A non-adapted sparse approximation of PDEs with stochastic inputs

Doostan, Alireza; Owhadi, Houman

doi:10.1016/j.jcp.2011.01.002

Cited by 428 publications

(532 citation statements)

References 65 publications

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“…To limit the number of terms, a sparse basis should be constructed which can be determined from the adaptive scheme or directly from a random sampling of the quantity of interest. In [25,27], the most significant terms of the PCE are extracted using iterative algorithm aiming at reducing not only the error of approximation but also the number of terms of the expansion. These methods are efficient if a small fraction of coefficients g i in the exact expression (10) of the quantity of interest are dominant.…”

Section: Non-intrusive Methodsmentioning

confidence: 99%

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Clénet

2016

Scientific Computing in Electrical Engineering

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. To cite this version :Stephane CLÉNET -Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics -2016Any correspondence concerning this service should be sent to the repository Administrator : archiveouverte@ensam.eu Approximation Methods to solve Stochastic Problems in Computational ElectromagneticsStéphane ClénetAbstract To account for uncertainties on model parameters, the stochastic approach can be used. The model parameters as well as the outputs are then random fields or variables. Several methods are available in the literature to solve stochastic models like sampling methods, perturbation methods or approximation methods. In this paper, we propose an overview on the solution of stochastic problems in computational electromagnetics using approximation methods. Some applications will be presented in order to illustrate the possibilities offered by the approximation methods but also their current limitations due to the curse of dimensionality. Finally, recent numerical techniques proposed in the literature to face the curse of dimensionality are presented for non-intrusive and intrusive approaches.

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Section: Non-intrusive Methodsmentioning

confidence: 99%

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Clénet

2016

Scientific Computing in Electrical Engineering

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“…For underdetermined systems, solving a regularized least squares problem is preferred (Doostan and Owhadi, 2011).…”

Section: Regressionmentioning

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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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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“…Hence the coupled system (7) is an excellent candidate for MOR. Moreover, a high potential for reduction appears, because sparse representations are often observed in a PC expansion, see [31,32]. If the number M refers to all polynomials up to a fixed degree, then a sparse representation implies that a smaller number of basis functions would also yield an approximation of the same quality.…”

Section: Mor For the Sg Systemmentioning

confidence: 99%

Stochastic Galerkin Methods and Model Order Reduction for Linear Dynamical Systems

Pulch

Maten

2015

Int. J. UncertaintyQuantification

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Linear dynamical systems are considered in the form of ordinary differential equations or differential algebraic equations. We change their physical parameters into random variables to represent uncertainties. A stochastic Galerkin method yields a larger linear dynamical system satisfied by an approximation of the random processes. If the original systems own a high dimensionality, then a model order reduction is required to

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A non-adapted sparse approximation of PDEs with stochastic inputs

Cited by 428 publications

References 65 publications

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

Approximation Methods to Solve Stochastic Problems in Computational Electromagnetics

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

Stochastic Galerkin Methods and Model Order Reduction for Linear Dynamical Systems

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