The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena

Reiß, Julius; Schulze, Philipp; Sesterhenn, Jörn; Mehrmann, Volker

doi:10.1137/17m1140571

Cited by 147 publications

(180 citation statements)

References 43 publications

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“…One of the challenges in Galerkin projection is the deformation of POD modes. As recently discussed by Reiss et al 75 , transport-dominated phenomena are usually a challenge for modal methods, since their dynamics cannot be captured accurately by a few dominant spatial modes. If we include more number of modes to better recover the embedded structures in the underlying system, the computational expense increases and the ROM might not be efficient from practical point of view.…”

Section: Introductionmentioning

confidence: 99%

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

et al. 2019

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In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various deep neural network architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equations, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a non-intrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly non-intrusive, since any prior information about the underlying governing equations is not required for generating the reduced order model. Our a posteriori analysis shows that the proposed data-driven approach is remarkably accurate, and can be used as a robust predictive tool for non-intrusive model order reduction of complex fluid flows.

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Section: Introductionmentioning

confidence: 99%

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

et al. 2019

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“…Moreover, the sPOD modes clearly reveal the front profiles of the different transports, whereas the POD is not suitable for identifying structures in this test case. In comparison to the heuristic sPOD algorithm proposed in [20], the new algorithm is based on a residual minimization and hence at least locally optimal. A drawback of the new algorithm is that it is more expensive than the POD and the original sPOD approach of [20].…”

Section: Discussionmentioning

confidence: 99%

“…where for the moment we assume that the shift frames ∆ are available or can be approximated before the optimization for the modes ψ and their time amplitudes α is carried out. Methods to estimate these shifts based on given snapshot data have been discussed in [20]. A crucial difference between (9) and (8) is that (9) is an unconstrained optimization problem without the orthonormality restriction for the modes ψ j .…”

Section: Optimal Spod Approximationmentioning

confidence: 99%

Model Reduction for a Pulsed Detonation Combuster via Shifted Proper Orthogonal Decomposition

Schulze

Reiß

Mehrmann

2018

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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We propose a new algorithm to compute a shifted proper orthogonal decomposition (sPOD) for systems dominated by multiple transport velocities. The sPOD is a recently proposed mode decomposition technique which overcomes the poor performance of classical methods like the proper orthogonal decomposition (POD) for transport-dominated phenomena. This is achieved by identifying the transport directions and velocities and by shifting the modes in space to track the transports. Our new algorithm carries out a residual minimization in which the main computational cost arises from solving a nonlinear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detonation combuster and observe that very few sPOD modes are sufficient to obtain a good approximation. For the same accuracy, the common POD needs ten times as many modes and, in contrast to the sPOD modes, the POD modes do not reflect the moving front profiles properly.

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“…It is easy to see that a linear projection of this solution to a low-dimensional basis would not yield a good approximation of the solution. Naturally, methods to remove translational symmetry [36,32,34] are being actively explored. This is also intimately related to the concept of displacement interpolation, a term we borrow from the optimal transport literature [40], in which one aims to minimize the Wasserstein distance, although we will not make the connection more explicit here.…”

Section: Displacement Interpolationmentioning

confidence: 99%

Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform

Rim¹

2018

SIAM J. Sci. Comput.

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We introduce a dimensional splitting method based on the intertwining property of the Radon transform, with a particular focus on its applications related to hyperbolic partial differential equations (PDEs). This dimensional splitting has remarkable properties that makes it useful in a variety of contexts, including multi-dimensional extension of large time-step (LTS) methods, absorbing boundary conditions, displacement interpolation, and multi-dimensional generalization of transport reversal [34].

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The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena

Cited by 147 publications

References 43 publications

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

Model Reduction for a Pulsed Detonation Combuster via Shifted Proper Orthogonal Decomposition

Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform

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