A sequential sparse polynomial chaos expansion using Voronoi exploration and local linear approximation exploitation for slope reliability analysis

Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

In this study, a hybrid metamodel using the orthogonal constraints of radial basis function and sparse polynomial chaos expansions is proposed for the global sensitivity analysis of time-consuming models. Firstly, the orthogonal conditions of radial basis functions (RBF) and polynomial chaos expansions (PCE) were derived to construct the hybrid metamodel. Then, the variance of the metamodel was decoupled into the variances of the RBF and PCE independently by using the orthogonal condition. Furthermore, the analytical formulations of Sobol indices for the hybrid metamodel were derived according to the orthogonal decomposition. Thus, the interaction items of radial basis function and polynomial chaos expansions were eliminated, which significantly simplifies the Sobol indices. Two analytical cases were employed to investigate the influence of the number of the polynomial chaos expansions items, and several analytical and engineering cases were tested to demonstrate the accuracy and efficiency of the proposed method. In the engineering cases, the proposed method yielded significant improvements in terms of both accuracy and efficiency comparing with the existing global sensitivity analysis approaches, which indicates that the proposed method is more appropriate to the global sensitivity analysis of time-consuming engineering problems.

Sparse Polynomial Chaos Expansion (PCE) is widely used in various engineering fields to quantitatively analyse the influence of uncertainty, while alleviating the problem of dimensionality curse. However, current sparse PCE techniques focus on choosing features with the largest coefficients, which may ignore uncertainties propagated with high order features. Hence, this paper proposes the idea of selecting polynomial chaos basis based on information entropy, which aims to retain the advantages of existing sparse techniques while considering entropy change as output uncertainty. A novel entropy-based optimization method is proposed to update the state-of-the-art sparse PCE models. This work further develops an entropy-based synthetic sparse model, which has higher computational efficiency. Two benchmark functions and a computational fluid dynamics (CFD) experiment are used to compare the accuracy and efficiency between the proposed method and classical methods. The results show that entropy-based methods can better capture the features of uncertainty propagation, improving accuracy and reducing sparsity while avoiding over-fitting problems.

Global sensitivity analysis: A Bayesian learning based polynomial chaos approach

Polynomial chaos expansion have received much attention in the past ten years in the domain of computational stochastic mechanics. One keypoint in this context is the problem of the curse of dimensionality. In order to bypass this difficulty, the present paper proposes the use of sparse polynomial chaos basis built from a prior choice of monomials by the so-called hyperbolic q-norms and a selection algorithm known as Least Angle regression. The approach is illustrated by solving a structural reliability problem of a nuclear reactor vessel submitted to a pressurized thermal shock.

Global sensitivity analysis for multivariate outputs using generalized RBF-PCE metamodel enhanced by variance-based sequential sampling

A robust and efficient stepwise regression method for building sparse polynomial chaos expansions

Bayesian sparse polynomial chaos expansion for global sensitivity analysis

Using sparse polynomial chaos expansions for the global sensitivity analysis of groundwater lifetime expectancy in a multi-layered hydrogeological model

Isogeometric dual reciprocity BEM for solving non-Fourier transient heat transfer problems in FGMs with uncertainty analysis

PLS-based adaptation for efficient PCE representation in high dimensions

This paper presents an algorithm for efficient uncertainty quantification (UQ) in the presence of many uncertainties that follow a nonstandard distribution (e.g., lognormal). Using the polynomial chaos expansion (PCE), the algorithm builds surrogate models of uncertainty as functions of a standard distribution (e.g., Gaussian variables). The key to build these surrogate models is to calculate PCE coefficients of model outputs, which is computationally challenging, especially when dealing with models defined by complex functions (e.g., nonpolynomial terms) under many uncertainties. To address this issue, an algorithm that integrates the PCE with the generalized dimension reduction method (gDRM) is utilized to convert the high-dimensional integrals, required to calculate the PCE coefficients of model predictions, into several lower-dimensional ones that can be rapidly solved with quadrature rules. The accuracy of the algorithm is validated with four examples in structural reliability analysis and compared to other existing techniques, such as Monte Carlo simulations and the least angle regression-based PCE. Our results show our algorithm provides accurate UQ results and is computationally efficient when dealing with many uncertainties, thus laying the foundation to address UQ in complex control systems.

Uncertainties widely existing in composite aircraft structures make flutter reliability and sensitivity analysis highly necessary. In this work, a probabilistic approach is proposed for reliability and sensitivity analysis of composite all-movable fin flutter. To be specific, an adaptive method based on basis-adaptive polynomial chaos expansion (PCE) is developed to estimate the probability of failure, and the basis-adaptive PCE-based Sobol indices are used for measuring the contributions of input variables to the reliability over their entire uncertainty ranges. A composite beam and a composite plate are analyzed to test the accuracy and efficiency of the proposed approach. The finite element model of composite all-movable fin flutter is approximated by the basis-adaptive PCE model, which is adaptively updated by the FBR learning function. The weighed K-means strategy is added to accelerate the convergence speed and reduce the number of iterations. Then the basis-adaptive PCE model is combined with numerical simulation for reliability and sensitivity analysis of composite all-movable fin flutter. Based on the results of Monte Carlo simulation, compared with adaptive support vector regression, kriging, and the traditional PCE method, it is found that the accuracy and effectiveness of the proposed method are the best for reliability and sensitivity analysis of composite all-movable fin flutter.

Performing uncertainty quantification (UQ) and sensitivity analysis (SA) is vital when developing a patient-specific physiological model because it can quantify model output uncertainty and estimate the effect of each of the model's input parameters on the mathematical model. By providing this information, UQ and SA act as diagnostic tools to evaluate model fidelity and compare model characteristics with expert knowledge and real world observation. Computational efficiency is an important part of UQ and SA methods and thus optimization is an active area of research. In this work, we investigate a new efficient sampling method for least-squares polynomial approximation, weighted approximate Fekete points (WAFP). We analyze the performance of this method by demonstrating its utility in stochastic analysis of a cardiovascular model that estimates changes in oxyhemoglobin saturation response. Polynomial chaos (PC) expansion using WAFP produced results similar to the more standard Monte Carlo in quantifying uncertainty and identifying the most influential model inputs (including input interactions) when modeling oxyhemoglobin saturation, PC expansion using WAFP was far more efficient. These findings show the usefulness of using WAFP based PC expansion to quantify uncertainty and analyze sensitivity of a oxyhemoglobin dissociation response model. Applying these techniques could help analyze the fidelity of other relevant models in preparation for clinical application.

Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach