On dynamic mode decomposition:  Theory and applications

Tu, Jonathan H.; Rowley, Clarence W.; Luchtenburg, Dirk M.; Brunton, Steven L.; Kutz, J. Nathan

doi:10.3934/jcd.2014.1.391

“…The Koopman modes arise from the spectral analysis of the Koopman operator, which is an infinitedimensional operator that evolves functions of the system's state. Koopman mode decomposition is closely related to dynamic mode decomposition (DMD) [7][8][9][10][11][12], which is a popular way to * a.sharma@soton.ac.uk …”

mentioning

confidence: 99%

“…The DMD modes are the eigenvectors of M, which may be found by various methods [7,[9][10][11][12]. Another way to look at this problem is to introduce a time-translation operator acting on the matrix of snapshots,…”

mentioning

confidence: 99%

Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

Sharma

¹

,

Mezić

²

,

McKeon

³

2016

View full text Add to dashboard Cite

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known. DOI: 10.1103/PhysRevFluids.1.032402This paper presents a unifying view of a range of methods used to characterize nonlinear solutions of the Navier-Stokes equations. Ordered by roughly decreasing complexity in their treatment of nonlinearity, these are invariant solutions, a generalized form of Koopman mode decomposition, dynamic mode decomposition, and resolvent mode decomposition. The instances in which the four methods of analysis coincide are identified here. We hope to provide a more rigorous basis on which to interpret these methods and to inform the potential user of the appropriate tool for a particular problem.Invariant solutions are exact (nonlinear) solutions of the Navier-Stokes equations that remain the same after certain symmetry operations are applied (such as reflection, rotation, or shifts in space or time). They are sometimes called exact coherent structures and are implicated in the formation of apparently repeating highly ordered spatiotemporal patterns in turbulence. These solutions are the subject of an active and long-standing area of research; see, for example, [1-3] for reviews. From the viewpoint of dynamical systems, turbulence may be understood in terms of a state trajectory visiting the neighborhood of various invariant solutions.Koopman modes are a general way of analyzing the dynamics of a nonlinear system [4][5][6][7]. The Koopman modes arise from the spectral analysis of the Koopman operator, which is an infinitedimensional operator that evolves functions of the system's state. Koopman mode decomposition is closely related to dynamic mode decomposition (DMD) [7][8][9][10][11][12], which is a popular way to * a.sharma@soton.ac.uk

show abstract

“…where / j are the scaled DMD modes such that the sum of the modes equals the first data vector (Tu et al 2013). One can follow the same steps to approximate the pressure field as well.…”

Section: Calculate the Matrixã Asmentioning

confidence: 99%

“…DMD was recently proposed by Schmid (2010). The DMD-based approach involves the following steps (Tu et al 2013;Ghommem et al 2013):…”

Section: Model Order Reductionmentioning

confidence: 99%

Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media

Ghommem

¹

,

Gildin

²

,

Ghasemi

³

2016

View full text Add to dashboard Cite

SummaryIn this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in heterogeneous porous media. We propose intrusive and nonintrusive model-reduction approaches that enable a significant reduction in the size of the subsurface flow problem while capturing the behavior of the fully resolved solutions. In one approach, we use the dynamic mode decomposition. This approach does not require any modification of the reservoir simulation code but rather postprocesses a set of global snapshots to identify the dynamically relevant structures associated with the flow behavior. In the second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their properorthogonal-decomposition modes. Furthermore, we use the discrete empirical interpolation method to approximate the mobilityrelated term in the global-system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE-10 benchmark permeability field, and present a numerical example in two-phase flow. One can efficiently use the proposed model-reduction methods in the context of uncertainty quantification and production optimization. IntroductionHigh-fidelity reservoir-simulation models are shown to yield better predictions in optimization problems and in planning for new reservoir developments. However, the computational time of such large-scale models becomes the bottleneck of fast turnarounds in the decision-making process and the assimilation of real-time data into reservoir models (namely, the closed-loop reservoir management) to improve the accuracy of such models (Gildin and Lopez 2011). Even in the case of unconventional reservoirs, numerical reservoir simulation was used with several modifications with the current models, particularly in the nature of the flow. To this end, natural fractures and pore-space networks were incorporated in the simulation process to account for flow in the fractures and matrix (Moridis et al. 2010;Yan et al. 2013;Alfi et al. 2014).Despite the great advances in reservoir-modeling tools and the advent of high-performance computing, high-fidelity physicsbased numerical simulation still remains a challenging step in understanding the physics of the reservoir because of the largescale nature of the discretization of the underlying partial-differential equations. Computationally intensive simulations, such as in the case of history matching (Afra et al. 2014), optimization, and uncertainty quantification (Doren et al. 2006), become impractical to be performed in a timely manner if real-time data need to be assimilated into the model (Jansen et al. 2009;Gildin and Lopez 2011).The importance of obtaining a simpler model that can represent the physics of the full system is paramount to speed up the work flows that require several (from dozens to thousands) calls of the forward model. This is usu...

show abstract

“…The simulation was run for 300 days, and the saturation is saved every Dt ¼ 1 day; thus, we have 300 snapshots. Note that here we applied uniform-timestep DMD, but one can refer to Tu et al (2013) for nonuniform time sampling DMD.…”

Section: Case Studymentioning

confidence: 99%

Complexity Reduction of Multi-Phase Flows in Heterogeneous Porous Media

Ghommem

¹

,

Calo

²

,

Efendiev

³

et al. 2013

View full text Add to dashboard Cite

SummaryIn this paper, we apply mode decomposition and interpolatory projection methods to speed up simulations of two-phase flows in heterogeneous porous media. We propose intrusive and nonintrusive model-reduction approaches that enable a significant reduction in the size of the subsurface flow problem while capturing the behavior of the fully resolved solutions. In one approach, we use the dynamic mode decomposition. This approach does not require any modification of the reservoir simulation code but rather postprocesses a set of global snapshots to identify the dynamically relevant structures associated with the flow behavior. In the second approach, we project the governing equations of the velocity and the pressure fields on the subspace spanned by their properorthogonal-decomposition modes. Furthermore, we use the discrete empirical interpolation method to approximate the mobilityrelated term in the global-system assembly and then reduce the online computational cost and make it independent of the fine grid. To show the effectiveness and usefulness of the aforementioned approaches, we consider the SPE-10 benchmark permeability field, and present a numerical example in two-phase flow. One can efficiently use the proposed model-reduction methods in the context of uncertainty quantification and production optimization. IntroductionHigh-fidelity reservoir-simulation models are shown to yield better predictions in optimization problems and in planning for new reservoir developments. However, the computational time of such large-scale models becomes the bottleneck of fast turnarounds in the decision-making process and the assimilation of real-time data into reservoir models (namely, the closed-loop reservoir management) to improve the accuracy of such models (Gildin and Lopez 2011). Even in the case of unconventional reservoirs, numerical reservoir simulation was used with several modifications with the current models, particularly in the nature of the flow. To this end, natural fractures and pore-space networks were incorporated in the simulation process to account for flow in the fractures and matrix (Moridis et al. 2010;Yan et al. 2013;Alfi et al. 2014).Despite the great advances in reservoir-modeling tools and the advent of high-performance computing, high-fidelity physicsbased numerical simulation still remains a challenging step in understanding the physics of the reservoir because of the largescale nature of the discretization of the underlying partial-differential equations. Computationally intensive simulations, such as in the case of history matching (Afra et al. 2014), optimization, and uncertainty quantification (Doren et al. 2006), become impractical to be performed in a timely manner if real-time data need to be assimilated into the model (Jansen et al. 2009;Gildin and Lopez 2011).The importance of obtaining a simpler model that can represent the physics of the full system is paramount to speed up the work flows that require several (from dozens to thousands) calls of the forward model. This is usu...

show abstract

On dynamic mode decomposition: Theory and applications

Cited by 1,214 publications

References 39 publications

Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

Complexity Reduction of Multiphase Flows in Heterogeneous Porous Media

Complexity Reduction of Multi-Phase Flows in Heterogeneous Porous Media

Contact Info

Product

Resources

About