500

1000

1500

2000

2500

3000

3500

4000

4500

x [km]

DMD computed (u,v) field at T=24h

y [k

m]

Figure 10. a. DMD computed geopotential height field at T = 24h; b. DMD computed (u, v) field atT = 24h.

a.

1800018000

1850018500

18500

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x [km]

y [k

m]

POD computed geopotential height field at T=24h

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

b.0 1000 2000 3000 4000 5000 6000

0

500

1000

1500

2000

2500

3000

3500

4000

4500

x [km]

POD computed (u,v) field at T=24h

y [k

m]

Figure 11. a. POD computed geopotential height field at T = 24h; b. POD computed (u, v) field at T = 24h.

The Galerkin projection gives the DMD-ROM, i.e., a dynamical system for temporal coefficientsaj (t)j=1..rDMD

:

ai (t) =

⟨ϕi (·) , f

(t,

rDMD∑j=1

λjϕj (·) aj (t)

)⟩, (56)

with the initial condition

ai (t0) = ⟨ϕi (·) , w0⟩ , for i = 1..rDMD. (57)

To derive the POD-ROM, we construct the flow from the POD basis by writting

w (x, y, t) ≈ wPOD (x, y, t) = Wb (x, y) +

rPOD∑j=1

bj (t)Φj (x, y), (58)

where Wb is the centering trajectory, rPOD is the number of POD basis functions and Φj (x, y)represents the POD basis functions. We seek for the coefficients bj projecting the SWE equations(53) onto the subspace XPOD = span Φ1(·),Φ2(·), ...,ΦrPOD (·) spanned by the POD basis:⟨

Φi (·) ,rPOD∑j=1

Φj (·) bj (t)

⟩=

⟨Φi (·) , f

(t,

rPOD∑j=1

Φj (·) bj (t)

)⟩, (59)

⟨Φi (·) ,

rPOD∑j=1

Φj (·) bj (t0)

⟩= ⟨Φi (·) , w0⟩ . (60)



Table I. The average relative errors of reduced order models

DMD-ROM POD-ROM

ehDMD = 0.0119 ehPOD = 0.0042euDMD = 0.1770 euPOD = 0.0929evDMD = 0.1534 evPOD = 0.0456

The POD-ROM given by the Galerkin projection reduces to the solution of the following systemof ODEs, for the temporal coefficients bj (t)j=1..rPOD

:

bi (t) =

⟨Φi (·) , f

(t,

rPOD∑j=1

Φj (·) bj (t)

)⟩, (61)

with the initial condition

bi (t0) = ⟨Φi (·) , w0⟩ , for i = 1..rPOD. (62)

The resulting autonomous system has linear and quadratic terms parameterized by cim and cimn

respectively:

ai (t) =

r∑m=1

r∑n=1

cimnam (t) an (t) +

r∑m=1

cimam (t) , i = 1..r. (63)

In the following, we emphasize the performances of the optimized DMD-ROM method and POD-ROM approach for 2D flows in comparison with the numerical solution of the full SWE model.To judge the quality of the DMD/POD reduced order models developed here, an error estimate isprovided. Using the following norms

eDMD =1

Nt + 1

Nt∑j=0

∥∥w0 (x, y, j)− wDMD−ROM (x, y, j)∥∥F

∥w0 (x, y, j)∥F, (64)

ePOD =1

Nt + 1

Nt∑j=0

∥∥w0 (x, y, j)− wPOD−ROM (x, y, j)∥∥F

∥w0 (x, y, j)∥F, (65)

we calculated the average relative errors in terms of Frobenius norm, for the variables ofthe full SWE model w0 = (u, v, h) (x, y, t) and the variables of the reduced ordel modelswDMD−ROM/POD−ROM . The results are presented in Table 1. The maximum error of numericalPOD-ROM solutions is less than the error of numerical DMD-ROM solutions, but the benefit ofemploying the optimized DMD prevails in the case of large time step observations. Although thePOD-ROM provides higher precision, the DMD-ROM method is less expensive with respect tothe numerical implementation costs, i.e. numerical results are obtained for a considerably smallernumber of expansion terms to derive the reduced order model.

The flow energy conservation is used as an additional metric to evaluate the quality of the tworeduced order models. Table 2 presents the total energy of the flow described by the two reducedorder models involved in this study and the absolute errors with respect to flow energy obtainedfrom the full model. The results indicate that both reduced order models will preserve the flow totalenergy.

Figure 12 illustrates the temporal coefficients of the two reduced order models. In Figure 12a,the coefficients of the DMD-ROM model, corresponding to the first four optimal Koopman modesare vizualized. Figure 12b plots the coefficients corresponding to the first ten POD modes of thePOD-ROM mdel.

A comparison of the geopotential height field between the full and DMD/POD models is providedin Figure 13. The geopotential height field computed at time level T = 10h by the two reduced ordermodels exhibits an overall good agreement with that from the full model.



Table II. Energy conserving test

Flow Energy E0 DMD-ROM Flow Energy EDMD POD-ROM Flow Energy EPOD

6.2000e+ 10 6.1800e+ 10 6.2100e+ 10Absolute error:

∣∣E0 − EDMD∣∣ = 1.9565e− 04

∣∣E0 − EPOD∣∣ = 7.4363e− 05

a.0

12

34

0

50

100−2

0

2

Time [s]

DMD−ROM temporal coefficients

aj

b.

Figure 12. a. DMD-ROM computed temporal coefficients; b. POD-ROM computed temporal coefficients.

a.

1800018000

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Solution of geopotential height field at T=10h

x [km]

y [k

m]

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

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4500

b.

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20000

2050020500

20500

2100021000

21000

21500 21500

21500

22000

2200022000

x [km]

y [k

m]

DMD−ROM computed geopotential height field at T=10h

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

c.

1800018000

18000

1850018500

18500

1900019000

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19500

2000020000

20000

2050020500

20500

2100021000

21000

2150021500

21500

2200022000

22000

x [km]

y [k

m]

POD−ROM computed geopotential height field at T=10h

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

Figure 13. Comparison of the geopotential height field between full model and reduced order models: a.Solution of geopotential height field computed at time T = 10h ; b. DMD-ROM solution of geopotentialheight field computed at time T = 10h; c. POD-ROM solution of geopotential height field computed at time

T = 10h.



x [km]

y [k

m]

|h(x,y)−hDMD−ROM(x,y)|

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

1

2

3

4

5

6

7x 10

−4

x [km]

y [k

m]

|h(x,y)−hPOD−ROM(x,y)|

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

0.5

1

1.5

2

x 10−4

Figure 14. Local errors between DMD-ROM, POD-ROM SWE solutions and the full SWE solution at timeT = 10h.

The local error between DMD-ROM, POD-ROM SWE solutions and the full SWE solution attime T = 10h is presented in Figure 14.

The correlation coefficient and the root mean-squared error (RMSE) defined below are used asadditional metrics to evaluate the quality of the two reduced order models:

Ci =

(∥∥w0 (x, y, i) · w (x, y, i)∥∥F

)2∥∥∥(w0 (x, y, i))H · w0 (x, y, i)

∥∥∥F

∥∥∥(w (x, y, i))H · w (x, y, i)

∥∥∥F

, i ∈ 0, . . . Nt − 1 , (66)

RmseiDMD/POD =

√∑ni=1 (∥w0 (x, y, i)− w (x, y, i)∥F )

2

n, (67)

where w0 (x, y, i) means the solution of the SWE model at time Ti, w (x, y, i) represents thecomputed solution of the SWE model at time Ti by means of the reduced order model, (·) representsthe Hermitian inner product and H denotes the conjugate transpose. A comparison of the correlationcoefficient between the full and DMD/POD models is provided in Figure 15. The values of thecorrelation coefficients are greater than 99%, 97%, respectively, and confirm the validity of thetwo reduced order models. To further quantify the quality of the numerical results with the use ofthe DMD/POD reduced order models, the root mean-squared error (RMSE) of results between theDMD/POD models and the full model are plotted in Figure 16. It can be seen that the ROM errordecreases, while the correlation increases for both models, as the time accrues along the simulationwindow. All the methods performed well, leading to RMSE results of order O

(10−2

)and local

errors of order O(10−4

).

7. SUMMARY AND CONCLUSIONS

We have proposed a framework for dynamic mode decomposition of 2D flows, when numerical orexperimental data snapshots are captured with large time steps. Such problems originate for instancefrom meteorology, when a large time step acts like a filter in obtaining the significant Koopmanmodes, therefore the classic dynamic mode decomposition method is not effective. This studywas motivated by the need to further clarify the connection between Koopman modes and PODdynamic modes. We have applied dynamic mode decomposition (DMD) and proper orthogonaldecomposition (POD) to derive reduced-order models of the shallow water equations.

We proposed a new algorithm for extracting the dominant Koopman modes of the flow fieldand we proved that for large time step observables, dynamic mode decomposition of mean-flow-subtracted data and temporal DFT analysis are not equivalent. We emphasized the excellent behavior



Figure 15. Correlation coefficients for the SWE variables: a. DMD-ROM model vs. full SWE model; b.POD-ROM model vs. full SWE model.

Figure 16. RMSE results between the DMD/POD models and the full model: a. RMS of DMD-ROM modelvs. full SWE model; b. RMS of POD-ROM model vs. full SWE model.

of the optimized DMD method in computing the modal decomposition for large time step data andwe proposed a new criterion of selecting the optimal Koopman modes.

We perform a quantitative comparison of the spatial modes computed from the twodecompositions discussed here using the Modal Assurance Criterion (MAC), as a measure ofthe degree of linearity between Koopman and POD modes. This evaluation indicates that theDMD modal decomposition is more efficient than POD decomposition, because the DMD modaldecomposition is achieved with a smaller number of modes.

Additionally, we presented a rigorous error analysis for the ROM models obtained by the classicPOD and the optimized DMD and we compared the relative computational efficiency of above-mentioned ROM methods.

We found a very close agreement between the flow reconstruction computed with the ROMmodels and the solution provided by the high fidelity SWE model. But the benefit of employingthe optimized DMD method prevails in the case of modal decomposition of 2D flows with largetime step observations, when the classic DMD available in literature fails to compute the significantRitz values and Koopman modes.

The similarity between the correlation coefficient between the full and DMD/POD models andthe root mean-squared error (RMSE) of results between the DMD/POD models and the high fidelitymodel certify that the optimized DMD method can be applied successfully in parallel with the PODdecomposition to obtain reduced order models of potential relevance.

The question whether the proposed DMD methodology is a viable alternative to the linear stabilityanalysis available in the community, for hydrodynamic stability investigation of flows described bylinearized dynamical systems is a subject which will be addressed carefully in our future work. The



methodology presented here offers the main advantage of deriving a reduced order model capableto provide a variety of information describing the behavior of 2D flows. A future extension of thisresearch will address an efficient numerical approach for modal decomposition of swirling flows,where the reduced order mathematical model will imply more sophisticated relations at domainboundaries.

Projection based methods presented in this paper lead to reduced order models with dramaticallyreduced numbers of equations and unknowns. However, for parametrically varying problems or formodeling problems with strong nonlinearities, the cost of evaluating the reduced order models stilldepends on the size of the full order model and therefore is still expensive. The Discrete EmpiricalInterpolation Method (DEIM) described in detail in [64] further approximates the nonlinearity inthe projection based reduced order strategies. The application of a DEIM-ROM strategy for FEMmodels combined with the methods proposed in this paper represents a subject that we will furtheraddress in our studies. The resulting DEIM-DMD/POD-ROM will be evaluated efficiently at a costthat is independent of the size of the original problem.

ACKNOWLEDGEMENT

Prof. D.A. Bistrian acknowledges the support of strategic grant POSDRU/159/1.5/S/137070 (2014) of theMinistry of National Education, Romania, co-financed by the European Social Fund-Investing in People,within the Sectorial Operational Program Human Resources Development 2007-2013.Prof. I.M. Navon acknowledges the support of NSF grant ATM-0931198.

REFERENCES

1. Schmid PJ, Henningson DS. Stability and Transition in Shear Flows. First edition, Springer, 2001.2. Luchtenburg DM, Rowley CW. Model reduction using snapshot-based realizations. American Physical Society,

64th Annual Meeting of the APS Division of Fluid Dynamics, 2011; H19.004.3. Liberge E, Hamdouni A. Reduced order modelling method via proper orthogonal decomposition (POD) for flow

around an oscillating cylinder. Journal of Fluids and Structures 2010; 26:292311.4. Wang Z, Akhtar I, Borggaard J, Iliescu T. Proper orthogonal decomposition closure models for turbulent flows: A

numerical comparison. Computer Methods in Applied Mechanics and Engineering 2012; 237-240:10–26.5. Abramov RV, Majda AJ. Low-frequency climate response of quasigeostrophic wind-driven ocean circulation. J.

Phys. Oceanogr. 2012; 42:243260.6. Osth J, Noack BR, Krajnovic S, Barros D, Boree J. On the need for a nonlinear subscale turbulence term in POD

models as exemplified for a high-Reynolds-number flow over an Ahmed body. J. Fluid Mech. 2014; 747:518–544.7. Mariani R, Dessi D. Analysis of the global bending modes of a floating structure using the proper orthogonal

decomposition. Journal of Fluids and Structures 2012; 28:115134.8. Buljak V, Maier G. Proper orthogonal decomposition and radial basis functions in material characterization based

on instrumented indentation. Engineering Structures 2011; 33(492501).9. Du J, Fang F, Pain CC, Navon IM, Zhu J, Ham D. POD reduced order unstructured mesh modelling applied to 2D

and 3D fluid flow. Computers and Mathematics with Applications 2013; 65:362–379.10. Fang F, Pain CC, Navon IM, Piggott MD, Gorman GJ, Allison P, Goddard AJH. Reduced order modelling of an

adaptive mesh ocean model. International Journal for Numerical Methods in Fluids 2009; 59(8):827–851.11. Stefanescu R, Navon IM. POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations

model. Journal of Computational Physics 2013; 237:95–114.12. Winton C, Pettway J, Kelley CT, Howington S, Eslinger OJ. Application of proper orthogonal decomposition (POD)

to inverse problems in saturated groundwater flow. Advances in Water Resources 2011; 34:1519–1526.13. Chen X, Navon IM, Fang F. A dual-weighted trust-region adaptive POD 4D-VAR applied to a finite-element

shallow-water equations model. International Journal for Numerical Methods in Fluids 2011; 65:250–541.14. Chen X, Akella S, Navon IM. A dual-weighted trust-region adaptive POD 4-D VAR applied to a finite-volume

shallow water equations model on the sphere. International Journal for Numerical Methods in Fluids 2012; 68:377–402.

15. Cao Y, Zhu J, Navon I, Luo Z. A reduced order approach to four-dimensional variational data assimilation usingproper orthogonal decomposition. International Journal for Numerical Methods in Fluids 2007; 53(10):1571–1583.

16. Ilak M, Schlatter P, Bagheri S, Henningson DS. Bifurcation and stability analysis of a jet in cross-flow: onset ofglobal instability at a low velocity ratio. J. Fluid Mech. 2012; 696:94–121.

17. Moore C. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEETrans. Automat. Contr. 1981; 26:17–32.

18. Rowley CW. Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcat. Chaos2005; 15(3):997–1013.

19. Bagheri S. Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 2013; 726:596–623.20. Mezic I. Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 2013;

45(1):357–378.



21. Rowley CW, Mezic I, Bagheri S, Schlatter P, Henningson DS. Spectral analysis of nonlinear flows. J. Fluid Mech.2009; 641:115–127.

22. Koopman B. Hamiltonian systems and transformations in Hilbert space. Proc. Nat. Acad. Sci. 1931; 17:315–318.23. Semeraro O, Bellani G, Lundell F. Analysis of time-resolved PIV measurements of a confined turbulent jet using

POD and Koopman modes. Exp. Fluids 2012; 53:1203–1220.24. Soucasse L, Riviere P, Soufiani A, Xin S, Quere PL. Transitional regimes of natural convection in a differentially

heated cubical cavity under the effects of wall and molecular gas radiation. Physics of Fluids 2014; 26:024 105.25. Frederich O, Luchtenburg DM. Modal analysis of complex turbulent flow. The 7th International Symposium on

Turbulence and Shear Flow Phenomena (TSFP-7), Ottawa, Canada,, 2011.26. Muld TW, Efraimsson G, Henningson DS. Flow structures around a high-speed train extracted using proper

orthogonal decomposition and dynamic mode decomposition. Computers and Fluids 2012; 57:87–97.27. Hossen MJ, Navon IM, Fang F. A penalized four-dimensional variational data assimilation method for reducing

forecast error related to adaptive observations. International Journal for Numerical Methods in Fluids 2012;70:1207–1220.

28. Daescu DN, Navon IM. Adaptive observations in the context of 4D-VAR data assimilation. Meteorology andAtmospheric Physics 2004; 85:205–226.

29. San O, Staples A, Wang Z, Iliescu T. Approximate deconvolution large eddy simulation of a barotropic oceancirculation model. Ocean Modelling 2011; 40(2):120–132.

30. Iliescu T, Fischer PF. Large eddy simulation of turbulent channel flows by the rational large eddy simulation model.Physics of Fluids 2003; 15:3036.

31. Vreugdenhil CB. Numerical Methods for Shallow Water Flow. Kluwer Academic Publishers, 1994.32. Navon IM. FEUDX: A two-stage, high accuracy, finite-element Fortran program for solving shallow-water

equations. Computers and Geosciences 1987; 13(3):255–285.33. Bryden IG, Couch SJ, Owen A, Melville G. Tidal current resource assessment. Proc. IMechE Part A: J. Power and

Energy 2007; 221:125–135.34. Sportisse B, Djouad R. Reduction of chemical kinetics in air pollution modeling. Journal of Computational Physics

2000; 164:354–376.35. Koutitus C. Mathematics models in coastal engineering. Pentech Press, London, 1988.36. Grammeltvedt A. A survey of finite-diference schemes for the primitive equations for a barotropic fluid. Monthly

Weather Review 1969; 97(5):384–404.37. Cullen MJP, Morton KW. Analysis of evolutionary error in finite-element and other methods. J. Computational

Physics 1980; 34:245–267.38. Navon IM. Finite-element simulation of the shallow-water equations model on a limited area domain. Applied

Mathematical Modeling 1979; 3:337–348.39. Fang F, Pain CC, Navon IM, Cacuci DG, Chen X. The independent set perturbation method for efficient computation

of sensitivities with applications to data assimilation and a finite element shallow water model. Computers andFluids 2013; 76:33–49.

40. Navon IM, Phua PKH, Ramamurthy M. Vectorization of conjugate-gradient methods for large-scale minimizationin meteorology. Journal of Optimization Theory and Applications 1990; 66(1):71–93.

41. Navon IM, DeVilliers R, GUSTAF: a quasi-Newton nonlinear ADI Fortran IV program for solving the shallow-water equations with augmented Lagrangians. Computers and Geosciences 1986; 12:151–173.

42. Navon I. A Numerov-Galerkin technique applied to a finite-element shallow water equations model with enforcedconservation of integral invariants and selective lumping. Journal of Computational Physics 1983; 52:313–339.

43. Arakawa A, Hsu YJG. Energy conserving and potential-enstrophy dissipating schemes for the shallow waterequations. Monthly Weather Review 1990; 118:1960–1969.

44. Arakawa A. Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I. Journal of Computational Physics 1997; 135:103–114.

45. Fiedler M. A note on companion matrices. Linear Algebra and its Applications 2003; 372:325–331.46. Rowley CW, Mezic I, Bagheri S, Schlatter P, Henningson DS. Reduced-order models for flow control: balanced

models and Koopman modes. Seventh IUTAM Symposium on Laminar-Turbulent Transition, IUTAM Bookseries,vol. 18, 2010; 43–50.

47. Schmid P. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 2010; 656:5–28.48. Bagheri S. Computational hydrodynamic stability and flow control based on spectral analysis of linear operators.

Archives of Computational Methods in Engineering 2012; 19(3):341–379.49. Golub G, van Loan CF. Matrix Cotnputations, Third Edition. The Johns Hopkins University Press, 1996.50. Schmid PJ, Violato D, Scarano F. Decomposition of Time-Resolved Tomographic PIV. Springer-Verlag, 2012.51. Noack BR, Afanasiev K, Morzynski M, Tadmor G, Thiele F. A hierarchy of low-dimensional models for the

transient and post-transient cylinder wake. J. Fluid Mech. 2003; 497:335–363.52. Chen KK, Tu JH, Rowley CW. Variants of dynamic mode decomposition: boundary condition, Koopman and

Fourier analyses. J Nonlinear Sci 2012; 22:887–915.53. Cammilleri A, Gueniat F, Carlier J, Pastur L, Memin E, Lusseyran F, Artana G. POD-spectral decomposition for

fluid flow analysis and model reduction. Theoretical and Computational Fluid Dynamics 2013; 27:787–815.54. Holmes P, Lumley JL, Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge

Univ. Press., 1996.55. Sirovich L. Turbulence and the dynamics of coherent structures, parts I-III. Q. Appl. Math. 1987; 3:561–590.56. Belson B, Tu JH, Rowley CW. Algorithm 945: Modred - a parallelized model reduction library. ACM Transactions

on Mathematical Software 2014; 40(4):Article 30.57. Jovanovic MR, Schmid PJ, Nichols JW. Low-rank and sparse dynamic mode decomposition. Center for Turbulence

Research Annual Research Briefs 2012; :139–152.58. Schmid PJ, Meyer KE, Pust O. Dynamic mode decomposition and proper orthogonal decomposition of flow in a

lid-driven cylindrical cavity. 8th International Symposium on Particle Image Velocimetry - PIV09, 2009.



59. Balajewicz MJ, Dowell EH, Noack BR. Low-dimensional modelling of high-Reynolds-number shear flowsincorporating constraints from the Navier-Stokes equation. J. Fluid Mech. 2013; 729:285–308.

60. Noack BR, Morzynski M, Tadmor G. Reduced-Order Modelling for Flow Control. Springer, 2011.61. Nyquist H. Certain topics in telegraph transmission theory. Transactions A. I. E. E. 1928; :617–644.62. Brown DL, Allemang RJ, Zimmerman RD, Mergeay M. Parameter estimation techniques for modal analysis. SAE

Transactions 1979; :SAE Paper Number 790 221.63. Mitchell LD. Increasing the sensitivity of the modal assurance criteria to small mode shape changes. The IMAC

Proceedings, International Modal Analysis Conference, 2001; 64–69.64. Antil H, Heinkenschloss M, Sorensen D. Reduced Order Methods for Modelng and Computational Reduction,

chap. Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear andParametric Systems. Springer International Publishing Switzerland, 2014; 101–136.


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