Global sensitivity analysis: A Bayesian learning based polynomial chaos approach

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.

A range of different techniques for determining the unknown coefficients of Polynomial Chaos Expansion (PCE) models for antenna structures, are presented. PCE models offer significant advantages over Monte Carlo analysis, for the modeling of the statistical behavior of structures, but the different approaches for calculating the PCE model coefficients exhibits a large variation in the number of required basis points, that number also being problem specific. A range of model coefficient calculation techniques are evaluated in this article, for the problem of modeling cross-polarization of an inset-fed patch antenna. This structure is one of the most widely used antenna structures, often produced in high-volume, and serves as an excellent example to emphasize the variation in performance between the different techniques, and problem-specific nature of finding an optimal solution. It is shown that the most optimal method requires fewer than 10% analysis points of the least optimal, and a factor of 100 fewer points than Monte-Carlo analysis.

Development of clustered polynomial chaos expansion model for stochastic hydrological prediction

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

Compressive sensing is a sub-Nyquist sampling technique for efficient signal acquisition and reconstruction of sparse or compressible signals. In order to account for the sparsity of the underlying signal of interest, it is common to use sparsifying priors such as Bernoulli–Gaussian-inverse Gamma (BGiG) and Gaussian-inverse Gamma (GiG) priors on the components of the signal. With the introduction of variational Bayesian inference, the sparse Bayesian learning (SBL) methods for solving the inverse problem of compressive sensing have received significant interest as the SBL methods become more efficient in terms of execution time. In this paper, we consider the sparse signal recovery problem using compressive sensing and the variational Bayesian (VB) inference framework. More specifically, we consider two widely used Bayesian models of BGiG and GiG for modeling the underlying sparse signal for this problem. Although these two models have been widely used for sparse recovery problems under various signal structures, the question of which model can outperform the other for sparse signal recovery under no specific structure has yet to be fully addressed under the VB inference setting. Here, we study these two models specifically under VB inference in detail, provide some motivating examples regarding the issues in signal reconstruction that may occur under each model, perform comparisons and provide suggestions on how to improve the performance of each model.

In this paper, surrogate models are built iteratively using polynomial chaos expansion (PCE) and detailed numerical simulations of a carbon sequestration system. Output variables from a numerical simulator are approximated as polynomial functions of uncertain parameters. Once generated, PCE representations can be used in place of the numerical simulator and often decrease simulation times by several orders of magnitude. However, PCE models are expensive to derive unless the number of terms in the expansion is moderate, which requires a relatively small number of uncertain variables and a low degree of expansion. To cope with this limitation, instead of using a classical full expansion at each step of an iterative PCE construction method, we introduce a mixed-integer programming (MIP) formulation to identify the best subset of basis terms in the expansion. This approach makes it possible to keep the number of terms small in the expansion. Monte Carlo (MC) simulation is then performed by substituting the va...

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

We present a global sensitivity analysis that quantifies the impact of parameter uncertainty on model outcomes. Specifically, we propose variance‐decomposition‐based Sobol' indices to establish an importance ranking of parameters and univariate effects to determine the direction of their impact. We employ the state‐of‐the‐art approach of constructing a polynomial chaos expansion of the model, from which Sobol' indices and univariate effects are then obtained analytically, using only a limited number of model evaluations. We apply this analysis to several quantities of interest of a standard real‐business‐cycle model and compare it to traditional local sensitivity analysis approaches. The results show that local sensitivity analysis can be very misleading, whereas the proposed method accurately and efficiently ranks all parameters according to importance, identifying interactions and nonlinearities. Computational techniques uncertainty quantification sensitivity analysis polynomial chaos expansion C60 C63

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.

We propose a set of equations for preliminary risk‐targeted seismic design of structures base‐isolated using single friction pendulum (SFP) bearings. The equations offer statistical estimates of the maximum displacement of the superstructure relative to the base and the maximum displacement of the isolated base response quantities of interest (RQIs), given ground motion intensity measures and the essential dynamic properties of the SFP bearing base‐isolated structure. The set of proposed equations enables preliminary seismic design of SFP base‐isolated structures using the mean annual frequency of exceeding displacement‐defined limit states related to the two RQIs. To develop the design equations, we define and use a two‐degree‐of‐freedom (2DOF) surrogate model that features a few essential dynamic parameters of the SFP bearing base‐isolated structure. We first verify that the 2DOF surrogate model is accurate enough to represent the base and superstructure displacements compared to the complete multiple‐degree‐of‐freedom model of the prototype SFP bearing base‐isolated structure. Next, we perform more than 5 million non‐linear dynamical analyses of 4900 distinct 2DOF surrogates subjected to 210 different recorded ground motions scaled five times. We fit the response data of each 2DOF surrogate to a linear model and use two different modeling techniques to derive the design equations. One model is based on Polynomial Chaos Expansion (PCE), and the other model is based on Linear Regression (LR). The PCE model is more accurate than the LR model, with the trade‐off regarding its simplicity. Nevertheless, both the PCE and the LR models are accurate enough to estimate the design quantities of interest of the SFP bearing base‐isolated structure for preliminary design purposes. Lastly, we exemplify the use of the proposed design equations and we show that the estimate of base displacement is conservative in the cases when the superstructure of the SPF bearing base‐isolated structure yields.

Soils show inherent variability due to various physio-chemical, geological and environmental processes. It is necessary to consider soil spatial variability in evaluation and probabilistic design of foundations. However probabilistic frameworks become computationally expensive and infeasible for a complex realistic scenario. This study develops an approach to evaluate the bearing capacity factor of strip footing resting on a spatially varying clayey slope using a Sparse polynomial chaos expansion (SPCE) surrogate model. Non-stationary soil condition is adopted in this study. Initially random fields are generated using KL expansion method considering 95% energy variance terms. This helps to reduce the terms for a random field as input to the SPCE model. The combined model of SPCE with Global Sensitivity analysis (GSA) reduced computational time and cost. This approach is very efficient because the predicted bearing capacity factor is accurate and needs only a smaller number of realizations to train. This makes it computationally time efficient. Meanwhile using any traditional approach, the probabilistic analysis is computationally inefficient in terms of cost and time.

Extreme learning machine (ELM) is suitable for nonlinear soft sensor development. Yet it faces an overfitting problem. To overcome it, this work integrates bound optimization theory with variational Bayesian (VB) inference to derive novel L1 norm-based ELMs. An L1 term is attached to the squared sum cost of prediction errors to formulate an objective function. Considering the nonconvexity and nonsmoothness of the objective function, this article uses bound optimization theory, and constructs a proper surrogate function to equivalently convert a challenging L1 norm-based optimization problem into easy one. Then, VB inference is adopted for optimizing the converted problem. Thus, an L1 norm-based ELM can be efficiently optimized by an alternating optimization algorithm with a proved convergence. Finally, a soft sensor is developed based on the proposed algorithm. An industrial case study is carried out to demonstrate that the proposed soft sensor is competitive against recent ones.

Global sensitivity analysis in high dimensions with PLS-PCE

The paper presents a hierarchical polynomial chaos expansion-based probabilistic approach to analyze the single diode solar cell model under Gaussian parametric uncertainty. It is important to analyze single diode solar cell model response under random events or factors due to uncertainty propagation. The optimal values of five electrical parameters associated with the single diode model are estimated using six deterministic optimization techniques through the root-mean-square minimization approach. Values corresponding to the best objective function response are further utilized to describe the probabilistic design space of each random electrical parameter under uncertainty. Adequate samples of each parameter corresponding Gaussian uncertain distribution are generated using Latin hypercube sampling. Furthermore, a multistage probabilistic approach is adopted to evaluate the model response using low-cost polynomial chaos series expansion and perform global sensitivity analysis under specified Gaussian distribution. Coefficients of polynomial basis functions are calculated using least square and least angle regression techniques. Unlike the highly non-linear and complex single diode representation of solar cells, the polynomial chaos expansion model provides a low computational burden and reduced complexity. To ensure reproducibility, probabilistic output response computed using proposed polynomial chaos expansion model is compared with the true model response. Finally, a multidimensional sensitivity analysis is performed through Sobol decomposition of polynomial chaos series representation to quantify the contribution of each parameter to the variance of the probabilistic response. The validation and assessment result shows that the output probabilistic response of the solar cell under Gaussian parametric uncertainty correlates to a Rayleigh probability distribution function. Output response is characterized by a mean value of 0.0060 and 0.0760 for RTC France and Solarex MSX83 solar cells, respectively. The standard deviation of pm 0.0034 and pm 0.0052 was observed in the probabilistic response for RTC France and Solarex MSX83 solar cells, respectively.

Here, we examine the suitability of truncated Polynomial Chaos Expansions (PCE) and truncated Gram-Charlier Expansions (GrChE) as possible methods for uncertainty quantification (UQ) in nonlinear systems with intermittency and positive Lyapunov exponents.These two methods rely on truncated Galerkin projections of either the system variables in a fixed polynomial basis spanning the 'uncertain' subspace (PCE) or a suitable eigenfunction expansion of the joint probability distribution associated with the uncertain evolution of the system (GrChE).Based on a simple, statistically exactly solvable non-linear and non-Gaussian test model, we show in detail that methods exploiting truncated spectral expansions, be it PCE or GrChE, have significant limitations for uncertainty quantification in systems with intermittent instabilities or parametric uncertainties in the damping.Intermittency and fat-tailed probability densities are hallmark features of the inertial and dissipation ranges of turbulence and we show that in such important dynamical regimes PCE performs, at best, similarly to the vastly simpler Gaussian moment closure technique utilized earlier by the authors in a different context for UQ within a framework of Empirical Information Theory.Moreover, we show that the non-realizability of the GrChE approximations is linked to the onset of intermittency in the dynamics and it is frequently accompanied by an erroneous blow-up of the second-order statistics at short times.These limitations of the two types of truncated spectral expansions arise from the following: (i) Non-uniform convergence in time of PCE and GrChE resulting in a rapidly increasing number of terms necessary for a good approximation of the random process as time evolves, (ii) Fundamental problems with capturing the constant flux of randomness due to white Gaussian noise forcing via finite truncations of the spectral representation of the associated Wiener process, (iii) Slow decay of PCE and GrChE coefficients in the presence of intermittency, hampering implementation of sparse truncation methods which have been widely used in nearly elliptic problems or in low Reynolds number flows.Rigorous justification of these limitations is richly illustrated by straightforward tests exploiting a simple nonlinear and non-Gaussian but statistically exactly solvable test model which is proposed here as a challenging benchmark for algorithms for UQ in systems with intermittency.