SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

Ahlfeld, Richard; Belkouchi, Badia; Montomoli, Francesco

doi:10.1016/j.jcp.2016.05.014

Cited by 98 publications

(81 citation statements)

References 35 publications

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“…Existing studies [25,37,38] have reported that the SGNI is suitable for uncertainty analysis due to its good performance, especially in estimating the low order moments of the system response. However, most researchers are using the SGNI with the traditional univariate integration points, such as the Gauss-Hermite or Gauss-Legendre integration nodes.…”

Section: The Proposed Methods For Up Analysismentioning

confidence: 99%

Uncertainty propagation analysis by an extended sparse grid technique

Jia

Jiang

et al. 2018

Front. Mech. Eng.

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In this paper, an uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy of the probability density function (PDF) of the system response. The proposed method incorporates the extended Gauss integration into the uncertainty propagation analysis. Moreover, assisted by the Rosenblatt transformation, the various types of extended integration points are transformed into the extended Gauss-Hermite integration points, which makes the method suitable for any type of continuous distribution. Subsequently, within the sparse grid numerical integration framework, the statistical moments of the system response are obtained based on the transformed points. Furthermore, based on the maximum entropy principle, the obtained first four-order statistical moments are used to fit the PDF of the system response. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed method, which includes two mathematical problems with explicit expressions and an engineering application with a black-box model.

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Section: The Proposed Methods For Up Analysismentioning

confidence: 99%

Uncertainty propagation analysis by an extended sparse grid technique

Jia

Jiang

et al. 2018

Front. Mech. Eng.

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“…Here, only the the main relations are summarised. Full details are described in Ahlfeld [17]. The Hankel matrix of the moments is formed by…”

Section: A Optimal Gaussian Quadrature Rules For Arbitrary Random Vamentioning

confidence: 99%

“…Non-intrusive polynomial chaos methods can be applied to propagate any probability distribution or random data set, as long as they have finite moments and their moment matrix is determinate in the Hamburger sense [17]. To circumvent the fact that heavy-tailed distributions have infinite moments, the distributions need to be truncated [18].…”

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confidence: 99%

“…In this work, a Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos, SAMBA (PC), is used [17]. SAMBA combines a simple mathematical framework to determine the PCE using the statistical moments of the input distributions with an adaptive and anisotropic version of Smolyak's algorithm to alleviate the curse of dimensionality for multiple input variables.…”

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confidence: 99%

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Uncertainty Quantification for Fat-Tailed Probability Distributions in Aircraft Engine Simulations

Ahlfeld

Montomoli

Scalas

et al. 2017

Journal of Propulsion and Power

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“…Moreover, being nested by definition, they allow for the efficient construction of sparse grids, thus enabling high-dimensional UQ studies. We note that this paper focuses on continuous PDFs of arbitrary shapes and does not consider data-driven approaches which define a PDF through its moments without any further assumptions about it [31,32].Due to a number of desirable properties, e.g., granularity, interpolation, and quadrature stability, and nestedness, weighted Leja nodes have been getting increasing attention in the context of approximation and UQ [16,30,[33][34][35][36]. Nonetheless, with the exceptions of [34,37,38] where beta distributions are considered for the parameters, the use of Leja nodes has been limited to uniform and log-normal parameter distributions.…”

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Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

Loukrezis

Gersem

2020

Algorithms

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Approximation and uncertainty quantification methods based on Lagrange interpolation are typically abandoned in cases where the probability distributions of one or more system parameters are not normal, uniform, or closely related distributions, due to the computational issues that arise when one wishes to define interpolation nodes for general distributions. This paper examines the use of the recently introduced weighted Leja nodes for that purpose. Weighted Leja interpolation rules are presented, along with a dimension-adaptive sparse interpolation algorithm, to be employed in the case of high-dimensional input uncertainty. The performance and reliability of the suggested approach is verified by four numerical experiments, where the respective models feature extreme value and truncated normal parameter distributions. Furthermore, the suggested approach is compared with a well-established polynomial chaos method and found to be either comparable or superior in terms of approximation and statistics estimation accuracy.Journal Not Specified 2020, xx, 1 2 of 21 interpolation schemes [5,6] or generalized polynomial chaos (gPC) [7], where the latter approach is typically based on least squares (LS) regression [8][9][10] or pseudo-spectral projection methods [11][12][13], or, less commonly, on interpolation [14]. Assuming a smooth input-output dependence, spectral UQ methods provide fast convergence rates, in some cases even of exponential order. However, their performance decreases rapidly for an increasing number of input parameters due to the curse of dimensionality [15]. To overcome this bottleneck, state-of-the-art approaches rely on algorithms for the adaptive construction of sparse approximations [8][9][10][16][17][18].Adaptive sparse approximation algorithms for Lagrange interpolation methods typically employ nested sequences of univariate interpolation nodes. While not strictly necessary [19], nested node sequences are helpful for the efficient construction of sparse grid interpolation algorithms. Nested node sequences are readily available for the well studied cases of uniformly or normally distributed random variables (RVs) [20,21], but not for more general probability measures. In principle, one could derive nested interpolation (or quadrature) rules tailored to an arbitrary probability density function (PDF) [22,23], however, this is a rather cumbersome task for which dedicated analyses are necessary each and every time a new PDF is considered.Diversely, the main requirement in gPC approximations, either full or sparse, is to find a complete set of polynomials that are orthogonal to each other with respect to the input parameter PDFs. In the case of arbitrary PDFs, such polynomials can be numerically constructed [24][25][26]. As a result, despite the fact that according to a number of studies Lagrange interpolation methods enjoy an advantage over gPC-based methods with respect to the error-cost ratio [27][28][29], they are abandoned whenever the input parameter PDFs are not uniform, or normal, or closely ...

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SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

Cited by 98 publications

References 35 publications

Uncertainty propagation analysis by an extended sparse grid technique

Uncertainty propagation analysis by an extended sparse grid technique

Uncertainty Quantification for Fat-Tailed Probability Distributions in Aircraft Engine Simulations

Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

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