Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark

Lüthen, Nora; Marelli, Stefano; Sudret, Bruno

doi:10.1137/20m1315774

Cited by 156 publications

(55 citation statements)

References 105 publications

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“…The construction of a surrogate model of implicit function h c by using a deterministic approach, such as the meshless methods [27,28,29,30,31,32], is not adapted taking into account a possible high dimension of the space on which h c is defined. Similarly, the construction of a representation on the chaos (Gaussian or another probability measure) [33,34,35,36,37,38,39,40,41,42,43] would not be at all effective in our case for the same reasons related to the possible high dimension. To circumvent this difficulty, we generalize the approach proposed in [26], which consists in constructing a statistical surrogate model ĥN of h c , depending on the number N of points generated in the constrained learned set, for which its gradient has an explicit algebraic representation.…”

Section: Introductionmentioning

confidence: 91%

Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints

Soize

2022

Preprint

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In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology in a parallel computing framework. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.

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Section: Introductionmentioning

confidence: 91%

Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints

Soize

2022

Preprint

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“…The method of the SPCE method has attracted much attention in the community of probabilistic analysis and uncertainty quantification [19,20]. In this section, the SPCE method is introduced as follows.…”

Section: Sparse Polynomial Chaos Expansionsmentioning

confidence: 99%

“…The first order reliability method requires an explicit expression of the limit-state function, which is, however, not always available. The method of sparse polynomial chaos expansions combined with Monte Carlo Simulations has been widely used in probabilistic analysis of classical geotechnical problems, such as slopes, tunnels, retaining walls, and foundations [14,15,19]. The benefit of the sparse polynomial chaos expansions (SPCE) is that it has higher computational efficiency compared with common polynomial chaos expansions, making it applicable to high-dimensional problems with accurate results of probabilistic analysis.…”

Section: Introductionmentioning

confidence: 99%

The Effects of Seismic Coefficient Uncertainty on Pseudo-Static Slope Stability: A Probabilistic Sensitivity Analysis

Sun

et al. 2021

Sustainability

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The method of pseudo-static analysis has been widely used to perform seismic slope stability, in which a seismic coefficient is used to represent the earthquake shaking effect. However, it is important but difficult to select the magnitude of seismic coefficients, which are inevitably subjected to different levels of uncertainties. This paper aimed to study the influences of seismic coefficient uncertainties on pseudo-static slope stability from the perspective of probabilistic sensitivity analysis. The deterministic critical slope height was estimated by the method of upper-bound limit analysis with the method of pseudo-static analysis. The soil shear strength parameters, the slope geometrical parameters (including slope inclinations, slope heights, and the slope widths), the horizontal seismic acceleration coefficient, and the unit weight of soil masses were considered as random variables. The influences of their uncertainty degrees, the correlation relations, and the distribution types of random variables on probabilistic density functions, failure probabilities, and sensitivity analysis were discussed. It was shown that the uncertainty degrees greatly impact the probability density distributions of critical slope heights, the computed failure probabilities, and Sobol’ index, and the horizontal seismic coefficient was the second most important variable compared to the soil shear strength parameters.

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“…A powerful class of methods are regression-based approaches that rely on an initial input sample X , called experimental design, and corresponding model evaluations M(X ) (See, e.g. Lüthen et al (2020) for a recent survey). Additionally, it is possible to design adaptive algorithms that choose the truncated basis size (Blatman and Sudret, 2011;Jakeman et al, 2015).…”

Section: Spectral Function Decompositionmentioning

confidence: 99%

Bayesian model inversion using stochastic spectral embedding

Wagner,

Marelli,

Sudret

2020

Preprint

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In this paper we propose a new sampling-free approach to solve Bayesian model inversion problems that extends the recently introduced spectral likelihood expansions (SLE) method.The latter solves the inverse problem by expanding the likelihood function onto a global polynomial basis orthogonal w.r.t. the prior distribution. This gives rise to analytical expressions for key statistics of the Bayesian posterior distribution, such as evidence, posterior moments and posterior marginals by simple post-processing of the expansion coefficients.It is well known that in most practically relevant scenarios, likelihood functions have close to compact support, which causes the global SLE approach to fail due to the high polynomial degree required for an accurate spectral representation. To solve this problem, we herein replace the global polynomial expansion from SLE with a recently proposed method for local spectral expansion refinement called stochastic spectral embedding (SSE). This surrogatemodeling method was developed for functions with high local complexity. To increase the efficiency of SSE, we enhance it with an adaptive sample enrichment scheme. We show that SSE works well for likelihood approximations and retains the relevant spectral properties of SLE, thus preserving analytical expressions of posterior statistics.To assess the performance of our approach, we include three case studies ranging from low to high dimensional model inversion problems that showcase the superiority of the SSE approach compared to SLE and present the approach as a promising alternative to existing inversion frameworks.

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Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark

Cited by 156 publications

References 105 publications

Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints

Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints

The Effects of Seismic Coefficient Uncertainty on Pseudo-Static Slope Stability: A Probabilistic Sensitivity Analysis

Bayesian model inversion using stochastic spectral embedding

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