Visualization and Selection of Dynamic Mode Decomposition Components for Unsteady Flow
Tim Krake, Stefan Reinhardt, Marcel Hlawatsch, Bernhard Eberhardt, Daniel Weiskopf
VVisualization and Selection of Dynamic Mode DecompositionComponents for Unsteady Flow
Tim Krake , , Stefan Reinhardt , , Marcel Hlawatsch , Bernhard Eberhardt , and Daniel Weiskopf VISUS, University of Stuttgart, Germany Hochschule der Medien, Germany.
Fig. 1: An overview of our proposed Dynamic Mode Decomposition (DMD) approach for the investigation of unsteady flow. Forthe data analysis process, we introduce improved DMD components and visualizations that respect the spatio-temporal characterof DMD and, thus, describe the flow appropriately. Moreover, two new clustering approaches allow the user to aggregate flowcomponents that segment the flow into physically relevant sections. These can therefore be used for the selection of DMDcomponents. The combination of these new techniques enables new insights with DMD, as the resulting temporal developmentssuggest.
Abstract — Dynamic Mode Decomposition (DMD) is a data-driven and model-free decomposition technique. It is suitable for revealing spatio-temporal features of both numerically and experimentally acquired data. Conceptually, DMD performs a low-dimensional spectral decomposition ofthe data into the following components: The modes, called DMD modes, encode the spatial contribution of the decomposition, whereas the DMDamplitudes specify their impact. Each associated eigenvalue, referred to as DMD eigenvalue, characterizes the frequency and growth rate of the DMDmode. In this paper, we demonstrate how the components of DMD can be utilized to obtain temporal and spatial information from time-dependentflow fields. We begin with the theoretical background of DMD and its application to unsteady flow. Next, we examine the conventional process withDMD mathematically and put it in relationship to the discrete Fourier transform. Our analysis shows that the current use of DMD components hasseveral drawbacks. To resolve these problems we adjust the components and provide new and meaningful insights into the decomposition: We showthat our improved components describe the flow more adequately. Moreover, we remove redundancies in the decomposition and clarify the interplaybetween components, allowing users to understand the impact of components. These new representations ,which respect the spatio-temporalcharacter of DMD, enable two clustering methods that segment the flow into physically relevant sections and can therefore be used for the selectionof DMD components. With a number of typical examples, we demonstrate that the combination of these techniques allow new insights with DMDfor unsteady flow.
Keywords —Dynamic Mode Decomposition, spectral decomposition, visualization of components, selection of components
NTRODUCTION
Dynamic Mode Decomposition (DMD) is a data-driven and model-free technique to decompose complex flows into fundamental spectralcomponents. These components correspond to spatio-temporal featuresthat characterize periodicity, damping, (temporal) segmentation, andlong-time behavior of the flow. Basically, the algorithm results in threecomponents: DMD modes, amplitudes, and eigenvalues. Whereasthe modes represent spatial contribution to the flow and the ampli- tudes specify their impact, each associated eigenvalue characterizes thetemporal development. The objective of this paper is to gain a betterunderstanding of these components such that a more insightful analysisis achieved. Moreover, as the visualization community has not paidmuch attention to DMD so far, we want to make DMD more accessiblefor both its users and the visualization research community.DMD is supposed to identify spatial patterns associated with fre-quencies and growth rates that determine the behavior of a system. So a r X i v : . [ phy s i c s . f l u - dyn ] D ec ar, the investigation via DMD has been performed by the study ofindividual DMD components. In addition, spatial and temporal proper-ties of components are treated independently. Since DMD is based onthe interplay of spatio-temporal components, this traditional analysisprocess is insufficient. It has several negative implications: First, therelevance of the components to the entire system is not clearly specified.Thus, an appropriate selection of components (for the analysis process)is not possible. Second, the existing DMD visualizations could bemisleading as the mutual dependencies of the components are not takeninto account.To address these problems, we focus on the representation of thecomponents and their visualization. Our approach is guided by theneeds of unsteady flow but could be extended to general time-dependentgrid-based data. Figure 1 illustrates the analysis process following ourapproach. Our contributions can be summarized as follows:• Conceptual contribution: We clarify drawbacks of the traditionalDMD components and provide a new perspective on DMD basedon a comparison with the discrete Fourier transform (DFT).• New visualizations: We improve DMD components and theirrepresentations using novel visualizations that respect the spatio-temporal character of DMD.• Data analysis contribution: We introduce two clustering ap-proaches to aggregate components that segment the flow intophysically relevant sections and can therefore be used for theselection of DMD components.We also discuss the mathematical foundation of DMD by providing aderivation and a specific formulation of DMD. Moreover, with artificialand simulated examples, we show that our approach is able to identifycharacteristic features of unsteady flow fields. ELATED W ORK
The visualization and analysis of unsteady flow are a challenging re-search topic. A variety of decomposition techniques has been proposedto extract different kind of features from a flow. The characteristics of afeature strongly depend on the method and are often difficult to define.In the context of unsteady flow, we distinguish between two types ofdecomposition techniques.The first type directly operates on the vector field, such as theHelmholtz Hodge decomposition (HHD) [5] or the Morse decompo-sition [9]. The HHD decomposes the vector field into two spatialcomponents that are divergence- and curl-free, respectively. Recentwork [22] deals with an extension of the HHD based on Fourier trans-formation. Wiebel et al. [34] propose a similar decomposition into apotential flow from the boundary and a localized flow to extract fea-tures. Rojo and G¨unter’s [23] splitting method decomposes the flowinto a steady and ambient part, enabling the description of the motionof topological elements and feature curves. However, these decom-position techniques do not encode temporal patterns like DMD. TheMorse decomposition divides a vector field into disjoint invariant sets,called Morse sets. The connection of those Morse sets is illustratedby a directed graph giving an overview of the topological skeleton ofthe flow field. While this approach highlights only spatial relations ofthe decomposition, we address spatio-temporal patterns. Bujack et al.’sstate-of-the-art report [8] interprets physical features of several decom-position techniques of that type in terms of mathematical properties.We follow a similar approach, however, DMD is of another type andtherefore extracts different features.The second type of decompositions makes use of temporal coherenceby performing the decomposition on the full time series, instead of con-sidering each step individually. Besides DMD, Principal ComponentAnalysis (PCA) [4, 15], also kwown as Proper Orthogonal Decomposi-tion (POD), is a technique of this type. PCA hierarchically decomposesthe data into an orthogonal basis of spatially correlated modes, calledprincipal components (or POD modes), modulated by appropriate ran-dom time-coefficients. Therefore, the dynamic information is oftenneglected and particular emphasis is put on the spatial components.
Algorithm 1
Exact Dynamic Mode Decomposition function DMD( x ,..., x m ) X = (cid:2) x ... x m − (cid:3) , Y = (cid:2) x ... x m (cid:3) Calculate the reduced SVD X = U Σ V ∗ with rank ( X ) = r . Calculate S = U ∗ YV Σ − . Calculate λ ,..., λ r and v ,..., v r of S . for λ i (cid:54) = do ϑ i = λ i YV Σ − v i end for Λ = diag ( λ , λ ,..., λ r ) with λ , λ ,..., λ r (cid:54) = Θ = (cid:2) ϑ ϑ ... ϑ r (cid:3) Calculate a = Λ − Θ + x with a = ( a ,..., a r ) Calculate if exist c = − ∑ r j = λ j ∏ k (cid:54) = j λ j − λ k return λ j , a j , ϑ j , c end function Pobitzer et al’s work [21] deals with an extension of POD based onfeature detectors. The visualization of POD components and the addi-tional use of feature detectors do not take spatio-temporal properties ofthe decomposition into account, which is again the main difference toour DMD approach.DMD was introduced by Schmid and Sesterhenn [28]. Schmid [29]improved the DMD algorithm by using a reduced singular value decom-position (SVD). Tu et al. [32] formulated the latest version of DMD,called exact DMD. As described in the introduction, the componentsof DMD are used separately and important spatio-temporal relationsare not taken into account, especially not for the visualization. In thisway, DMD has been applied on diverse flow setups, e.g., the analysis ofwake flows [3,25,32,35], cavity flows [1,16], mixing layer flow [26,27],and jet flows [24, 30]. In the context of Lagrangian coherent structures,the interaction of DMD and Finite Time Lyapunov Exponent (FTLE)was considered to provide a feature-based description of the entireflow field [2, 19, 33]. Kou and Zhang [12] propose a new criterionfor the selection of dominant modes. However, this approach usestraditional DMD components and is based on the energy over the fulltime. Regarding visualization and computer graphics, DMD was usedfor background/foreground video separation [11], background mod-eling [10, 20], edge detection [6], and the visualization of large-scalepower systems [17]. However, all above mentioned publications use thetraditional DMD components. With our novel visualizations of the im-proved components, we show that our techniques overcome drawbacksof the conventional DMD approach for visual flow analysis.
ATHEMATICAL F OUNDATION
We concentrate on time-dependent flow fields defined on a grid andsampled uniformly in time. More precisely, for a 2D velocity fieldon a grid with N points at time t k , the data is characterized by u ( t k ) , v ( t k ) ,..., u N ( t k ) , v N ( t k ) ∈ R , where u and v represent the veloc-ity components. For DMD, the snapshots x ,..., x m ∈ R N are givenby | | | x x ... x m | | | = u ( t ) u ( t ) u ( t m ) v ( t ) v ( t ) ... v ( t m ) ... ... ... ... u N ( t ) u N ( t ) u N ( t m ) v N ( t ) v N ( t ) ... v N ( t m ) . A 3D flow field is defined analogously with three instead of two com-ponents.Before formulating the algorithm of DMD, we summarize the con-cepts of DMD [13,14] and compare them with DFT. These aspects helpunderstand the visualization techniques and their principles.
DMD performs a spectral decomposition on arbitrary data. Therefore,we generally consider complex-valued snapshots x , x ,..., x m ∈ C n , i m e P o w e r F r e q u e n c y T i m e D o m a i n T i m e P o w e r F r e q u e n c y T i m e D o m a i n C o m p l e x F r e q u e n c y D o m a i n R e a l F r e q u e n c y D o m a i n DMD DFT
Fig. 2: A comparison of DMD (left) with DFT (right). Both methods lead to a structurally similar decomposition. However, DMD computes itsown complex frequencies (depending on a frequency and growth rate) by the eigenvalues of a least-squares fit matrix, whereas DFT uses fixedreal frequencies given by the (complex-valued) roots of unity. The components of DMD can therefore converge or diverge over time, whichcreates new opportunities for investigating data as the simple complex frequency domain highlights.which are usually uniformly sampled in time. The size of a snapshot istypically substantially greater than the number of snapshots, i.e., n (cid:29) m .For the snapshots, we consider the following minimization problem:min A ∈ C n × n m − ∑ j = (cid:107) Ax j − x j + (cid:107) , (1)where (cid:107)·(cid:107) denotes the Euclidean norm. In other words, we search amatrix A that optimally connects the subsequent snapshots in a leastsquares sense. The idea of DMD is to calculate a low-dimensionalrepresentation of A and to perform an eigenvalue decomposition onit to detect frequency patterns in the data. To this end, we insert thesnapshots as column vectors into the following two matrices: X = | | x ... x m − | | , Y = | | x ... x m | | . (2)The optimization problem in Equation 1 can now be rewritten asmin A ∈ C n × n (cid:107) AX − Y (cid:107) F , where (cid:107) M (cid:107) F : = ( ∑ ni = ∑ mj = | m i j | ) / denotesthe Frobenius norm. Consequently, a best-fit matrix is explicitly givenby A = Y X + ∈ C n × n , (3)where X + is the Moore-Penrose pseudoinverse, a generalized inverse[18] of the non-square matrix X . Note that A is a (large) matrix of size n × n . We assume that A is diagonalizable (which is almost alwaysthe case, if n (cid:29) m ). This means that there exists an invertible matrix V = (cid:2) v v ... v n (cid:3) ∈ C n × n consisting of eigenvectors, as well as adiagonal matrix Λ = diag ( λ , λ ,..., λ n ) ∈ C n × n with correspondingeigenvalues, such that V − AV = Λ . Therefore, we can approximate the k th snapshot by x k ≈ A k x = V Λ k V − x = n ∑ j = b j λ kj v j , (4)where b = (cid:0) b b ... b n (cid:1) T = V − x . Note that entries of b arecoefficients of the linear combination of x in the eigenvector basis.Given that n ≥ m , the rank of A cannot be higher than m due to thedimension restriction and, hence, the number of non-zero eigenvaluesis at most m . Consequently, the dynamic behavior will be capturedby m summands, consisting of the triple ( b j , λ j , v j ) ∈ C × C × C n for j = , ,..., m . DMD calculates exactly these (non-zero) eigenvaluesand eigenvectors referred to as DMD eigenvalues and DMD modes,respectively. For simplicity, we denote them as eigenvalues and modesthroughout the rest of the paper. The coefficients b ,..., b m in the de-composition of Equation 4 need to be computed by DMD in a differentway. We call them DMD amplitudes. However, as the modes usuallydo not form a basis, an error will occur in the reconstruction of the firstsnapshot x . Algorithm 1 shows the DMD method derived from the previous section.However, it differs in the calculation of amplitudes from the standardliterature. The new definition of amplitudes, which uses the secondsnapshot instead of the first one for the reconstruction, was introducedby Krake et al. [13]. We make use of this new formulation by improvingthe representation of DMD components and integrating them to arriveat novel and more adequate DMD visualizations.In Algorithm 1, a triple ( a j , λ j , ϑ j ) similar to the one in Equation 4is determined without explicitly computing A . This is achieved byusing a reduced SVD applied to the data-matrix X , which results intwo unitary matrices U ∈ C n × r and V ∈ C m × r (which are real-valuedfor real-valued data) as well as a real-valued diagonal matrix Σ ∈ R r × r with r = rank ( X ) (lines 2–3). The low-dimensional representation of A is given by (line 4) S = U ∗ AU = U ∗ YV Σ − . (5)Next, we compute the eigenvalues λ j and eigenvectors v j of S for j = ,..., r (line 5). Then, we transform the eigenvectors with non-zero eigenvalues into modes ϑ j (lines 6–8). Finally, the amplitudes a j and the error scaling c are calculated (lines 9–12). By this choiceof amplitudes, Krake et al. [13] proved the following reconstructionproperty: x = m ∑ j = a j ϑ j + c · q , x k = m ∑ j = λ kj a j ϑ j , (6)for k = ,... , m and q = x m − XX + x m . This property holds when both x ,... , x m − and x ,..., x m are linearly independent and λ ,..., λ m aredistinct (in this case r = r = m ). The assertion states that DMD inheritsthe property of Equation 4 with appropriate coefficients. This versionof DMD captures the entire system by providing a structured spectraldecomposition into temporal and spatial aspects. Thus, we are able toclarify the impact of (aggregated) components precisely. Moreover, theinterplay of the eigenvalues, modes, and amplitudes can be interpretedin a new and clearer way. Based on these insights, we create novelvisualizations that respect the spatio-temporal character of DMD. Discrete Fourier transform (DFT) is a well-understood tool to analyzetime-dependent data. It is able to extract frequency-based features likeperiodicity. Since DFT and DMD lead to a structurally similar decom-position of data, it is possible to translate properties and proceduresfrom DFT to DMD. More precisely, DFT yields x k = m ∑ j = µ kj ˆ x j , (7)where µ j = e π jm + are the roots of unity and ˆ x is the Fourier transform.Thus, the DFT converts 1D snapshots x k into complex numbers ˆ x j ig. 3: Traditional visualization of a 2D mode ϑ j with arrow glyphsand color-coded velocity magnitude (blue = low, red = high).(for vector-valued data x k into complex vectors ˆ x j ) each dependingon a root of unity µ j . Using the exponential form, i.e., µ j = e i ϕ j , weobserve that the components µ j have magnitude 1. Thus, they are onlydetermined by a real frequency f j = ϕ j π . This leads to a real frequencydomain representation as shown in Figure 2 on the right side, where themagnitude of a summand µ kj ˆ x j does not change over time.In contrast, DMD computes the triplets: eigenvalues λ j , modes ϑ j ,and amplitudes a j . A direct comparison of the components is difficultsince the interplay of DMD components is more complex than forDFT. Nonetheless, we can bring DMD and DFT in accordance usingEquations 6 and 7. In the decompositions, the temporal componentsare given by λ j and µ j , respectively. Each set of temporal componentscharacterize a decomposition entirely as the spatial contributions, i.e.,the Fourier transformed vectors ˆ x j or the scaled modes a j ϑ j , are simplyfitted to those in a unique way. Therefore, the decompositions onlydiffer in the choice of temporal components. In fact, the eigenvalues λ j computed by DMD are variable, whereas the DFT uses roots of unity µ j . Hence, the eigenvalues λ j = r j e i ϕ j are characterized by both afrequency f j = ϕ j π and a magnitude r j , or, in other words, by a complexfrequency. Thus, DMD produces a complex frequency representationof the data as illustrated in Figure 2 on the left side.The DMD procedure can be summarized as a two-stage method:First, DMD computes appropriate complex frequencies based on thedata. Then, the data is transformed into complex numbers (for vector-valued data into complex vectors) that depend on those complex fre-quencies. From a certain point of view, DMD can thus be seen asan extension of DFT. This opens up new possibilities to extract com-plex frequency-based features, like periodicity, damping, and temporalsegmentation. In Figure 2, we observe that DMD needs three (non-vanishing) complex frequencies to reconstruct the signal, whereas DFTa number of real frequencies. In a more complex scenario, the challengeis to find relevant DMD components. This task is more complicatedthan in the case of DFT, since the interplay of eigenvalues λ j , modes ϑ j , and amplitudes a j needs to be taken into account in order to respectthe spatio-temporal character of DMD.Starting with the eigenvalues λ j (complex frequencies), these shouldbe used to highlight the impact of modes over time. Previously, themodes ϑ should be adjusted by their amplitudes a j according toDFT such that the scaled modes a j ϑ j (analogously to the Fourier-transformed vectors) characterize the spatial contribution and the im-pact. Based on these observations, we provide improved componentsand new visualizations clarifying the interplay of components. More-over, a cluster method is developed on the principles of DFT (aggre-gation of harmonics) that is used for the selection of relevant DMDcomponents. NVESTIGATION A PPROACH
In this section, we first describe the conventional DMD approach withthe traditional DMD components and visualizations by highlightingtheir benefits and drawbacks. Then, we show how these componentscan be improved and visualized in a novel way to resolve these issues.On this basis, we present two clustering methods that segment the flowinto physically relevant sections. Finally, an approach for the selectionof DMD components is proposed.
In the following, we point out the pros and cons of the conventionalDMD approach. DMD computes the following triplets: modes ϑ j , Fig. 4: The traditional visualization of the amplitudes (left) and eigen-values (right) for the 2D Karman vortex street. The absolute valuesof amplitudes | a j | are visualized in a bar diagram that is sorted by thefrequency of their corresponding eigenvalues. The visualization neithertakes temporal aspects nor connections to the eigenvalue visualizationinto account. The eigenvalues are represented in the complex planeencoded by their corresponding amplitudes, where light gray indicatesinsignificant components. For real-valued data (which is the case forflow data), the eigenvalues occur in complex conjugate pairs. Hence, re-dundant information is visualized. Moreover, the radial representationhas several drawbacks for the identification of patterns.amplitudes a j , and eigenvalues λ j . Traditionally, the three groups ofcomponents are mainly considered and visualized separately. DMD Modes
The modes represent the spatial contribution to theflow field and are supposed to display local and global features, likesymmetry, mixing, transient response, and long-time behavior. Eachentry of a vector-valued mode ϑ j corresponds to a spatial location ofthe velocity field as shown in Figure 3. For the investigation with DMD,traditionally, the modes are considered individually. However, as DMDis based on the superposition principle, modes that are inspected mustbe selected carefully. Otherwise, the validity of spatial features is notensured. Moreover, the modes are complex-valued and therefore mostlyvisualized by their real and imaginary part such that an interpretationis even more complicated. A noteworthy mode is the one with corre-sponding eigenvalue λ j =
1. If it exists, it represents the constant flow,often called time-averaged flow, which does not change over time.
DMD Amplitudes
The amplitude a j determines the influence ofthe mode ϑ j to the flow. As mentioned before, the amplitudes aretraditionally computed by a = Θ + x instead of a = Θ + x (compareAlgorithm 1) such that a reconstruction like in Equation 6 does nothold. These complex numbers are then visualized by their absolutevalue, i.e., | a j | , in a bar diagram sorted by the frequency of theircorresponding eigenvalues as illustrated in Figure 4 on the left. Thevisualization gives an overview of the distribution of modes’ influence.On that basis, the selection of relevant modes is performed. Moreprecisely, a choice of the k -th most dominant modes is made by the k highest absolute values of amplitudes. This approach does not taketemporal patterns into account, since the amplitudes only preciselyreflect the influence of modes at time t . Additionally, the influence isnot precisely exact for the traditional amplitudes and the correspondingmodes are not necessarily normalized. Therefore, a misleading choiceof modes could be made that do not represent any features of theflow, especially the ones that evolve over time. Moreover, redundantcomponents are highlighted in Figure 4 (as we will prove later) thatmay lead to unnecessary selection and visualization of modes. DMD Eigenvalues
An eigenvalue of a corresponding mode de-scribes its temporal behavior. The position within the complex planeprovides information about frequency and growth rate. These quantitiesare given by their arguments and magnitudes, respectively. Usually, theeigenvalues are (gray) scaled by the absolute value of their correspond-ing amplitudes, as illustrated in Figure 4 on the right. For real-valueddata, the eigenvalues occur in complex conjugate pairs, i.e., if λ = r e i ϕ is an eigenvalue, then λ = r e − i ϕ is an eigenvalue, too. Therefore, thetraditional visualization in the complex plane, as shown in Figure 4 onthe right, displays redundant information.ig. 5: Our proposed visualizations of the eigenvalues (bottom) andthe dominance structure (top) for the 2D Karman vortex street. Thenon-redundant eigenvalues are represented by their frequencies (argu-ments) and growth rates (magnitudes) and are grayscaled by the normsof their associated scaled modes, where light gray indicates insignif-icant components. The dominance structure visualizes the influenceof non-redundant scaled modes. For this, the norms of non-redundantscaled modes are illustrated in a bar diagram that is sorted by the fre-quencies of their corresponding eigenvalues. In addition, the temporaldevelopment of each scaled mode at specific time steps is integratedinto the representation.Using the exponential form λ = r e i ϕ with frequency f = ϕ π andmagnitude r as well as Equation 6, the eigenvalues can be categorizedin the following way:• r j <
1: These components describe transient responses, becausethe potentiation of the eigenvalues will cause the component tovanish.• r j >
1: Such components show contrary behavior: due to thepotentiation of the eigenvalue, the components will grow anddiverge.• r j =
1: In this case, potentiation of the eigenvalue will causea rotation on the unit circle with a specific frequency. Thesecomponents characterize the steady state.Even though the categorization into these three cases is possible andthe interpretation of eigenvalue potentiation is more accessible (seeFigure 4), the detection of recurring and salient patterns suffer from theradial representation.
In this subsection, we present an improvement to the conventionalDMD components and visualizations that resolves the issues explainedin the previous subsection.
DMD Eigenvalues
For real-valued data (as given in many appli-cations), the eigenvalues occur in complex conjugate pairs (compareFigure 4 on the left). This well known fact follows from a real-valuedSVD resulting in the real-valued matrix S (see Equation 5 or appendix).Therefore, the representation of the eigenvalues can be restricted tothe upper half plane (complex numbers with non-negative imaginarypart), which hides redundant components and leads to a clearer view.Furthermore, the visualization in the complex plane has some disadvan-tages as well. It is difficult to detect recurring and salient patterns dueto the radial representation of the eigenvalues. Therefore, we proposea characterization with more emphasis on the frequency and growthrate. To this end, we remove the redundant eigenvalues and plot theremaining ones in a coordinate system with axes set by argument and Fig. 6: The traditional DMD mode to the eigenvalue λ = Scaled DMD Modes
The DFT can be linked to DMD becauseDMD can be seen as a two-stage method where eigenvalues computedfirst and the modes and amplitudes are fitted subsequently to those.The Fourier-transformed vectors are the counterpart to the modes ϑ j multiplied with their amplitudes a j (see Section 3.3). Thus, we pro-pose scaling the modes for the analysis and denoting these new objects a j ϑ j as scaled DMD modes. Due to this combination, the new repre-sentation contains both the spatial contribution to the decompositionand the influence to the system. Before we discuss these two aspects,some advantageous mathematical properties of the representation arepresented.One important property of the scaled modes is that they occur incomplex conjugate pairs, i.e., the scaled modes of a complex conjugatepair of eigenvalues λ , λ are given by a ϑ , a ϑ . A detailed mathematicalproof of this fact can be found in the appendix. Hence, the superpositionof these two scaled modes can be expressed by twice the real part, since a ϑ + a ϑ = ℜ ( a ϑ ) . Furthermore, the spatio-temporal development isgiven by λ k a ϑ + λ k a ϑ = ℜ ( λ k a ϑ ) , k ∈ N . (8)As a result, we can combine these complex conjugate pairs, whichfacilitates the analysis approach. In particular, the number of eigenval-ues λ j and scaled modes a j ϑ j (with an eigenvalue having a non-zeroimaginary part) is reduced by a factor of two. This aspect is crucial forall following steps. (a) Spatial Properties: The multiplication of a mode ϑ j by its (com-plex) amplitude a j corrects its orientation. To demonstrate this, thetraditional mode to the eigenvalue λ = ϑ j needs to be linked to thedata by multiplying it with its amplitude a j . In Figure 6 (bottom), thescaled mode to the eigenvalue λ = Reconstruction errors over time Considered EigenvaluesHarmonic Clustering ApproachDistance-based Clustering Approach
IIIIIIIIIII t E rr o r Fig. 7: The reconstruction error over time (left) of different subsets of components (I, II, and III) from the 2D Karman vortex street data set.The eigenvalues of the respective clusters are shown on the right. Whereas the subset III consists of all components, leading to an error-freereconstruction of the flow, the subsets I and II are thinned out and used for the cluster approaches that lead to a selection of modes. Differentthresholds facilitate the identification of patterns and support the clustering of components (red and blue on the right). (b) Influence Properties:
The traditional amplitudes (typically usedfor the selection of modes) suffer from an imprecise computation asEquation 6 does not hold exactly. As a result, the influence of modesis not reflected correctly. Using the proposed improved representation,i.e., the scaled mode a j ϑ j computed by Algorithm 1, the influence isgiven by its norm (cid:107) a ϑ (cid:107) . This formula is in accordance to the influenceof Fourier-transformed vectors and is more precise.Even though a correct choice of the most dominant components isnow more likely, the norm only represents the influence at time t . Aselection of components only based on the norm of scaled modes (orsimply on the absolute value of amplitudes) is insufficient as a temporalencoding is not included (compare Figure 4 on the left). To integratethis, we propose a visual representation, referred to as dominancestructure, that uses both the scaled modes and the eigenvalues. Thisrepresentation is illustrated in Figure 5 (top). Basically, the norms ofthe scaled DMD modes (cid:107) a ϑ (cid:107) are visualized in a bar diagram that issorted by the frequency of their corresponding eigenvalues. However,it also takes the following aspects into account:• Since the scaled modes occur in complex conjugate pairs (like theeigenvalues), it is sufficient to display one of them, as they havethe exact same impact.• Due to the sorting according to the frequency (argument), it ispossible to differentiate between the impact of low and highfrequencies, which is in the spirit of DFT (see Figure 2) andthe eigenvalue visualization (see Figure 5). Therefore, the twovisualizations can be linked as both rely on the same quantities.More precisely, the frequency is represented in an ordinal wayfor the dominance structure (i.e., sequence of numbers) and aquantitative way for the eigenvalues (i.e., exact positions).• The temporal development of a scaled mode a j ϑ j (according tothe norm) is integrated into the representation. This developmentis given by the values (cid:107) λ kj a j ϑ j (cid:107) at specific time steps k , whichare visualized by color-coded bars. The corresponding time steps k are illustrated by a color map on the right. For the sake ofvisibility, a color-coded bar should be always either placed in theforeground (if | λ j | <
1) or background (if | λ j | > k = DMD decouples time-dependent flow into spatial and temporal com-ponents. For an appropriate selection of components, however, thespatio-temporal character of DMD needs to be taken into account.Mathematically, a selection of components can be formalized by a subset C ⊆ { ,..., r } of all components. Having defined a subset C ,we denote the temporal development of it at step k as: λ k C a C ϑ C : = ∑ i ∈ C λ ki a i ϑ i . (9)If the traditional dominance-based approach for the selection ofcomponents is consulted, then the components are chosen accordingto the norms of scaled modes (or actually to the absolute values of theamplitudes). Figure 7 (left) shows the reconstruction error over time(i.e., (cid:107) x k − λ k C a C ϑ C (cid:107) plotted for k = ,..., m ) for three examples ofsubsets C ∈ { C I , C II , C III } . The non-redundant eigenvalues belongingto the three subsets are illustrated in Figure 7 on the right (compareFigure 5). The subset C III contains all components and is the maxi-mally achievable order of accuracy (the reconstruction is exact, sincethe conditions of Equation 6 are satisfied). The other two subsets C I and C II contain the components with the k highest influence, where dif-ferent thresholds have been chosen. However, this traditional approachfor the selection of components neither clarifies the interplay of thechosen components nor classifies them appropriately. In addition, dueto the superposition principle, an incoherently selection may lead tocomponents that eliminate each other. In sum, incoherently selected(scaled) components may not describe any relevant spatial and temporalfeatures of the flow.Another approach for the selection of components can be performedon the basis of the improved components and visualizations. Sincetemporal aspects are encoded in the visualization of the dominancestructure, it probably provides a better tool for the selection of com-ponents. In general, it can be conducted in the following way (wherewe always select only non-redundant components): First, the compo-nents with a high influence at a certain time step k are selected. Thisis achieved by selecting every component whose value (cid:107) λ k j a j ϑ j (cid:107) ishigher than a chosen fixed threshold. If k =
0, then the procedureis equal to the traditional one. In a second step, undesirable compo-nents are sorted out manually. Using the new dominance structure,non-selected components that represent important steady state partscan be added. In contrast, selected components that vanish extremelyfast can be excluded. With the traditional visualization (Figure 4 on theleft), this is not possible, even if the eigenvalue visualization (Figure 4on the right) is consulted additionally. To assist this procedure, wekeep attention to the reconstruction error over time (see Figure 7) usingthe selected ones as well as all modes as a reference. This shows theprecision of the chosen components and helps substantiate the selectionof components.The proposed approach based on our visualizations is useful fora first impression or an explorative analysis. Our experiences haveshown that it is expedient for simpler data sets. For complex systemsthat exhibit several different time-dependent phenomena, the interplay,interpretation, and classification of selected components still may re-main unclear. So far, the selected (scaled) modes were classified by theig. 8: The procedure of the harmonic cluster approach applied to the superposed quadgyre data set: On the left, selected snapshots fromthe two unsteady flow fields (Flow I and II) are shown that characterize the superposed quadgyre. These data sets consist in each case offour vortices moving periodically from left to right (Flow I) or from top right to the bottom right in a crescent-shaped move (Flow II). Theeigenvalue representation in the middle reveals two different frequency patterns that can be captured by the harmonic clustering approach. Foreach aggregation, the three most dominant modes are represented by their temporal development. It can be observed that each aggregationcharacterizes one respective base flow, which verifies the usefulness of the cluster method.location of their corresponding eigenvalues. This can lead to the factthat components eliminate each other for certain sections of the flow.Therefore, an appropriate criterion for the selection of components is toclassify those which represent a certain section of the flow accurately.Mathematically, a discrete optimization problem can be formulated:For a certain time section 0 ≤ k < ..., k ≤ m , we look for a subset C M with 0 < M < r elements that satisfiesmin C M k ∑ k = k (cid:107) x k − λ k C a C ϑ C (cid:107) . (10)This technique can be obviously used for the selection of components,since each computed subset may characterize a section of the flowand the components its features. For (almost) periodic data sets, thesegmentation of the flow into sections is irrelevant, however, a classi-fication into different base frequencies is of importance. The discreteoptimization problem (Equation 10) can practically not be solved asthe computational complexity grows with (cid:0) r M (cid:1) (for non-rank-deficientdata (cid:0) mM (cid:1) ) for the selection of M components.Instead of using this time-consuming (PCA-like energy sorted) se-lection technique, we propose two fast clustering approaches for ag-gregating components that will detect the same components (as wewill show later). As shown above, the aggregated components revealrelevant physical features classifying the flow into segments such as thetransient response or the steady state. Subsequently, the componentscan be investigated individually with regard to the specific detectedfeature.The following two clustering approaches operate on the eigenvaluesand their new representation, i.e., we aggregate on the basis of temporalpatterns. Therefore, every aggregation include the complex conjugate counterpart of a components such that the temporal development shouldbe evaluated using the real part as in Equation 8. Distance-Based Clustering
Several flow phenomena like damp-ing processes are characterized by modes that exhibit very similarfrequencies and growth rates. Therefore, we propose aggregating com-ponents with closely located eigenvalues. For this procedure, we firstthin out the potential eigenvalues such that the flow can be reconstructedadequately by them. Then, a distance-based clustering method is ap-plied to those eigenvalues. The methodology is highlighted in Figure 7by (I). It often leads to multiple clusters C , C ,... that may revealfeatures. However, the eigenvalue λ = C is shown in Figure 1 on the right. Harmonic Clustering
For the identification of temporal patterns,DFT uses harmonics, i.e., multiples of frequencies. We adapt thisconcept and aggregate components that exhibit patterns of harmonics.The clustering approach uses again an appropriately thinned out setof eigenvalues whose components reconstruct the flow adequately.Then, we search for multiples in the eigenvalue representation, eithermanually or algorithmically. The choice depends on the distribution andcomplexity of eigenvalues. The process is demonstrated in Figure 7 by(II) (or Figure 8). If there is more than one cluster, the eigenvalue λ = C is shown in Figure 1 on the right.In sum, we propose using the two clustering approaches for theselection of components. A selection consists of united clusters (foundfrom the clustering approaches), each reveals features of the flow byig. 9: The dominance structure of the superposed quadgyre dataset.Besides of decreasing behavior for higher frequencies, we immediatelyobserve no converging and diverging parts. We conclude the periodicityof the flow. Moreover, the representation indicates the existence of twodifferent decay patterns.inspecting the respective individual components. This is due to thefact that the aggregations segment the flow into relevant sections andtherefore classify the components. In Figure 1, the two clusteringapproaches are illustrated representing different phenomena of the flow. ESULTS
To demonstrate the usefulness of our proposed techniques, we ap-ply them to unsteady flow fields to identify different frequency-basedfeatures. The first example is a generated synthetic flow, called su-perposed quadgyre. It is an overlay of two artificial 2D flow fields,called quadgyre, which are extensions of the double gyre [31] flowfield This scenario demonstrates the application of DMD to periodicflows and illustrates the correctnesss of our aggregation approaches.The next example is a simulated von Karman vortex street that is amore complex unsteady flow resulting in an equilibrium state. Ourimproved techniques allows us to identify the relevant components thatare associated to the transient response and steady state. Finally, weconsider a 3D von Karman vortex street to show how the approachcarries over to 3D.
When applying DMD to overlapping periodic phenomena, such asthe superposed quadgyre data set, we obtain characteristic featureswith DMD. Our improved techniques identify these and the clustersrepresent each individual periodic phenomena as we will demonstratein the following: The superposed quadgyre data set is a superposition oftwo unsteady flow fields with different base movements and frequencies.A full period of each flow is illustrated in Figure 8 (left). The analyticalformula of both flows is given by u ( x , y , t ) = − π A sin ( π f ( x , t )) cos ( π g ( y , t )) dgdy ( y , t ) , v ( x , y , t ) = π A cos ( π f ( x , t )) sin ( π g ( y , t )) d fdx ( x , t ) , where f ( x , t ) = ε sin ( ω f t + s f ) x + x − ε sin ( ω f t + s f ) x and g ( y , t ) = ε sin ( ω g t + s g ) y + y − ε sin ( ω g t + s g ) y . Both data sets consist of fourvortices that move periodically. The vortices in the first dataset moveright to left and back with parameters A = ε = ω f = π , ω g = s f = π , and s g =
0. The vortices from Flow II move from top rightto bottom right in a crescent-shaped motion with parameters A = ε = ω f = ω g = π , and s f = s g = π . The flow fields are sampledat a resolution of 201 ×
201 cells (resulting in a snapshot dimension of n = t Fig. 10: The plot shows the error between the temporal developmentof the two aggregations found by the harmonic clustering approachapplied to the superposed quadgyre data set and the respective originalflows, which are mean subtracted in order to eliminate constant parts(compare Figure 8). Since the error is very small, each cluster capturesthe dynamic behavior of one original flow accurately.(Figure 9) as all bars are colored red. The next step is to select relevantcomponents. As mentioned before, for periodic systems, it is more ofimportance to classify the data into different base frequencies. Hence,instead of using the traditional dominance-based approach for the selec-tion of modes, we use the proposed harmonic clustering approach. Twoclusters C I and C II are detected accurately as demonstrated in Figure 8.To verify the relevance and correctness of the two found aggregations,where each belong to a different base frequency phenomena, we com-pare each of them with the suitable analytical base flow. More precisely,we consider the error between the temporal development of an aggrega-tion (without the constant part) and the respective base flow, where wesubtracted the mean. The error plot is depicted in Figure 10. It can beobserved that the relative error is approximately 4% for both clusters.Hence, each found cluster characterizes one of the base flows (whichform the superposed quadgyre by superposition). This shows that ourclustering approach can extract and classify overlapping phenomenawith different frequency patterns. Therefore, the components of thetwo aggregations can now be analyzed separately from each other.In Figure 8 (right), the three most important modes of each clusterare depicted as well as their temporal development. The upper aggrega-tion belongs to the standard quadgyre. A similar flow was investigatedby Brunton et al. [7], however, our approach suppresses the negligiblecomplex conjugate scaled modes. The upper three modes show sym-metric vanishing and recurring vortices indicating a periodic flow inthe horizontal direction. The lower three modes indicate a crescentshape movement of vortices. The temporal development of the firstmode shows the two vortices in the top and bottom right as well as theemerging vortex on the left. This vortex can be found in the other twomodes as well. Whereas the second one mainly describes the crescent-shaped movement (since the frequency is twice as the base frequency),the last one highlights the vortices in the left upper and lower corners.The temporal evolution of the modes (Figure 8 right) shows that thesymmetry properties of both systems are conserved.If the DFT is consulted for such a frequency-based investigation,a classification like this would not work as the frequencies cannot beidentified directly. As the DFT uses uniformly distributed frequenciesdepending on the total number of snapshots, the mixed frequenciesof the superposed quadgyre will not be determined. Therefore, thefrequency detection with DFT is blurred and none of the original flowsis precisely detected. To compare our improved components and visualizations with the tradi-tional ones, a flow past a circular object forming a von Karman vortexstreet is investigated (see Figure 1 on top). This typical analysis exam-ple for DMD was simulated on a grid with 881 ×
166 points. ApplyingDMD to the 251 snapshots results in 250 DMD components. In Fig-ure 5, the dominance structure and eigenvalues are visualized with ournew approach. In this example, we are faced with a rather complicateddominance structure. It can be observed that a lot of scaled modeshave a large influence on the entire system. However, we can alsoig. 11: Visualization of scaled modes from the 2D Karman vortexstreet. These are selected from an aggregation that represent the steadystate. The aggregation was found by the harmonic clustering approach.immediately observe that nearly all components characterize dampingphenomena, in particular those with a high impact at the beginning.After about 50 time steps, many components have vanished and partsof the steady state (highlighted by pure red bars) will become moredominant. Hence, by the temporal encoding in the dominance visual-ization, we get a precise understanding of the component’s impact. Inaddition, as the two axes of the dominance and eigenvalue visualizationare linked, we observe a similar wave-shaped contour. The traditionalvisualizations in Figure 4 do not detect or highlight any of these aspects.Based on the observations of our improved visualizations, it may bepossible to make an adequate selection of components, however, wewant to demonstrate the usefulness of our cluster approaches for theselection of components.In Figure 7, the reconstruction error over time for different subsetsof components is shown where the corresponding eigenvalues are rep-resented on the right. For the aggregation of components, we use bothclustering approaches on these different representations that facilitatethe detection of patterns. The whole procedure is demonstrated inFigure 7 and the two chosen clusters C , and C , segment the flowinto physically relevant parts. Figure 1 shows the errors over time ofthe components belonging to the two aggregations C1,1 and C2,1. Forthe first aggregation, we observe a small error at the beginning of thesimulation. Therefore, the transient response can be described by theindividual components of C1,1, whereas aggregation C2,1 character-izes the steady state as the error decreases over time. Thus, we havedetected two important phenomena that classify the components.A comprehensive time-continuous and -discrete analysis of wakeflows with DMD was conducted by Bagheri [3]. In this context, a clas-sification was achieved that equals the result of our cluster approaches.To validate the cluster approaches once again, we computed the discreteminimization problem in Equation 10. We applied it both to a sectionin the beginning and in the end, where we search for a subset C M with M = λ = DMD is independent of the dimensionality of the spatial domain. Foran analysis with DMD, we just need to adjust the visualization of thescaled modes to the dimensionality of the spatial domain. Therefore,our investigation approach with DMD can be equally applied to 3Dunsteady flow, except for the computational time. To demonstrate this,we analyze a 3D von Karman vortex street representing an extension tothe 2D counterpart. The grid has a resolution of 228 × ×
44 and 381snapshots were considered. The flow field starts analogously as the 2Dscenario in an emerging phase of the vortex street.In Figure 12, the dominance structure and eigenvalue visualizationare depicted. In both visualizations, we observe the same structure as inthe 2D counterpart in Figure 5. An evaluation can be indeed conductedanalogously.For the investigation of spatial properties, we exemplarily visualizea mode in Figure 13. Our visualization for the 3D scaled mode doesnot aim to highlight features, it is used to show similarities with the 2Dcounterpart. We use a volume based representation as well as a crosssection that shows the interaction inside. The cross section exhibitssimilar structures as the modes from the 2D counterpart (Figure 11).For 3D data sets, the computation time is affected moderately as onlythe SVD of the matrix X and the computation of the amplitudes scaleslinearly with the additional dimension. Furthermore, our visualizationsand clustering approaches are only based on multiple calculations withthe DMD eigenvalues. Therefore, no overhead is produced. ONCLUSION
In this paper, we have thoroughly recapped the mathematical foundationof DMD that allows us to combine and improve the DMD componentsig. 13: Visualization of a scaled mode for the 3D von Karman vortexstreet. A volume-based representation with a cross section is used toshow the connection to the 2D case. The frequency of the second 2Dmode in Figure 11 is approximately equal and, therefore, the samespatial behavior can be observed in the cross section.such that the underlying physics is represented more adequately, e.g.,by the construction of scaled modes. Moreover, the interplay of thecomponents is clarified and a new view on DMD is given by comparingit to DFT. These new insights are used to design appropriate visualiza-tions such that the spatio-temporal character of DMD is respected andredundant parts are hidden. Therefore, a more adequate selection ofcomponents can be made and the identification of specific patterns andfeatures is facilitated. The two novel clustering approaches should beadditionally consulted for the selection of components as these segmentthe flow into physically relevant sections. In sum, a deeper understand-ing of the DMD components is gained that make DMD more accessiblefor users.These new techniques may also be combined with other visualizationmethods that highlight further features in the components. For instance,we could apply methods like FTLE to the temporal development ofscaled modes. In this case, FTLE could be seen as a post-processingstep to gain another view on the spatial components. Another interestingresearch direction might be the application of our techniques to otherdata. So far, we only used data represented by a grid that consistsof velocity components. It would be interesting to choose a differentdata representation, like particle-data, and evaluate different quantities,e.g., pressure or vorticity. We plan to apply our proposed techniquesto other (real-valued) data and evaluate the effectiveness for differentapplications again. For example, the clustering approaches are onlyuseful, if the data exhibits similar frequency patterns such as fluid flow. A CKNOWLEDGMENTS
This work is partly supported by “Kooperatives Promotionskolleg Digi-tal Media” at Hochschule der Medien and the University of Stuttgart. R EFERENCES [1] S. Abu and J. S. Hyung. Dynamic mode decomposition of turbulent cavityflows for self-sustained oscillations.
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Journal of Fluids and Structures , 49:53–72, 2014. A PPENDIX : C
OMPLEX C ONJUGATED
DMD C
OMPONENTS
Lemma
Let β ∈ C and z ∈ C n with ℜ ( z ) and ℑ ( z ) are linearly inde-pendent. Then, the following assertion holds: β (cid:54) = = ⇒ z + β z / ∈ R . Proof:
First, the imaginary part of z + β z is calculated: z + β z = ℜ ( z ) + i ℑ ( z ) + ( ℜ ( β ) + i ℑ ( β ))( ℜ ( z ) − i ℑ ( z ))= ℜ ( z ) + i ℑ ( z ) + ℜ ( β ) ℜ ( z ) + ℑ ( β ) ℑ ( z )+ i ℑ ( β ) ℜ ( z ) − i ℜ ( β ) ℑ ( z )=[( + ℜ ( β )) ℜ ( z ) + ℑ ( β ) ℑ ( z )]+ i [ ℑ ( β ) ℜ ( z ) + ( − ℜ ( β )) ℑ ( z )] . The proof is done by contraposition, i.e., let us assume that z + β z ∈ R .Then, the imaginary part has to equal zero, i.e., ℑ ( β j ) · ℜ ( z j ) + ( − ℜ ( β j )) · ℑ ( z j ) = . Since ℜ ( z ) and ℑ ( z ) are linearly independent, we conclude that1 − ℜ ( β j ) = , ℑ ( β j ) = , which implies β j =
1. This proves the statement by contraposition.
Theorem
Consider real-valued data x , x ,..., x m ∈ R n . If bothx ,..., x m − and x ,..., x m are linearly independent and the DMDeigenvalues λ ,..., λ m are distinct, then Algorithm 1 produces pairsof complex conjugate eigenvalues, i.e., if λ ∈ C \ R is a pure complexDMD eigenvalue then the complex conjugate λ is a DMD eigenvalue,too, and the associated scaled DMD modes are given by a ϑ and a ϑ ,respectively. Proof:
We use the notation from Algorithm 1. Since we considerreal-valued data, the matrices X , Y ∈ R n × m will be real-valued. Forreal-valued matrices the singular value decomposition can be chosenreal-valued and hence the DMD matrix S ∈ R m × m is real-valued. As aresult, the eigenvalues of S (which are the DMD eigenvalues) occur incomplex conjugate pairs. Let λ , λ = λ , λ , λ = λ ,..., λ k − , λ k = λ k − ∈ C \ R be the complex conjugate pairs and λ k + ,..., λ m ∈ R the remaining real-valued eigenvalues. The corresponding eigenvectorsare given by v , v ,..., v k − , v k , v k + ,..., v m . Since the eigenvaluesare distinct, which implies a one-dimensional eigenspace, and occur incomplex conjugate pairs, the following relations hold v = c v , ... v k = c k − v k − , for some scaling factors c , c ,..., c k − ∈ C with | c | = ··· = | c k − | =
1. The DMD modes are calculated by ϑ j = λ j YV Σ − v j . Hence, theDMD modes maintain the structure of the eigenvectors, i.e., we candenote those analogously by ϑ , ϑ ,..., ϑ k − , ϑ k , ϑ k + ..., ϑ m with ϑ = c ϑ , ... ϑ k = c k − ϑ k − . Using the reconstruction property, i.e., Equation 6 for the secondtime step, we get the following relationship: x = k ∑ l = λ l a l ϑ l + m ∑ l = k + λ l a l ϑ l = k ∑ l = λ l − a l − ϑ l − + k ∑ l = λ l a l ϑ l + m ∑ l = k + λ l a l ϑ l = k ∑ l = λ l − a l − ϑ l − + k ∑ l = λ l − a l c l − ϑ l − + m ∑ l = k + λ l a l ϑ l , where a , a ,..., a m are the DMD amplitudes. As we assume real-valued data, which implies in particular x ∈ R n , the above sum hasto be real-valued. Let us express the DMD amplitudes a , a ..., a k (belonging to the complex conjugate counterpart) as a = b a , ... a k = b k − a k − , for some appropriate scaling factors b , b ..., b k − ∈ C . If we define β = b c , β = b c ,..., β k − = b k − c k − ∈ C , we can express theabove sum as k ∑ l = λ l − a l − ϑ l − + k ∑ l = β l − λ l − a l − ϑ l − + m ∑ l = k + λ l a l ϑ l ∈ R n . Now, the proof is complete, if we are able to show that β = β = ··· = β k − =
1, since every scaled DMD mode a l ϑ l (belonging to a DMDeigenvalue with a complex conjugate pair of eigenvalues) is given by a j ϑ j = b j − c j − a j − ϑ j − = β j − a j − ϑ j − = a j − ϑ j − . To prove the statement, consider the real-valued sum from above,however, for simplicity we use z j = λ j a j ϑ j : k ∑ l = z l − + k ∑ l = β l − z l − + m ∑ l = k + z l ∈ R n . Assume that there is at least one coefficient β l (cid:54) =
1. Sincethe DMD modes are linearly independent (because thesnapshots x ,..., x m are linearly independent), the vec-tors ϑ , ϑ ,..., ϑ k − , ϑ k − , ϑ k + ,..., ϑ m are linearly in-dependent. In addition, a simple calculation shows that ℜ ( ϑ ) , ℑ ( ϑ ) , ℜ ( ϑ ) , ℑ ( ϑ ) ,..., ℜ ( ϑ k − ) , ℑ ( ϑ k − ) , ϑ k + ,..., ϑ m are linearly independent as well. Finally, ℜ ( z j ) and ℑ ( z j ) are linearlyindependent and by the previous proven Lemma, we conclude that z j + β j z j / ∈ R n . However, there is no possibility to eliminate theupcoming imaginary part, though, as the full sum has to be real-valued.Consequently, the assumption is wrong and β = ··· ==
1. Sincethe DMD modes are linearly independent (because thesnapshots x ,..., x m are linearly independent), the vec-tors ϑ , ϑ ,..., ϑ k − , ϑ k − , ϑ k + ,..., ϑ m are linearly in-dependent. In addition, a simple calculation shows that ℜ ( ϑ ) , ℑ ( ϑ ) , ℜ ( ϑ ) , ℑ ( ϑ ) ,..., ℜ ( ϑ k − ) , ℑ ( ϑ k − ) , ϑ k + ,..., ϑ m are linearly independent as well. Finally, ℜ ( z j ) and ℑ ( z j ) are linearlyindependent and by the previous proven Lemma, we conclude that z j + β j z j / ∈ R n . However, there is no possibility to eliminate theupcoming imaginary part, though, as the full sum has to be real-valued.Consequently, the assumption is wrong and β = ··· == β k − ==