Decay of the Kolmogorov <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e145" altimg="si13.svg"><mml:mi>N</mml:mi></mml:math>-width for wave problems

Greif, Constantin; Urban, Karsten

doi:10.1016/j.aml.2019.05.013

Cited by 112 publications

(85 citation statements)

References 9 publications

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“…Theoretically, the decay rate of Kolmogorov n-width [18,19] can measure how fast the POD spatial bases change in time for a dynamical system. The Kolmogorov n-width is defined as [20]:…”

Section: Criteria For Furthest Forecast Timementioning

confidence: 99%

“…The Kolmogorov n-width as in Eq. ( 19) provides the worst-case error resulting from a projection onto the best-possible linear subspace of a given dimension n [20]. If the Kolmogorov n-width decays faster with respect to n, the reduced-order solution in the subspace constructed by the first R POD bases can be an accurate approximation of the full-order solution for longer time beyond the range of snapshot data.…”

Section: Criteria For Furthest Forecast Timementioning

confidence: 99%

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Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using Gaussian process regression

Pan

2021

Computer Methods in Applied Mechanics and Engineering

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We present a data-driven nonintrusive model order reduction method for dynamical systems with moving boundaries. The proposed method draws on the proper orthogonal decomposition, Gaussian process regression, and moving least squares interpolation. It combines several attributes that are not simultaneously satisfied in the existing model order reduction methods for dynamical systems with moving boundaries. Specifically, the method requires only snapshot data of state variables at discrete time instances and the parameters that characterize the boundaries, but not further knowledge of the full-order model and the underlying governing equations. The dynamical systems can be generally nonlinear. The movements of boundaries are not limited to prescribed or periodic motions but can be free motions. In addition, we numerically investigate the ability of the reduced order model constructed by the proposed method to forecast the full-order solutions for future times beyond the range of snapshot data. The error analysis for the proposed reduced order modeling and the criteria to determine the furthest forecast time are also provided. Through numerical experiments, we assess the accuracy and efficiency of the proposed method in several benchmark problems. The snapshot data used to construct and validate the reduced order model are from analytical/numerical solutions and experimental measurements.

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“…Theoretically, the decay rate of Kolmogorov n-width [18,19] can measure how fast the POD spatial bases change in time for a dynamical system. The Kolmogorov n-width is defined as [20]:…”

Section: Criteria For Furthest Forecast Timementioning

confidence: 99%

Section: Criteria For Furthest Forecast Timementioning

confidence: 99%

Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using Gaussian process regression

Pan

2021

Computer Methods in Applied Mechanics and Engineering

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“…We consider here a prototypical example similar to the one given in [11] (see also [59,31] for other examples). Consider the univariate transport equation…”

Section: A Pure Transport Problemmentioning

confidence: 99%

Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

et al. 2020

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We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been based on the approximation of the solution set by linear spaces on Hilbert or Banach spaces. This approach can be expected to be successful only when the Kolmogorov width of the set decays fast. While this is the case on certain parabolic or elliptic problems, most transport-dominated problems are expected to present a slow decaying width and require to study nonlinear approximation methods. In this work, we propose to address the reduction problem from the perspective of general metric spaces with a suitably defined notion of distance. We develop and compare two different approaches, one based on barycenters and another one using tangent spaces when the metric space has an additional Riemannian structure. As a consequence of working in metric spaces, both approaches are automatically nonlinear. We give theoretical and numerical evidence of their efficiency to reduce complexity for one-dimensional conservative PDEs where the underlying metric space can be chosen to be the L 2 -Wasserstein space. arXiv:1909.06626v1 [math.NA]

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“…The Kolmogorov n-width is a concept from approximation theory that determines the linear reducibility of a system [65,66]. Mathematically, it is defined as [66][67][68] where S n is a linear n-dimensional subspace, M is the solution manifold, and Π S n is the orthogonal projector onto S n . In other words, d n (M) quantifies the maximum possible error that might arise from the projection of solution manifold onto the best-possible n-dimensional linear subspace.…”

mentioning

confidence: 99%

“…The Kolmogorov n-width is a concept from approximation theory that determines the linear reducibility of a system [65,66]. Mathematically, it is defined as [66][67][68]…”

mentioning

confidence: 99%

Breaking the Kolmogorov Barrier in Model Reduction of Fluid Flows

Ahmed

San

2020

Fluids

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Turbulence modeling has been always a challenge, given the degree of underlying spatial and temporal complexity. In this paper, we propose the use of a partitioned reduced order modeling (ROM) approach for efficient and effective approximation of turbulent flows. A piecewise linear subspace is tailored to capture the fine flow details in addition to the larger scales. We test the partitioned ROM for a decaying two-dimensional (2D) turbulent flow, known as 2D Kraichnan turbulence. The flow is initiated using an array of random vortices, corresponding to an arbitrary energy spectrum. We show that partitioning produces more accurate and stable results than standard ROM based on a global application of modal decomposition techniques. We also demonstrate the predictive capability of partitioned ROM through an energy spectrum analysis, where the recovered energy spectrum significantly converges to the full order model's statistics with increased partitioning. Although the proposed approach incurs increased memory requirements to store the local basis functions for each partition, we emphasize that it permits the construction of more compact ROMs (i.e., of smaller dimension) with comparable accuracy, which in turn significantly reduces the online computational burden. Therefore, we consider that partitioning acts as a converter which reduces the cost of online deployment at the expense of offline and memory costs. Finally, we investigate the application of closure modeling to account for the effects of modal truncation on ROM dynamics. We illustrate that closure techniques can help to stabilize the results in the inertial range, but over-stabilization might take place in the dissipative range. Fluids 2020, 5, 26 2 of 25Challenges in the development of ROM for turbulent flows are also related to the Kolmogorov n-width for these systems. The Kolmogorov n-width is a concept from approximation theory that determines the linear reducibility of a system [65,66]. Mathematically, it is defined as [66][67][68] where S n is a linear n-dimensional subspace, M is the solution manifold, and Π S n is the orthogonal projector onto S n . In other words, d n (M) quantifies the maximum possible error that might arise from the projection of solution manifold onto the best-possible n-dimensional linear subspace. When the decay of d n (M) with increasing n is fast, this means that the energy cascade and modal hierarchy allow the reduction of the system and the solution trajectory can be adequately approximated to evolve on a reduced linear subspace. However, for systems with strong nonlinearity, like turbulent flows, the intense interactions between different modes reduce the decay rate of the Kolmogorov n-width.Consequently, the linear reducibility is hindered. In ROM context, this is denoted as the "Kolmogorov barrier" because it necessitates involving more modes in ROM to guarantee sufficient approximation of the underlying dynamics. In short, turbulent flows are generally characterized by a slow decay of the Kolmogorov n-width, which ra...

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Decay of the Kolmogorov N-width for wave problems

Cited by 112 publications

References 9 publications

Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using Gaussian process regression

Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using Gaussian process regression

Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces

Breaking the Kolmogorov Barrier in Model Reduction of Fluid Flows

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