POD-Galerkin Modeling and Sparse-Grid Collocation for a Natural Convection Problem with Stochastic Boundary Conditions

Ullmann, Sebastian; Lang, Jens

doi:10.1007/978-3-319-04537-5_13

Cited by 9 publications

(16 citation statements)

References 18 publications

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“…To our best knowledge, little attempt has been made to use the sparse grid method in ROMs with exception of Peherstorfer [37], Cheng [38], Ullmann [39] and Lang, and Sumant [40]. Peherstorfer [37] presented a reduced-order model of parametrised systems by employing a sparse grid machine learning method and applied this new ROM to thermal conduction and chemical reaction simulations.…”

Section: Introductionmentioning

confidence: 99%

Non-intrusive reduced order modelling of the Navier–Stokes equations

Xiao

Fang²,

Buchan³

et al. 2015

Computer Methods in Applied Mechanics and Engineering

136

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Section: Introductionmentioning

confidence: 99%

Non-intrusive reduced order modelling of the Navier–Stokes equations

Xiao

Fang²,

Buchan³

et al. 2015

Computer Methods in Applied Mechanics and Engineering

136

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“…Model order reduction is a tool to decrease the computational cost for applications where a parametrized PDE problem needs to be solved multiple times for different parameter values. Therefore model order reduction is often studied in the context of optimal control [7,15,20] or uncertainty quantification [5,8,22]. Snapshot-based model order reduction requires a set of representative samples of the solution, which need to be computed in advance.…”

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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“…In many applications, one is interested in computing the probability density function (PDF) of a certain "quantity of interest" (output) of the model [1,6,7,21,33,42,54]. Often, density estimation is performed using standard uncertainty propagation methods and surrogate models [22,48], such as Stochastic Finite Element and generalized Polynomial Chaos (gPC) [23,36,47,60], hp-gPC [57], and Wiener-Haar expansion [32], since these methods can approximate moments with spectral accuracy [61,62].…”

mentioning

confidence: 99%

“…The PDF of the pressure and of the velocity were computed in [54] when θ 1 (y; α) = θ 1 (α) and α is uniformly distributed in [α min , α max ], and in [21] when θ 1 (y; α α α) is a Gaussian random process.…”

mentioning

confidence: 99%

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

Ditkowski¹,

Fibich²,

Sagiv³

2020

SIAM/ASA J. Uncertainty Quantification

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The effect of uncertainties and noise on a quantity of interest (model output) is often better described by its probability density function (PDF) than by its moments. Although density estimation is a common task, the adequacy of approximation methods (surrogate models) for density estimation has not been analyzed before in the uncertainty-quantification (UQ) literature. In this paper, we first show that standard surrogate models (such as generalized polynomial chaos), which are highly accurate for moment estimation, might completely fail to approximate the PDF, even for one-dimensional noise. This is because density estimation requires that the surrogate model accurately approximates the gradient of the quantity of interest, and not just the quantity of interest itself. Hence, we develop a novel spline-based algorithm for density-estimation whose convergence rate in L q is polynomial in the sampling resolution. This convergence rate is better than that of standard statistical density-estimation methods (such as histograms and kernel density estimators) at dimensions 1 ≤ d ≤ 5 2 m, where m is the spline order. Furthermore, we obtain the convergence rate for density estimation with any surrogate model that approximates the quantity of interest and its gradient in L ∞ . Finally, we demonstrate our algorithm for problems in nonlinear optics and fluid dynamics.1 g in L q , the corresponding density p g might not be a good approximation of p f . Indeed, for p g to approximate p f , then g ′ needs to be close to f ′ , and g ′ (α) should also vanish if and only if f ′ (α) does. These conditions might not be satisfied by some of the above-mentioned standard UQ methods. In contrast, spline interpolation approximates both the function and its gradient [3,25,41,45], and does not tend to produce spurious extremal points. Therefore, we construct a novel algorithm for density estimation in uncertainty-propagation problems using splines as our surrogate model. With cubic splines, our density-estimation algorithm has a guaranteed convergence rate of at least h 3 , where h is the maximal sampling spacing (resolution). More generally, with splines of order m, the convergence rate is at least h m . These rates are superior to those of the standard kernel density estimators [52,59], for noise dimension 1 ≤ d ≤ 5 2 m. Our choice of splines is motivated by the availability and efficiency of one-and multi-dimensional spline toolboxes. Nevertheless, other surrogate models can be used in this algorithm, and indeed this paper lays the theoretical framework for deriving the convergence rate of such methods (Corollaries 4.8 and 5.5). We show, essentially, that density estimation convergence can be performed with any surrogate model for which the L ∞ error of both the function and its gradient converge to zero as the spacing resolution h vanishes. Because we only rely on solving the underlying deterministic model (i.e., our method is non-intrusive), and because interpolation by spline is a standard numerical procedure, our proposed method is...

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POD-Galerkin Modeling and Sparse-Grid Collocation for a Natural Convection Problem with Stochastic Boundary Conditions

Cited by 9 publications

References 18 publications

Non-intrusive reduced order modelling of the Navier–Stokes equations

Non-intrusive reduced order modelling of the Navier–Stokes equations

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

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

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