Sparse grids

Bungartz, Hans‐Joachim; Griebel, Michael

doi:10.1017/cbo9780511569975.003

Cited by 354 publications

(644 citation statements)

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“…and the vectorf contains all coefficient approximations; the subscript p on U n p is for pseudospectral. Note that we have overloaded the notation by defininĝ f i as the pseudospectral coefficient (15), instead of the true Fourier coefficient in (5). We next state two lemmas about the relationship between the spectral collocation and pseudospectral approximations for future reference.…”

Section: Gaussian Quadrature Collocation Pseudospectral Methodsmentioning

confidence: 99%

“…In practice, one may wish to take advantage of the relatively small number of points in the sparse grid quadrature rule when computing a pseudospectral approximation. This is often done by first truncating the Fourier series representation of f (s) (see (5)), and then approximating its spectral coefficients with a sparse grid quadrature rule. Unfortunately, choosing the parameters of the sparse grid rule that will accurately approximate the integral formulation of the Fourier coefficient is not straightforward.…”

Section: Sparse Pseudospectral Approximation Methodsmentioning

confidence: 99%

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Sparse pseudospectral approximation method

Constantine

Eldred²,

Phipps³

2012

Computer Methods in Applied Mechanics and Engineering

172

170

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Multivariate global polynomial approximations -such as polynomial chaos or stochastic collocation methods -are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials.By reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.

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Section: Gaussian Quadrature Collocation Pseudospectral Methodsmentioning

confidence: 99%

Section: Sparse Pseudospectral Approximation Methodsmentioning

confidence: 99%

Sparse pseudospectral approximation method

Constantine

Eldred²,

Phipps³

2012

Computer Methods in Applied Mechanics and Engineering

172

170

View full text Add to dashboard Cite

show abstract

“…However, this method suffers from the curse of dimensionality, i.e. the number of needed evaluations rises exponentially with the number of uncertain parameters [17], such that this method is only usable for scenario C at the very most, cf. third column in Tab.…”

Section: Robust Optimizationmentioning

confidence: 99%

“…I. Instead, Smolyak sparse grids are employed, which represent subsets of the full tensor grids [17]. Furthermore, Clenshaw-Curtis knots are used to lower the number of needed PDE evaluations further [18].…”

Section: Robust Optimizationmentioning

confidence: 99%

Robust Optimization of a Permanent-Magnet Synchronous Machine Considering Uncertain Driving Cycles

D'Angelo¹,

Bontinck²,

Schöps³

et al. 2020

IEEE Trans. Magn.

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This work focuses on the robust optimization of a permanent magnet (PM) synchronous machine while considering a driving cycle. The robustification is obtained by considering uncertainties of different origins. Firstly, there are geometrical uncertainties caused by manufacturing inaccuracies. Secondly, there are uncertainties linked to different driving styles. The final set of uncertainties is linked to ambient parameters such as traffic and weather conditions. The optimization goal is to minimize the PM's volume while maintaining a desired machine performance measured by the energy efficiency over the driving cycle and the machine's maximal torque. The magnetic behavior of the machine is described by a partial differential equation (PDE) and is simulated by the finite-element method employing an affine decomposition to avoid reassembling of the system of equations due to the changing PM geometry. The Sequential Quadratic Programming algorithm is used for the optimization. Stochastic collocation is applied to compute moments of stochastic quantities. The robustness of the optimized configurations is validated by a Monte Carlo sampling. It is found that the uncertainties in driving style and road conditions have significant influence on the optimal PM configuration.

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“…The sparse grid (SG) stochastic collocation methods are used in this study to develop the surrogate models, because it has been demonstrated that the SG methods are efficient and effective for groundwater flow models (Zeng et al, ) and solute transport models (Zhang et al, ). The SG surrogate models are built based on Smolyak collocation method (Smolyak, ) and hierarchical interpolation method (Bungartz & Griebel, ). The key idea of the SG method is to generate a sparse grid in the model parameter space with a small number of SG nodes, each of which is a point in the parameter space (Barthelmann et al, ).…”

Section: Introductionmentioning

confidence: 99%

Improved Nested Sampling and Surrogate‐Enabled Comparison With Other Marginal Likelihood Estimators

Zeng¹,

Wu³

et al. 2018

Water Resources Research

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Estimating marginal likelihood is of central importance to Bayesian model selection and/or model averaging. The nested sampling method has been recently used together with the Metropolis‐Hasting (M‐H) sampling algorithm for estimating marginal likelihood. This study develops a new implementation of nested sampling by using the DiffeRential Evolution Adaptive Metropolis (DREAMzs) sampling algorithm. The two implementations of nested sampling are evaluated for four models of a synthetic groundwater flow modeling. The DREAMzs‐based nested sampling outperforms the M‐H‐based nested sampling in terms of their accuracy, convergence, efficiency, and stability, which is attributed to the fact that DREAMzs is more robust than M‐H for parameter sampling. The nested sampling method is also compared with four other methods (arithmetic mean, harmonic mean, stabilized harmonic mean, and thermodynamic integration) commonly used for estimating marginal likelihood and posterior probability of the four groundwater models. The comparative study requires hundreds of millions of model executions, which would not be possible without using surrogate models to replace the original models. Using the arithmetic mean estimator as the reference, the comparison reveals that thermodynamic integration outperforms nested sampling in terms of accuracy and stability, whereas nested sampling is more computationally efficient to reach to convergence. The harmonic mean and stabilized harmonic mean methods give the worst marginal likelihood estimation. Accurate marginal likelihood estimation is important for accurate estimation of posterior model probability and better predictive performance of Bayesian model averaging.

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Sparse grids

Cited by 354 publications

References 0 publications

Sparse pseudospectral approximation method

Sparse pseudospectral approximation method

Robust Optimization of a Permanent-Magnet Synchronous Machine Considering Uncertain Driving Cycles

Improved Nested Sampling and Surrogate‐Enabled Comparison With Other Marginal Likelihood Estimators

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