Polynomial Chaos Expansion Coefficients Research Articles

An enhanced learning function for bootstrap polynomial chaos expansion-based enhanced active learning algorithm for reliability analysis of structure

Sparse polynomial chaos expansion (PCE) combined with the bootstrap resampling method is a viable alternative to obtain an active learning algorithm for reliability analysis. The existing learning functions in PCE-based active learning algorithms do not consider the joint probability density function (PDF) information. The present study explores a sparse PCE-based active learning algorithm based on a newly proposed learning function that maintains a balance between the misclassification probability and the joint PDF information of sample points. In doing so, the coefficients of the sparse PCE are estimated using a Bayesian compressive sensing regressor, as it is noted to be one of the best-performing regression solvers for PCE, irrespective of sampling schemes. The proposed learning function considers the weight of the joint PDF with the local accuracy measure of bootstrap PCE (bPCE) to add new samples iteratively in the existing training set. The convergence is achieved when the ten consecutive failure estimates are within a negligible discrepancy and also checks the confidence bounds of the bPCE estimates. The effectiveness of the proposed approach is demonstrated using two structural engineering examples and one well-known analytical test function and is found to be quite efficient and accurate in estimating reliability.

A fully Bayesian sparse polynomial chaos expansion approach with joint priors on the coefficients and global selection of terms

Polynomial chaos expansion (PCE) is a versatile tool widely used in uncertainty quantification and machine learning, but its successful application depends strongly on the accuracy and reliability of the resulting PCE-based response surface. High accuracy typically requires high polynomial degrees, demanding many training points especially in high-dimensional problems through the curse of dimensionality. So-called sparse PCE concepts work with a much smaller selection of basis polynomials compared to conventional PCE approaches and can overcome the curse of dimensionality very efficiently, but have to pay specific attention to their strategies of choosing training points. Furthermore, the approximation error resembles an uncertainty that most existing PCE-based methods do not estimate. In this study, we develop and evaluate a fully Bayesian approach to establish the PCE representation via joint shrinkage priors and Markov chain Monte Carlo. The suggested Bayesian PCE model directly aims to solve the two challenges named above: achieving a sparse PCE representation and estimating uncertainty of the PCE itself. The embedded Bayesian regularizing via the joint shrinkage prior allows using higher polynomial degrees for given training points due to its ability to handle underdetermined situations, where the number of considered PCE coefficients could be much larger than the number of available training points. We also explore multiple variable selection methods to construct sparse PCE expansions based on the established Bayesian representations, while globally selecting the most meaningful orthonormal polynomials given the available training data. We demonstrate the advantages of our Bayesian PCE and the corresponding sparsity-inducing methods on several benchmarks.

Least Squares Polynomial Chaos Regression for Stochastic Analysis of Transmission Lines Without and With Noise

Stochastic transmission line (TL) analysis is often challenging due to the difficulty of fully identifying the probability distributions of all randomly varying parameters, especially in the presence of noise. These problems cannot be directly addressed by the existing polynomial chaos expansion (PCE) framework. To overcome the limitations, this paper proposes an improved PCE method for analyzing the stochastic TL model with noise. A regression-based non-intrusive approach termed least squares polynomial chaos regression (LSPCR) is employed to determine the PCE coefficients. Six algorithms for LSPCR are developed by respectively combining three sampling strategies, i.e., standard sampling (SS), asymptotic sampling (AS), and coherence-optimal sampling (COS), with two types of norm problems: the least-squares optimization problem (LSO) and the l1-minimum problem (l1-M). Numerical experiments are conducted on noiseless and noisy problems. The results reveal that the LSO-based algorithms (LSOs) are much faster than the l1-M-based algorithms (l1-Ms), regardless of the sampling strategy. In the absence of noise, these six algorithms require only a relatively small number of samples to achieve higher accuracy in solution moments compared to the Stochastic Galerkin (SG) method. When the sample size is large, the AS-based algorithms (ASs) and the COS-based algorithms (COSs) yield more accurate results than the SS-based algorithms (SSs), regardless of the norm problems. In the presence of noise, the LSOs outperform the l1-Ms for small sample sizes. However, no evident differences are observed among the six algorithms with large sample sizes.

Recursive Feasibility and Stability for Stochastic MPC based on Polynomial Chaos

This paper deals with stochastic model predictive control (SMPC) based on polynomial chaos expansion (PCE) for linear systems with time-invariant stochastic parametric uncertainties and time-varying stochastic additive disturbances subject to chance constraints on states and inputs. Exploiting terminal ingredients in the SMPC problem and a hybrid update strategy, a recursively feasible optimization problem is formulated. Moreover, stability of the system of PCE coefficients can be shown. Furthermore, in the paper the performance and computational complexity of SMPC based on PCEs is compared to tube-based SMPC and robust model predictive control (RMPC) is analyzed and benefits are demonstrated in simulation.

Stochastic Quantification of Array Antennas With Random Feeding Errors Using an Improved Polynomial Chaos Expansion Method

Electromagnetic properties of an array antenna are inevitably affected by random magnitude and phase errors in the feeding network. Polynomial chaos expansion (PCE) method can analyze such problems, but the number of numerical quadrature grows rapidly with stochastic dimensionality for solving the PCE coefficients. An improved PCE algorithm is proposed to improve the computational efficiency. A large array is decomposed into several small groups those can be efficiently treated by the PCE method. Stochastic quantification of the whole array is achieved by the superposition of grouped results. In addition, the proposed method can integrate with the method of moments to treat real-world arrays. The results computed by this method agree with those of Monte Carlo, and the computational time is reduced by two orders of magnitude for a dipole array with 64 elements.

On the use of sparse Bayesian learning-based polynomial chaos expansion for global reliability sensitivity analysis

Global reliability sensitivity analysis determines the effects of input uncertain parameters on the failure probability of a system. Usually, the global reliability sensitivity analysis can be performed by the conventional Monte Carlo simulation (MCS) approach. However, the MCS approach requires a large number of model evaluations which limits MCS to apply for realistic problems. For that reason, a sparse polynomial chaos expansion (PCE) model is used in the present work based on a variational Bayesian (VB) inference. More specifically, the PCE coefficients are computed by the VB inference and the important terms in the PCE basis are selected by an automatic relevance determination (ARD) approach. Therefore, the VB inference is fully connected with the ARD approach. Global reliability sensitivity analysis is performed for some numerical examples using the sparse PCE model and all the results are compared with the MCS and the least angle regression (LARS)-based PCE model predicted results. The 95% confidence interval is also obtained by the VB approach to measure the prediction uncertainty. It is found that a very good result is obtained with the sparse PCE model using much less number of model evaluations as compared to the MCS approach. The accuracy of obtaining the PCE coefficients is higher by the VB inference than the LARS approach. Further, the required number of terms is small for the VB-PCE model and therefore, the number of PCE coefficients is also small.

Uncertainty quantification of large-eddy simulation results of riverine flows: a field and numerical study

We present large-eddy simulations (LES) of riverine flow in a study reach in the Sacramento River, California. The riverbed bathymetry was surveyed in high-resolution using a multibeam echosounder to construct the computational model of the study area, while the topographies were defined using aerial photographs taken by an Unmanned Aircraft System (UAS). In a series of field campaigns, we measured the flow field of the river river across multiple transects throughout the field site using an acoustic Doppler current profiler (ADCP) and estimated using large-scale particle velocimetry of the videos taken during the operation UAS. We used the measured data of the river flow field to evaluate the accuracy of the LES-computed hydrodynamics. The propagation of uncertainties in the LES results due to the variations in the riverbed’s effective roughness height and the river’s inflow discharge was studied and showed that both parameters redistributed the flow distribution laterally and vertically in the velocity profile. For the uncertainty quantification (UQ) analyses, the polynomial chaos expansion (PCE) method was used to develop a surrogate model, which was randomly sampled sufficiently by the Monte Carlo Sampling (MCS) method to generate confidence intervals for the LES-computed velocity field. Also, Sobol indices derived from the PCE coefficients were calculated to help understand the relative influence of different input parameters on the global uncertainty of the results. The UQ analysis showed that uncertainties of LES results in the shallow near bank regions of the river were mainly related to the roughness, while the variation of inflow discharge leads to uncertainty in the LES results throughout the river, indiscriminately. Changes in discharge and bed roughness altered the flow distribution laterally and vertically in the vertical profile. Polynomial chaos expansions provided a practical approach for characterizing uncertainty in LES results of a natural river. Uncertainty in simulation results near the bank is mainly due to parameter uncertainty in bed roughness characterization. Uncertainty related to the inflow discharge dominated the overall uncertainty in LES results of the river.

Uncertainty propagation and sensitivity analysis of three-phase flow in porous media using polynomial chaos expansion

In this study, polynomial chaos expansion (PCE) will be utilized to investigate propagation and evolution of uncertainty from uncertain model parameters to model outputs. Unknown PCE coefficients are calculated using probabilistic collocation method (PCM), which is a non-intrusive technique. The proposed approach is evaluated on a 2D reservoir under three-phase flow condition. By use of PCEs, dependent model outputs, such as well bottomhole pressure and production rates, are represented as functions of uncertain model parameters, including parameters defining the capillary pressure (Pc) and relative permeability (kr) curves. Accuracy, compatibility and efficiency of the proposed approach are compared with Monte Carlo (MC) method, which is considered as the reference case. Mean values, standard deviations and histograms for well bottomhole pressure and production rates estimated from the PCM-based PCE are closely matched to those obtained from MC method. This discloses the fact that the proposed approach can successfully restore the simulation model to represent propagation of uncertainty from uncertain model parameters toward model outputs and to perform sensitivity analysis (SA). Unlike the MC approach which requires an extremely large number of realizations to obtain accurate results, the PCM-based PCE requires to perform model evaluations only at the selected collocation points, thus the computational cost is significantly reduced. Subsequently, using the proposed PCE as the surrogate model of the reservoir, Sobol's sensitivity indices are calculated analytically. Results of SA reveal that those parameters defining Pc curves have negligible effect on the model outputs compared with those parameters defining kr curves, and thus may be ignored during history matching process. Moreover, it is observed that the effects caused by interactions between uncertain kr and Pc parameters have negligible contribution compared to the individual effect of each parameter.

Global self-optimizing control with active-set changes: A polynomial chaos approach

Global self-optimizing control (gSOC) aims to identify optimal controlled variables (CVs) that minimize the average economic cost when uncertainties vary in the whole distribution space. Rigorous numerical optimization for the global optimal CVs is mainly hampered by the intensive computation load, hence only approximate solutions were previously developed to afford tractable computations. These approximations may however be large in many cases. This paper revisits this challenging problem within the nonlinear programming (NLP) framework for gSOC and proposes a sequential solution strategy. Within the solution method, the polynomial chaos expansion (PCE) is introduced for the sake of fast computations of the statistics of the cost function. We also handle the active-set change problem by satisfying chance constraints. To this end, PCEs of the constrained variables are also constructed, to formulate conditions that should be respected for the CVs. To determine the PCE coefficients, the sparse grid collocation method is adopted to further reduce the computation burden, as the dimension of uncertainty could be large for most gSOC problems. Three case studies are provided to show the new gSOC approach, all of which result in more accurate CVs, while maintaining tractable numerical optimizations.

Uncertainty Quantification Across Design Space Using Spatially Accurate Polynomial Chaos

In the last decade, the demand for stochastic engineering results has increased substantially. Concurrently, improvements in computing hardware, access to large computing clusters, and efficient statistical methods have made uncertainty quantification studies feasible for complex, computationally expensive simulations such as computational fluid dynamics or finite element methods. In current practice, uncertainty may be quantified at particular locations within a flight envelope or design space, but an ongoing challenge remains to interpolate or extrapolate this information to predict the uncertainty at untested or unsimulated locations. The goal of this paper is to introduce a spatially accurate polynomial chaos method, which can interpolate both aleatory and epistemic uncertainty throughout a flight envelope or design space. An additive correction is also presented, which improves accuracy at a small additional sample cost. Interpolation of polynomial chaos expansion coefficients is achieved using established regression techniques, allowing for a spatially accurate uncertainty propagation method which is agnostic to the characterization of parametric input uncertainties. As a demonstration of the method, results are shown for an analytic test function and for simulations of the NASA Common Research Model. The results indicate that Spatially Accurate Polynomial Chaos and its additive correction can provide accurate interpolated estimates of uncertainty.

Modified Dimension Reduction-Based Polynomial Chaos Expansion for Nonstandard Uncertainty Propagation and Its Application in Reliability Analysis

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.

A comparison of techniques for finding coefficients of polynomial chaos models for antenna 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.

An extended polynomial chaos expansion for PDF characterization and variation with aleatory and epistemic uncertainties

This paper presents an extended polynomial chaos formalism for epistemic uncertainties and a new framework for evaluating sensitivities and variations of output probability density functions (PDF) to uncertainty in probabilistic models of input variables. An ”extended” polynomial chaos expansion (PCE) approach is developed that accounts for both aleatory and epistemic uncertainties, modeled as random variables, thus allowing a unified treatment of both types of uncertainty. We explore in particular epistemic uncertainty associated with the choice of prior probabilistic models for input parameters. A PCE-based Kernel Density (KDE) construction provides a composite map from the PCE coefficients and germ to the PDF of quantities of interest (QoI). The sensitivities of these PDF with respect to the input parameters are then evaluated. Input parameters of the probabilistic models are considered. By sampling over the epistemic random variable, a family of PDFs is generated and the failure probability is itself estimated as a random variable with its own PCE. Integrating epistemic uncertainties within the PCE framework results in a computationally efficient paradigm for propagation and sensitivity evaluation. Two typical illustrative examples are used to demonstrate the proposed approach.

Development of clustered polynomial chaos expansion model for stochastic hydrological prediction

This study introduced a clustered polynomial chaos expansion (CPCE) model to reveal random propagation and dynamic sensitivity of uncertainty parameters in hydrologic prediction. In the CPCE model, the random characteristics of the streamflow simulations resulting from parameter uncertainties are characterized through the polynomial chaos expansion (PCE) model based on the probabilistic collocation method. At the same time, a multivariate discrete non-functional relationship between PCE coefficients and hydrological model inputs is established based on stepwise cluster analysis. Therefore, compared with traditional PCE method, the developed CPCE model cannot only reflect uncertainty propagation in stochastic hydrological simulation, but also have the capability of random forecasting. Moreover, the dynamic sensitivities of model parameters are investigated through the multilevel factorial analyses. The developed approach was applied for streamflow forecasting for the Ruihe watershed, China. Results showed that with effective quantification for the random characteristics of hydrological processes, the CPCE model can directly predict runoff series and generate the associated probability distributions at different time periods. The dynamic sensitivity analysis indicates that the maximum soil moisture capacity within the catchment plays a key role in the accuracy of the low-flow forecasting, while the degree of spatial variability in soil moisture capacities has a remarkable impact on the accuracy of the high-flow forecasting in the studied watershed.

Geo-acoustic inversion using polynomial chaos expansion

In order to improve the computational efficiency of algorithms while exploring the method to overcome the ambiguity problems in underwater geo-acoustic inversion, we use the data of transmission losses at the broadband sound frequencies and multiple propagating distances with the matrix of polynomial chaos expansion coefficients of transmission losses to invert the speed (<i>c</i>), attenuation (<i>α</i>) of compression sound wave and the density ratio of seabed to seawater (<i>ρ</i>) in their prior searching intervals. When approximating the transmission loss with the polynomial chaos expansion, the expansion coefficients are the functions of parameters including sound frequency, source and hydrophone’s position while the polynomial bases are functions of the above geo-acoustic parameters which are uniformly distributed in their respective intervals. The expansion coefficients are calculated by embedding the orthogonal polynomial bases into the acoustic wide-angle parabolic equation model. After that, the coefficients are deduced using the Galerkin projection and least angel regression. Under the situations of low sound frequency, short or medium sound propagation distance and short or medium length of intervals of geo-acoustic parameters, the polynomial chaos expansion can approximate the transmission losses accurately with the relatively error less than 1%. In the simulation case, with the high signal to noise ratio and the low errors of relative distances between source and receivers, the geo-acoustic parameters can be inverted accurately when the appropriate truncated powers are chosen. And the time cost is reduced by at least an order of magnitude compared with that of traversal grids searching procedure.

UNCERTAINTY QUANTIFICATION IN STEADY STATE SIMULATIONS OF A MOLTEN SALT SYSTEM USING POLYNOMIAL CHAOS EXPANSION ANALYSIS

Uncertainty Quantification (UQ) of numerical simulations is highly relevant in the study and design of complex systems. Among the various approaches available, Polynomial Chaos Expansion (PCE) analysis has recently attracted great interest. It belongs to nonintrusive spectral projection methods and consists of constructing system responses as polynomial functions of the stochastic inputs. The limited number of required model evaluations and the possibility to apply it to codes without any modification make this technique extremely attractive. In this work, we propose the use of PCE to perform UQ of complex, multi-physics models for liquid fueled reactors, addressing key design aspects of neutronics and thermal fluid dynamics. Our PCE approach uses Smolyak sparse grids designed to estimate the PCE coefficients. To test its potential, the PCE method was applied to a 2D problem representative of the Molten Salt Fast Reactor physics. An in-house multi-physics tool constitutes the reference model. The studied responses are the maximum temperature and the effective multiplication factor. Results, validated by comparison with the reference model on 103 Monte-Carlo sampled points, prove the effectiveness of our PCE approach in assessing uncertainties of complex coupled models.

A Simulation Study on Turnpikes in Stochastic LQ Optimal Control

Abstract This paper presents a simulation study on turnpike phenomena in stochastic optimal control problems. We employ the framework of Polynomial Chaos Expansions (PCE) to investigate the presence of turnpikes in stochastic LQ problems. Our findings indicate that turnpikes can be observed in the evolution of PCE coefficients as well as in the evolution of statistical moments. Moreover, the turnpike phenomenon can be observed for optimal realization trajectories and with respect to the optimal stationary distribution. Finally, while adding variance penalization to the objective alters the turnpike, it does not destroy the phenomenon.

Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems

Uncertainty quantification (UQ) is an important part of mathematical modeling and simulations, which quantifies the impact of parametric uncertainty on model predictions. This paper presents an efficient approach for polynomial chaos expansion (PCE) based UQ method in biological systems. For PCE, the key step is the stochastic Galerkin (SG) projection, which yields a family of deterministic models of PCE coefficients to describe the original stochastic system. When dealing with systems that involve nonpolynomial terms and many uncertainties, the SG-based PCE is computationally prohibitive because it often involves high-dimensional integrals. To address this, a generalized dimension reduction method (gDRM) is coupled with quadrature rules to convert a high-dimensional integral in the SG into a few lower dimensional ones that can be rapidly solved. The performance of the algorithm is validated with two examples describing the dynamic behavior of cells. Compared to other UQ techniques (e.g., nonintrusive PCE), the results show the potential of the algorithm to tackle UQ in more complicated biological systems.

Ship Robust Design Optimization Based on Polynomial Chaos Expansions

Considerable parameter perturbations occur owing to the influence of uncertain factors in actual ship transportation, resulting in a substantial decline in ship performance. These parameters should not be regarded as certain values but uncertain variables. Ship robust design optimization (RDO) is a method in which various uncertainties are fully considered in the early stages of ship design to ensure that the optimal case adapts to the perturbation of the uncertain parameters. In this study, instead of the commonly used Monte Carlo method, polynomial chaos expansions (PCEs) are adopted to quantify the uncertainty, and an improved probabilistic collocation method (PCM) based on the linear independence principle is proposed to select sample points for calculating polynomial coefficients of PCE, which not only reduces the number of collocation points compared with the traditional statistical sampling method but also avoids the problem that arises with the traditional PCM, which cannot maintain high calculation accuracy even with considerable collocation points. Finally, to ensure ship robustness, in comparison with deterministic optimization design, the proposed RDO framework is applied to minimum Energy Efficiency Design Index (EEDI) KRISO Container Ship hull form design.

Uncertainty Quantification of Stochastic Impact Dynamic Oscillator Using a Proper Orthogonal Decomposition-Polynomial Chaos Expansion Technique

Abstract A proper orthogonal decomposition (POD)-based polynomial chaos expansion (PCE) is utilized in this article for the uncertainty quantification (UQ) of an impact dynamic oscillator. The time-dependent nonsmooth behavior and the uncertainties are decoupled using the POD approach. The uncertain response domain is reduced using the POD approach, and the dominant POD modes are utilized for the UQ of the response quantity. Furthermore, the PCE model is utilized for the propagation of the input uncertainties. Two different cases of impact oscillator are considered, namely, single impact and multiple impact. The contact between two bodies is modeled by Hertz’s law. For both the cases, UQ is performed on the projectile displacement, projectile velocity, and contact force. A highly nonsmooth behavior is noticed for the contact force. For that reason, most number of POD modes are required to assess the UQ of contact force. All the results are compared with the Monte Carlo simulation (MCS) and time domain PCE results. Very good accuracies are observed for the PCE and the POD-PCE predicted results using much less number of model evaluations compared to MCS. As the PCE coefficients are dependent on time, the PCE model is computed at each time step. On the contrary, for the POD-PCE model, the PCE coefficients are computed for the number of POD modes only: it is much less than the PCE model.