Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion

Oladyshkin, Sergey; Nowak, Wolfgang

doi:10.1016/j.ress.2012.05.002

Cited by 385 publications

(257 citation statements)

References 44 publications

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“…6.2). Furthermore, the optimization with PC-R finds optimums comparable to those found with the rectangle rule that used roughly three times as many simulations 5 . We found that starting an optimization from a good layout will, in general, find better optimums than 5 starting from a random layout, but starting from the random layout can lead to novel layouts and possibly better layouts as the turbines explore more of the design space.…”

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confidence: 84%

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Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization

Padron¹,

Thomas²,

Stanley³

et al. 2018

Preprint

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Abstract.In this paper, we develop computationally-efficient techniques to calculate statistics used in wind farm optimization with the goal of enabling the use of higher-fidelity models and larger wind farm optimization problems. We apply these techniques to maximize the Annual Energy Production (AEP) of a wind farm by optimizing the position of the individual wind turbines. The AEP (a statistic itself) is the expected power produced by the wind farm over a period of one year subject to uncertainties in 5 the wind conditions (wind direction and wind speed) that are described with empirically-determined probability distributions.To compute the AEP of the wind farm, we use a wake model to simulate the power at different input conditions composed of wind direction and wind speed pairs. We use polynomial chaos (PC), an uncertainty quantification method, to construct a polynomial approximation of the power over the entire stochastic space and to efficiently (using as few simulations as possible) compute the expected power (AEP). We explore both regression and quadrature approaches to compute the PC coefficients. 10PC based on regression is significantly more efficient than the rectangle rule (the method most commonly used to compute the expected power). With PC based on regression, we have reduced by as much as an order of magnitude the number of simulations required to accurately compute the AEP, thus enabling the use of more expensive, higher-fidelity models or larger wind farm optimizations. We perform a large suite of gradient-based optimizations with different initial turbine locations and with different numbers of samples to compute the AEP. The optimizations with PC based on regression result in optimized 15 layouts that produce the same AEP as the optimized layouts found with the rectangle rule but using only one-third of the samples. Furthermore, for the same number of samples, the AEP of the optimal layouts found with PC is 1 % higher than the AEP of the layouts found with the rectangle rule.

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confidence: 84%

“…We showed that to properly compare methods an ensemble of 5 The rectangle rule used almost three times as many simulations to compute the AEP for each optimization iteration (see the number of samples in Table 2), which results in the optimization using roughly three times as many simulations.…”

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confidence: 99%

Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization

Padron¹,

Thomas²,

Stanley³

et al. 2018

Preprint

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show abstract

“…Moreover, Oladyshkin and Nowak presented a derivation of the optimal orthogonal polynomials from the moments. In the cases he reported, the convergence of the moment-based expansion was significantly better that for any other polynomial expansion [7] using fitted parametric PDFs.…”

Section: Introductionmentioning

confidence: 89%

“…In recent years, however, engineering applications have created a growing demand for the extension of Polynomial Chaos techniques to more general input distributions [7]. The gPC was extended to arbitrary input distributions by splitting the random space into piecewise elements and propagating them locally using the Askey scheme [8,9,10,11].…”

Section: Introductionmentioning

confidence: 99%

“…Oladyshkin and Nowak observed that as every set of random data, as well as a continuous or discrete PDF, can be described using the moments without making any assumptions about the shape or existence of a suitable probability distribution, the moments provided a very general approach to propagate data without requiring the determination of deterministic PDFs. Oladyshkin and Nowak [7] promoted this concept in the geo-sciences by successfully applying it to identify uncertainties in carbon dioxide storage in geological formations [15,16,17] and also for robust design [18]. Moreover, Oladyshkin and Nowak presented a derivation of the optimal orthogonal polynomials from the moments.…”

Section: Introductionmentioning

confidence: 99%

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SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

Ahlfeld

Belkouchi

Montomoli

2016

Journal of Computational Physics

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Please cite this article in press as: R. Ahlfeld et al., SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos, J. Comput. Phys. (2016), http://dx.doi.org/10.1016/j. jcp.2016.05.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. SAMBA AbstractA new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability.The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of

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Impact of space‐time mesh adaptation on solute transport modeling in porous media

Esfandiar

Porta

Perotto

et al. 2015

Water Resources Research

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We implement a space-time grid adaptation procedure to efficiently improve the accuracy of numerical simulations of solute transport in porous media in the context of model parameter estimation. We focus on the Advection Dispersion Equation (ADE) for the interpretation of nonreactive transport experiments in laboratory-scale heterogeneous porous media. When compared to a numerical approximation based on a fixed space-time discretization, our approach is grounded on a joint automatic selection of the spatial grid and the time step to capture the main (space-time) system dynamics. Spatial mesh adaptation is driven by an anisotropic recovery-based error estimator which enables us to properly select the size, shape, and orientation of the mesh elements. Adaptation of the time step is performed through an ad hoc local reconstruction of the temporal derivative of the solution via a recovery-based approach. The impact of the proposed adaptation strategy on the ability to provide reliable estimates of the key parameters of an ADE model is assessed on the basis of experimental solute breakthrough data measured following tracer injection in a nonuniform porous system. Model calibration is performed in a Maximum Likelihood (ML) framework upon relying on the representation of the ADE solution through a generalized Polynomial Chaos Expansion (gPCE). Our results show that the proposed anisotropic space-time grid adaptation leads to ML parameter estimates and to model results of markedly improved quality when compared to classical inversion approaches based on a uniform space-time discretization.

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Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion

Cited by 385 publications

References 44 publications

Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization

Polynomial chaos to efficiently compute the annual energy production in wind farm layout optimization

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

Impact of space‐time mesh adaptation on solute transport modeling in porous media

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