Reduced Basis Exact Error Estimates with Wavelets

Ali, Mazen; Urban, Karsten

doi:10.1007/978-3-319-39929-4_34

Cited by 8 publications

(17 citation statements)

References 7 publications

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“…k of the remaining singular values. However, caution must be taken when using this method instead of the SVD, because the condition number κ(U) := σ max (U)/σ min (U) of the associated snapshot matrix U is squared: κ(U T U) = κ(U) 2 . This can lead to inaccuracy of POD modes with small singular values (see the famous example of Lächli [33]).…”

Section: The Snapshot Pod and Randomized Svdmentioning

confidence: 99%

“…This fact has both advantages and drawbacks: On the one hand, adaptation techniques allow direct relations to the "truth", i.e., the exact solution of the PDE or the underlying physical system. For example, [1,2] obtain approximations of the exact solution of the PDE within a given tolerance by using dual wavelet expansions of the PDE systems' continuous residual together with greedy reduced basis methods. Furthermore, adaptation is advantageous over equidistant grids because it distributes computational efforts to places where relevant information is located.…”

Section: Introductionmentioning

confidence: 99%

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Wavelet adaptive proper orthogonal decomposition for large-scale flow data

et al. 2022

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The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we propose a wavelet-based adaptive version of the POD (the wPOD), in order to overcome this limitation. The amount of data to be analyzed is reduced by compressing them using biorthogonal wavelets, yielding a sparse representation while conveniently providing control of the compression error. Numerical analysis shows how the distinct error contributions of wavelet compression and POD truncation can be balanced under certain assumptions, allowing us to efficiently process high-resolution data from three-dimensional simulations of flow problems. Using a synthetic academic test case, we compare our algorithm with the randomized singular value decomposition. Furthermore, we demonstrate the ability of our method analyzing data of a two-dimensional wake flow and a three-dimensional flow generated by a flapping insect computed with direct numerical simulation.

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Section: The Snapshot Pod and Randomized Svdmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wavelet adaptive proper orthogonal decomposition for large-scale flow data

et al. 2022

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“…Then the error between the reduced-order solution and a corresponding discrete solution can be estimated with the help of the discrete residual. For the case of adaptive snapshot computations, [1,2] use wavelet techniques to estimate the required dual norm of the continuous residual. In contrast, [23] derives a bound for the dual norm of the continuous residual from a special mixed finite element and reduced basis formulation.…”

Section: Introductionmentioning

confidence: 99%

POD-Galerkin reduced-order modeling with adaptive finite element snapshots

Ullmann

Rotkvic

Lang

2016

Journal of Computational Physics

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We consider model order reduction by proper orthogonal decomposition (POD) for parametrized partial differential equations, where the underlying snapshots are computed with adaptive finite elements. We address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces. We propose a method to create a POD-Galerkin model without interpolating the snapshots onto their common finite element mesh. The error of the reduced-order solution is not necessarily Galerkin orthogonal to the reduced space created from space-adapted snapshot. We analyze how this influences the error assessment for POD-Galerkin models of linear elliptic boundary value problems. As a numerical example we consider a two-dimensional convection-diffusion equation with a parametrized convective direction. To illustrate the applicability of our techniques to non-linear timedependent problems, we present a test case of a two-dimensional viscous Burgers equation with parametrized initial data.

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“…The Allen-Cahn equation (1) was originally introduced to describe the phase of a binary mixture. Nowadays it is used as a model for interface problems in material science, fluid dynamics, image analysis, mean curvature flow, and pattern formation.…”

mentioning

confidence: 99%

“…The parameter ε is related to the interfacial width, capturing the dominating effect of reaction kinetics and stays for effective diffusivity. The non-linear term f (u) in (1) is given by f (u) = F (u). Depending on the choice of the potential functional F(u), i.e.…”

mentioning

confidence: 99%

Energy Stable Model Order Reduction for the Allen-Cahn Equation

Uzunca

Karasözen

2017

Model Reduction of Parametrized Systems

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The Allen-Cahn equation is a gradient system, where the free-energy functional decreases monotonically in time. We develop an energy stable reduced order model (ROM) for a gradient system, which inherits the energy decreasing property of the full order model (FOM). For the space discretization we apply a discontinuous Galerkin (dG) method and for time discretization the energy stable average vector field (AVF) method. We construct ROMs with proper orthogonal decomposition (POD)-greedy adaptive sampling of the snapshots in time and evaluating the nonlinear function with greedy discrete empirical interpolation method (DEIM). The computational efficiency and accuracy of the reduced solutions are demonstrated numerically for the parametrized Allen-Cahn equation with Neumann and periodic boundary conditions.

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Reduced Basis Exact Error Estimates with Wavelets

Cited by 8 publications

References 7 publications

Wavelet adaptive proper orthogonal decomposition for large-scale flow data

Wavelet adaptive proper orthogonal decomposition for large-scale flow data

POD-Galerkin reduced-order modeling with adaptive finite element snapshots

Energy Stable Model Order Reduction for the Allen-Cahn Equation

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