An Adaptive Minimum Spanning Tree Multielement Method for Uncertainty Quantification of Smooth and Discontinuous Responses

Halder, Yous van; Sanderse, Benjamin; Koren, Barry

doi:10.1137/18m1219643

Cited by 9 publications

(5 citation statements)

References 47 publications

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“…The present study does not aim to address numerical difficulties in handling discontinuities in random space using the gPC approximation, which may trigger the Gibbs phenomenon. In fact, such challenges can be addressed by machine learning to track discontinuities [42], adaptive level set methods for discontinuity detection [43], and adaptive minimum spanning tree multielement methods combined with vector machines for discontinuity identification [44].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification in shallow water-sediment flows: A stochastic Galerkin shallow water hydro-sediment-morphodynamic model

Cao

Borthwick

2021

Applied Mathematical Modelling

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Section: Introductionmentioning

confidence: 99%

Uncertainty quantification in shallow water-sediment flows: A stochastic Galerkin shallow water hydro-sediment-morphodynamic model

Cao

Borthwick

2021

Applied Mathematical Modelling

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“…A popular approach for moment estimation of high-dimensional noise is the use of sparse sampling grids [22,60]. Recently, a spline approximation based on sparse grids was used in the context of forward uncertainty propagation [55]. Most sparse-grid methods, however, are designed with moment estimation in mind.…”

Section: γ(R)γ(s)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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“…A combination of spatial and dimension-wise adaptivity has also been proposed. 11 Alternatively, the family of multi-element methods [12][13][14][15][16] presents a class of approximation techniques which have been successfully applied in the UQ context for addressing non-smooth stochastic response functions. In a multi-element method, the parameter space is decomposed into subdomains referred to as elements, where local polynomial basis functions are used.…”

Section: Introductionmentioning

confidence: 99%

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

Galetzka

Loukrezis

Georg

et al. 2023

Numerical Meth Engineering

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This article introduces an hp-adaptive multi-element stochastic collocation method, which additionally allows to re-use existing model evaluations during either h-or p-refinement. The collocation method is based on weighted Leja nodes. After h-refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub-element in a hierarchical manner.For p-refinement, the local polynomial approximations are based on total-degree or dimension-adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non-smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

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An Adaptive Minimum Spanning Tree Multielement Method for Uncertainty Quantification of Smooth and Discontinuous Responses

Cited by 9 publications

References 47 publications

Uncertainty quantification in shallow water-sediment flows: A stochastic Galerkin shallow water hydro-sediment-morphodynamic model

Uncertainty quantification in shallow water-sediment flows: A stochastic Galerkin shallow water hydro-sediment-morphodynamic model

Density Estimation in Uncertainty Propagation Problems Using a Surrogate Model

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

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