Polynomial Chaos Expansion Approach Research Articles

An active sparse polynomial chaos expansion approach based on sequential relevance vector machine

Polynomial chaos expansion (PCE) is a popular surrogate modeling approach employed in uncertainty quantification for a variety of engineering problems. However, the challenges for full PCE lie in the “curse of dimensionality” of the expansion coefficients. In this paper, we propose a new sparse PCE approach by introducing active learning technique and sequence relevance vector machine (SRVM). As an active learning technique, efficient loss function based on expected improvement (EI-based ELF) is introduced to search for the best next sampling point and enrich the training samples. Relevance vector machine (RVM) is a superior machine learning technique due to the sparsity of its adopted model. SRVM leads to the maximization of marginal likelihood by sequentially selecting basis functions. To assess the performance of the proposed method, four examples are investigated, and the results show that the proposed method outperforms the classical sparse PCE in terms of both accuracy and efficiency.

Method study on failure probability and uncertainty analysis of stress corrosion cracking in fixed parts of PWR

To estimate the failure probability of stress corrosion cracking (SCC) in the fixed parts of pressurized water reactors (PWR), an approach based on the Paris model was developed and its uncertainty analysis was performed as well. In the study of SCC failure, the uncertainty of fracture toughness was translated into integral form to improve calculation efficiency. Moreover, uncertainty parameters such as crack size and Paris model coefficients were studied in the uncertainty analysis. On this basis, the thermowell of the pressurizer of PWR as an example was analyzed in this study. The Monte Carlo (MC) method and polynomial chaos expansion (PCE) approach were used to calculate the mean value, relative uncertainty, upper limit of tolerance interval of failure probability, and sensitivity coefficient of input parameter to compare the advantages and disadvantages of these methods. According to the results, compared with the other method, 300 sets of sampling with Latin Hypercube sampling (LHS) and Sobol sequences can converge the mean, and 300 sets of sampling with Halton sequence and Hammersley sequence can converge the relative uncertainty. When LHS with the 4th order Wilks' formula and Halton sequence with the 1st order Wilks' formula to compute the upper limit of tolerance interval, the accuracy and cost of the computation can be balanced. According to the results of the sensitivity analysis, crack aspect ratio is the most sensitive parameter. PCE surrogate model can accurately calculate the mean, relative uncertainty, the upper limit of tolerance interval and sensitivity coefficient, thereby reducing the computational cost.

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.

A Nonintrusive Nuclear Data Uncertainty Propagation Study for the ARC Fusion Reactor Design

It is widely recognized that the safe and robust design of a nuclear system requires an uncertainty propagation analysis concerning the various nuclear data used as input parameters, especially in view of the most recent design methodologies like the best-estimate plus uncertainty approach. The evaluation of the input uncertainties and their propagation to the design parameters of interest is particularly important in the case of nuclear fusion machines, such as the Affordable Robust Compact (ARC) reactor, which features the presence of uncommon isotopes in the nuclear engineering field, like fluorine, beryllium, and lithium. The uncertainties of the nuclear data of these nuclides can have a significant impact on the fundamental design parameters, such as the target tritium breeding ratio (TBR). Hence, in this work we investigate the application of different methods for propagating the nuclear data uncertainty to the parameters of interest, computed with the Serpent 2 Monte Carlo code. All the methods proposed in this work share the feature of being nonintrusive, implying that they can be profitably employed independently of the physical and/or computational model adopted. The methods discussed in this work are the fast Total Monte Carlo, the GRS, the unscented transform, and the polynomial chaos expansion. The first three methods led to similar values in terms of relative standard deviation of the TBR due to nuclear data and can be considered as fast alternatives to brute-force sampling methods. For these three methods, the present paper suggests how to select the best approach according to the kind of analysis to be performed and the nuclides considered in the study. The effect of the use of different nuclear data libraries and of different input covariance matrices is also examined. The main outcome of these analyses suggests that the uncertainties in the nuclear data of nickel, fluorine, beryllium, and lithium are sufficiently small (i.e., smaller than 1%) to prevent the TBR from assuming values below the design constraints. The overall uncertainty on the TBR of ARC due to the nuclides here considered was evaluated to be ~0.9%. Concerning the polynomial chaos expansion approach, this paper shows that its application is computationally inefficient compared to the other techniques when the input data dimensionality are very large, as for the case of nuclear data.

Non-Linear Partial-Least-Squares-based Polynomial Chaos Expansion (NLPLS-based PCE) approach for global sensitivity analysis of a High-Q Partially Air-Filled Pedestal Resonator Integrated in a Printed Circuit Board (PCB)

This paper presents a global sensitivity analysis of a high-Q partially air-filled pedestal resonator integrated in a printed circuit board. Nonlinear partial-least-squares-based polynomial chaos expansion (NLPLS-based PCE) approach is used for the global sensitivity analysis. Using NLPLS-based PCE a surrogate model is constructed with a reduced dimensionality, which enhances the performance of the algorithm. A standard PCE surrogate model, with all the system parameters, is created from the reduced NLPLS-based PCE surrogate model. The statistical information needed to perform the sensitivity analysis, i.e., variance, is extracted from the standard PCE surrogate model. A variance-based global sensitivity analysis is performed on the PCE model, each system parameter's sensitivity is quantified as the partial influence on the total variance of the performance variable S11. The chosen manufacturing technology involves a three-stage process: micromachining of the cavity, metallization, and thermos-diffusion stacking. During the three stages several problems may occur that can have an influence on the performance of the resonator, such as shape and size variation, and misalignment. The system parameters are set up according to these most common problems. The results show that, of the 8 system parameters chosen to evaluate, the height of the cavity and pedestal are the most sensitive parameters.

Hybrid Polynomial Chaos Expansion and proper generalized decomposition approach for uncertainty quantification problems in the frame of elasticity

Polynomial Chaos Expansion (PCE) is a powerful and flexible metamodeling technique, but suffers from the fact that the number of required training samples grows exponentially with the dimensionality of the problem. Recently, proper generalized decomposition (PGD) has become a popular way to address this challenge since its complexity grows linearly with the dimension of the problem. In this work, we present a new technique called PGD–PCE, which combines the advantages of both methods. The algorithm is based on separate representations and constructed by orthonormal polynomial functions. We test the proposed approach on simple toy problems and engineering finite element problems. The results show that PGD–PCE performs well in terms of accuracy and computational efficiency when dealing with large problems. • We propagate uncertainties in mechanical problems. • We develop a simple to implement and non-intrusive method. • We show that accuracy is preserved while cost drastically decrease. • We illustrate the capabilities of the approach on various numerical tests.

A time-dependent reliability analysis method based on multi-level meta-models for problems involving interval variables

This paper proposes a new time-dependent reliability analysis method based on a two-level meta-models technique for problems with interval variables, through which the upper and lower bounds of reliability indexes of structures within a given design lifetime period can be obtained efficiently. Firstly, the time parameter, interval and stochastic process parameters in the performance function are transformed into the corresponding random variables. Secondly, the polynomial chaos expansion (PCE) approach is adopted for using an approximate polynomial expression to accurately surrogate the complex original performance function. Thirdly, the Kriging meta-model of time-dependent reliability index about the interval variables is constructed based on the approximate expression, and the upper and lower bounds of the reliability index are calculated with the efficient global optimization (EGO) method. Finally, the effectiveness of the proposed method is verified by investigating three numerical examples.

A functional global sensitivity measure and efficient reliability sensitivity analysis with respect to statistical parameters

Sensitivity analysis and reliability assessment are two important aspects of structural and system safety. Epistemic uncertainty with respect to probabilistic model of input parameters due to lack of knowledge is present in many scarce-data applications and complicates the characterization of uncertainty in model response. In this article, we present two importance measures to evaluate the impact of distribution parameters on the probability distribution function (PDF) of the output and the failure probability. The epistemic uncertainty associated with the distribution parameters is modeled as random variables. A modified extended polynomial chaos expansion (MEPCE) approach is introduced in which aleatory and epistemic random variables are modeled and propagated simultaneously while allowing the separate assessment for any single epistemic variable. A MEPCE-based kernel density estimation (KDE) construction provides a composite map from each epistemic variable to the response PDF. The functional global sensitivity index of the PDF with respect to the distribution parameters is thus derived, as a function of output, which is both more informative and more efficient than standard scalar sensitivity measures. Reliability sensitivity indices can be readily evaluated by integrating the global sensitivity index function over the failure zone. Three illustrative examples are used to demonstrate the proposed methodology.

Data-Driven Calibration of Rough Heat Transfer Prediction Using Bayesian Inversion and Genetic Algorithm

The prediction of heat transfers in Reynolds-Averaged Navier–Stokes (RANS) simulations requires corrections for rough surfaces. The turbulence models are adapted to cope with surface roughness impacting the near-wall behaviour compared to a smooth surface. These adjustments in the models correctly predict the skin friction but create a tendency to overpredict the heat transfers compared to experiments. These overpredictions require the use of an additional thermal correction model to lower the heat transfers. Finding the correct numerical parameters to best fit the experimental results is non-trivial, since roughness patterns are often irregular. The objective of this paper is to develop a methodology to calibrate the roughness parameters for a thermal correction model for a rough curved channel test case. First, the design of the experiments allows the generation of metamodels for the prediction of the heat transfer coefficients. The polynomial chaos expansion approach is used to create the metamodels. The metamodels are then successively used with a Bayesian inversion and a genetic algorithm method to estimate the best set of roughness parameters to fit the available experimental results. Both calibrations are compared to assess their strengths and weaknesses. Starting with unknown roughness parameters, this methodology allows calibrating them and obtaining between 4.7% and 10% of average discrepancy between the calibrated RANS heat transfer prediction and the experimental results. The methodology is promising, showing the ability to finely select the roughness parameters to input in the numerical model to fit the experimental heat transfer, without an a priori knowledge of the actual roughness pattern.

Uncertainty Quantification of WRF Model for Rainfall Prediction over the Sichuan Basin, China

The mesoscale Weather Research and Forecasting (WRF) model has been widely employed to forecast day-ahead rainfalls. However, the deterministic predictions from the WRF model incorporate relatively large errors due to numerical discretization, inaccuracies in initial/boundary conditions and parameterizations, etc. Among them, the uncertainties in parameterization schemes have a huge impact on the forecasting skill of rainfalls, especially over the Sichuan Basin which is located east of the Tibetan Plateau in southwestern China. To figure out the impact of various parameterization schemes and their interactions on rainfall predictions over the Sichuan Basin, the Global Forecast System data are chosen as the initial/boundary conditions for the WRF model and 48 ensemble tests have been conducted based on different combinations of four microphysical (MP) schemes, four planetary boundary layer (PBL) schemes, and three cumulus (CU) schemes, for four rainfall cases in summer. Compared to the observations obtained from the Chinese ground-based and encrypted stations, it is found that the Goddard MP scheme together with the asymmetric convective model version 2 PBL scheme outperforms other combinations. Next, as the first step to explore further improvement of the WRF physical schemes, the polynomial chaos expansion (PCE) approach is then adopted to quantify the impacts of several empirical parameters with uncertainties in the WRF Single Moment 6-class (WSM6) MP scheme, the Yonsei University (YSU) PBL scheme and the Kain-Fritsch CU scheme on WRF rainfall predictions. The PCE statistics show that the uncertainty of the scaling factor applied to ice fall velocity in the WSM6 scheme and the profile shape exponent in the YSU scheme affects more dominantly the rainfall predictions in comparison with other parameters, which sheds a light on the importance of these schemes for the rainfall predictions over the Sichuan Basin and suggests the next step to further improve the physical schemes.

Polynomial chaos expansion for uncertainty analysis and global sensitivity analysis

Uncertainty analysis has received increasing attention across all kinds of scientific and engineering fields recently. Uncertainty analysis is often conducted by Monte Carlo simulation (MCS), while with low convergence rate. In this paper, numerical test examples as benchmarks and engineering problems in practice are studied by polynomial chaos expansion (PCE) and compared with the solutions got by MCS. Results show that PCE approach establishes accurate surrogate model for complicated original model with efficiency to conduct uncertainty analysis and global sensitivity analysis. What’s more, sparse PCE is able to tackle problem of high dimension with efficiency. Hence PCE approach can be applied in uncertainty analysis and global sensitivity analysis of engineering problems with efficiency and effectiveness.

A reduced modal subspace approach for damped stochastic dynamic systems

A novel method for characterising and propagating system uncertainty in structures subjected to dynamic actions is proposed, whereby modal shapes, frequencies and damping ratios constitute the random quantities. The latter, defined in the modal subspace rather than the full geometrical space, reduce the number of the random variables and the size of the dynamic problem. A numerical procedure is presented for their identification by calibrating their probabilistic definition in line with the geometrical space. A high-order perturbation technique is proposed for the multi-fidelity response quantification by means of an ad hoc extension of the conventional perturbation method. The approach involves a set of auxiliary deterministic differential equations to be adaptively solved with the piecewise exact method, and moment-cumulant relationships are employed to approximate high-order moments. Finally, a polynomial chaos expansion approach is adopted to complement the second-moment analysis for spectral quantification with the modal subspace reduction. Demonstrated on a multi-storey steel frame with semi-rigid connections and a simply supported bridge subjected to a moving load, the proposed variants exhibit improved performance with respect to the conventional second-order and improved perturbation, as well as increased flexibility, enabling the analyst to decide, on-demand, the level of fidelity, balancing accuracy and computational effort.

Global Sensitivity Analysis of Large Distribution System With PVs Using Deep Gaussian Process

Global sensitivity analysis (GSA) of voltage to uncertain power injection variations plays an important role for appropriate Volt-VAR optimization. This paper proposes a data-driven GSA method for large-scale distribution systems with a large number of uncertain sources. Specifically, the deep Gaussian process is used to identify the mapping relationship between uncertain power injections and voltages. This allows resorting to the analysis of variance framework to calculate the Sobol indices for GSA. Unlike the existing polynomial chaos expansion and Gaussian process-based approaches, our proposed method has much better scalability. Test results on the EPRI 1747-node K1 circuit with different numbers of uncertain sources with various uncertain levels and different PV distributions demonstrate that the proposed method can achieve accurate GSA.

Uncertainty Analysis for Hydrological Models With Interdependent Parameters: An Improved Polynomial Chaos Expansion Approach

AbstractThe use of polynomial chaos expansion (PCE) has gained a lot of attention due to its ability to efficiently estimate the effects of parameter uncertainty on model outputs. The traditional PCE technique requires the studied parameters to be independent. In hydrological modeling, although model parameters are often assumed to be independent for simplicity of computation, such an assumption is not always valid. Neglecting parameter correlations could significantly affect the analysis of uncertainty, leading to distorted modeling results. In this study, an improved PCE approach is proposed to address this issue and support the uncertainty analysis for hydrological models with correlated parameters. The proposed approach is based on the integration of principle component analysis (PCA) and PCE, where PCA is used to transform correlated parameters into orthogonal independent components. To demonstrate the applicability of this approach, the Soil & Water Assessment Tool (SWAT) model is applied to the Guadalupe River Watershed in Texas, US, and the integrated PCA‐PCE framework is used to assess the propagation of uncertainty of SWAT's interdependent parameters. A traditional Monte‐Carlo (MC) simulation is also used to address the uncertainty in the developed SWAT model. The results show that PCA‐PCE could generate similar probabilistic flow results compared to MC while maintaining a very high computational efficiency. The coefficients of determination (R2) for the mean and variance are 0.998 and 0.973, respectively, and the computational requirement is reduced by 99% using the developed PCA‐PCE approach. It is shown that the PCA‐PCE approach is reliable and efficient in assessing uncertainties in hydrological models with interdependent parameters.

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.

Structural uncertainty analysis with the multiplicative dimensional reduction–based polynomial chaos expansion approach

Structural uncertainty analysis with the multiplicative dimensional reduction–based polynomial chaos expansion approach

An adaptive Sequential Enhanced PCE approach and its application in aerodynamic uncertainty quantification

Uncertainty is common in the life cycle of an aircraft, and has great influence on flight quality. Among the approaches of uncertainty quantification, the polynomial chaos expansion (PCE) method has been more and more widely used. However, existing works indicate that the original PCE considering all the features is not so efficient, and has difficulties in its application in robust optimization considering uncertainty. Therefore, a novel PCE method, named as SEPCE, is proposed in this paper. A feature selecting module and a sequential sampling module is integrated in SEPCE. In this case, the SEPCE is capable of picking out the crucial features, and gradually appending new samples under requirements of model refinement. Three benchmark functions are utilized for testing the performance of SEPCE. After that, it is applied in the quantification of aerodynamic uncertainty of RAE2822 airfoil. The comparison between the SEPCE and the original PCE is also performed. Results show that the SEPCE model is more efficient and accurate. It is believed that this model will have better performance when applied to robust optimization.

Preliminary uncertainty and sensitivity analysis of the Molten Salt Fast Reactor steady-state using a Polynomial Chaos Expansion method

In this work, we present the results of a preliminary uncertainty quantification and sensitivity analysis study of the Molten Salt Fast Reactor (MSFR) behavior at steady-state performed by applying a non-intrusive Polynomial Chaos Expansion (PCE) approach. An in-house high-fidelity multi-physics simulation tool is used as reactor reference model. Considering several thermal-hydraulics and neutronics parameters as stochastic inputs, with a limited number of samples we build a PCE meta-model able to reproduce he reactor response in terms of effective multiplication factor, maximum, minimum, and average salt temperatures, and complete salt temperature distribution. The probability density functions of the responses are constructed and analyzed, highlighting strengths and issues of the current MSFR design. The sensitivity study highlights the relative importance of each input parameter, thus providing useful indications for future research efforts. The analysis on the whole temperature field shows that the heat exchanger can be a critical component, so its design requires particular care.

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.

Investigation on the influence of initial thermodynamic conditions and fuel compositions on gasoline octane number based on a data-driven approach

The objective of this work is to clarify the influence mechanism of initial thermodynamic conditions and fuel chemistry on gasoline’s octane numbers (ON), including research octane number (RON) and motor octane number (MON). Investigated fuels include three kinds of well-characterized gasoline, FACE (Fuels for Advanced Combustion Engines) A, C and J, which were simplified as toluene primary reference (TRF) mixtures in this work. The parameters of interest cover initial thermodynamic environment parameters and the ratios of each composition in gasoline. A hybrid analysis framework, combining the transient tracking method, data-driven modeling algorithm, global sensitivity and reaction analysis, was proposed to realize the fast forward prediction and analysis on fuel’s ONs. The effectiveness of this framework was verified by comparing with two empirical correlations. Different data-driven algorithms were fully compared to determine the optimal ON modeling scheme. The polynomial chaos expansion approach has proved to be the best modeling choice, which could substantially reduce computational costs by 2 orders of magnitude in comparison with the sampling-based method (SBM) for predicting ONs with a satisfactory accuracy (greater than 0.985). Global sensitivity analysis showed the initial thermodynamic parameters had more impact on fuel’s ONs than fuel compositions, while the chemical reasons provided by the reaction analysis. Furthermore, a slight change in fuel composition was found could result in a complex shift of reaction pathways of small-molecules under both RON/MON test standards. Therefore, it is suggested that a rigorous influence analysis should be conducted before new gasoline-like fuel is applied.