Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark

Lüthen, Nora; Marelli, Stefano; Sudret, Bruno

doi:10.48550/arxiv.2002.01290

Cited by 3 publications

(8 citation statements)

References 86 publications

(236 reference statements)

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“…• Sparse gPC expansion with a total degree truncation [51] (denoted as Sparse gPC). With the truncation scheme in Eq.…”

Section: A Baseline Methods For Comparisonmentioning

confidence: 99%

High-Dimensional Uncertainty Quantification via Active and Rank-Adaptive Tensor Regression

Zhang

2020

2020 IEEE 29th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS)

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Fabrication process variations can significantly influence the performance and yield of nano-scale electronic and photonic circuits. Stochastic spectral methods have achieved great success in quantifying the impact of process variations, but they suffer from the curse of dimensionality. Recently, low-rank tensor methods have been developed to mitigate this issue, but two fundamental challenges remain open: how to automatically determine the tensor rank and how to adaptively pick the informative simulation samples. This paper proposes a novel tensor regression method to address these two challenges. We use a q / 2 group-sparsity regularization to determine the tensor rank. The resulting optimization problem can be efficiently solved via an alternating minimization solver. We also propose a twostage adaptive sampling method to reduce the simulation cost. Our method considers both exploration and exploitation via the estimated Voronoi cell volume and nonlinearity measurement respectively. The proposed model is verified with synthetic and some realistic circuit benchmarks, on which our method can well capture the uncertainty caused by 19 to 100 random variables with only 100 to 600 simulation samples.

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“…• Sparse gPC expansion with a total degree truncation [51] (denoted as Sparse gPC). With the truncation scheme in Eq.…”

Section: A Baseline Methods For Comparisonmentioning

confidence: 99%

High-Dimensional Uncertainty Quantification via Active and Rank-Adaptive Tensor Regression

Zhang

2020

2020 IEEE 29th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS)

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“…, with M typically chosen in the range between 2 ⋅ D and 3 ⋅ D, [23] and the corresponding model evaluations…”

Section: Polynomial Chaos Expansionmentioning

confidence: 99%

Uncertainty quantification in three‐dimensional magnetohydrodynamic equilibrium reconstruction via surrogate‐assisted Bayesian inference

Köberl,

von Toussaint,

Bungartz

et al. 2023

Contributions to Plasma Physics

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In three‐dimensional (3D) equilibrium, reconstruction defining parameters of an ideal magneto‐hydrodynamic equilibrium are inferred from a set of plasma diagnostic measurements. For the reconstructed parameters, various forms of uncertainty estimates exist within common 3D reconstruction frameworks. These estimates often assume a Gaussian posterior distribution. The validity of this assumption is not obvious in such highly nonlinear inverse problems, and therefore the accuracy of the estimates cannot be guaranteed. In this work, we formulate the problem of 3D equilibrium reconstruction in a Bayesian sense and explore the posterior distribution of reconstruction parameters via Markov chain Monte Carlo (MCMC) sampling. The target reconstruction parameters, that is, shape and scaling factors of the pressure and toroidal current‐density profiles as well as the total toroidal flux, are taken from a reduced subspace of Wendelstein 7‐X equilibrium configurations. Since the corresponding forward model evaluations are computationally demanding, we replace the forward model via a polynomial chaos expansion surrogate. We compare the posterior distribution obtained via MCMC sampling to Laplace's approximation, which assumes a Gaussian posterior. We find that the two approaches provide similar results in regimes where the uncertainty on the plasma diagnostic signals is low. However, discrepancies between the posteriors are observed in cases of higher diagnostic signal uncertainty. Therefore, we provide further validation of the commonly used Laplace's approximation as a method of uncertainty quantification in 3D equilibrium reconstruction and showcase some of its limitations.

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“…To assess the predictive ability of the PCE surrogate, we employ an error metric known as the generalization error [9], which is defined as…”

Section: Surrogate Modeling Via Pcementioning

confidence: 99%

“…Accordingly, the total degree s max is chosen so that the validation error is minimized. In cases where a validation dataset cannot be generated due to computational constraints, alternative measures such as the k-fold cross validation can be considered [13,12,9]. However, such techniques introduce different computational costs, as they require the construction of multiple surrogates for different partitionings of training dataset, therefore a prior evaluation of the trade-off and respective costs is required.…”

Section: Surrogate Modeling Via Pcementioning

confidence: 99%

“…One of the main advantages of PCE methods is that UQ tasks such as moment estimation and sensitivity analysis can be easily performed by post-processing the terms of the PCE [6,7,8]. In the case of a high-dimensional input parameter space, sparse PCE methods have been successfully proposed in the literature, which take advantage of the so-called "sparsity of effects" principle to construct surrogates with only a small number of forward model evaluations [9,10,11]. In such cases however, a suitable model selection criterion must be employed for tuning the hyperparameters that are used to obtain the optimal model.…”

Section: Introductionmentioning

confidence: 99%

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Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models

Kontolati,

Loukrezis,

Santos

et al. 2021

Preprint

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In this work we introduce a manifold learning-based method for uncertainty quantification (UQ) in systems describing complex spatiotemporal processes. Our first objective is to identify the embedding of a set of high-dimensional data representing quantities of interest of the computational or analytical model. For this purpose, we employ Grassmannian diffusion maps, a two-step nonlinear dimension reduction technique which allows us to reduce the dimensionality of the data and identify meaningful geometric descriptions in a parsimonious and inexpensive manner. Polynomial chaos expansion is then used to construct a mapping between the stochastic input parameters and the diffusion coordinates of the reduced space. An adaptive clustering technique is proposed to identify an optimal number of clusters of points in the latent space. The similarity of points allows us to construct a number of geometric harmonic emulators which are finally utilized as a set of inexpensive pre-trained models to perform an inverse map of realizations of latent features to the ambient space and thus perform accurate out-of-sample predictions. Thus, the proposed method acts as an encoder-decoder system which is able to automatically handle very high-dimensional data while simultaneously operating successfully in the small-data regime. The method is demonstrated on two benchmark problems and on a system of advection-diffusion-reaction equations which model a first-order chemical reaction between two species. In all test cases, the proposed method is able to achieve highly accurate approximations which ultimately lead to the significant acceleration of UQ tasks.

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Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark

Cited by 3 publications

References 86 publications

High-Dimensional Uncertainty Quantification via Active and Rank-Adaptive Tensor Regression

High-Dimensional Uncertainty Quantification via Active and Rank-Adaptive Tensor Regression

Uncertainty quantification in three‐dimensional magnetohydrodynamic equilibrium reconstruction via surrogate‐assisted Bayesian inference

Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models

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