Full Polynomial Chaos Expansion 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.

Polynomial chaos expansion approximation for dimension-reduction model-based reliability analysis method and application to industrial robots

Polynomial chaos expansion (PCE) is considered an excellent method for accurately and efficiently reliability analysis in various engineering problems. However, it becomes practically infeasible in scenarios characterized by high-dimensional input random variables and thousands of training data. To solve this problem, this paper proposes a new PCE-based surrogate-assisted method. To start with, the interacting variables are screened out from original high-dimensional input random variables by contribution-degree analysis (CDA). The original high-dimensional performance function is decomposed into a lower-dimensional component function composed of the interacting variables and multiple one-dimensional component function containing only one non-interacting variable. Then, PCE combined with a sample points selection strategy is used to fit the lower-dimensional component function, and a full PCE is utilized for fitting each one-dimensional component function. Three methods, namely the Hadamard's inequality of the design matrix, the rank revealing QR factorization of the design matrix and the maximum entropy, are implemented to realize sample point selection. Four examples are investigated to demonstrated the effectiveness of the proposed method.

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

In this paper, the precise integration multi-patch isogeometric dual reciprocity boundary element method (IG-DRBEM) of the non-Fourier transient heat transfer problem in functionally graded materials (FGMs) is proposed, and the stochastic analysis of multidimensional material uncertainty by hyperbolic truncated adaptive sparse polynomial chaos expansion (PCE) is established based on the determine theory. Up to now, most research results of the isogeometric analysis boundary element method (IGABEM) in transient heat transfer problems are based on Fourier's law. The main reason is the higher-order derivative term with respect to time in the non-Fourier heat transfer control equation leads to additional domain integrals in the boundary-domain integral equation. The dual reciprocity method (DRM) can transform the domain integral to the boundary conveniently and maintain the dimension reduction advantage of the BEM. The introduction of the precise integration method (PIM) can improve the stability and accuracy of the time-domain solution. In the random analysis based on IG-DRBEM, a sparse PCE strategy with hyperbolic truncation is constructed to overcome the dimensional disaster difficulty of traditional PCE. Through the proposed adaptive iterative process, a relatively small number of PCE terms are retained compared with the full PCE on the premise of meeting the error estimation requirements. The presented method is verified to have good accuracy and high efficiency by the complex model similar to actual engineering, which further expands the application extent of IG-DRBEM.

Sparse polynomial chaos expansion based on Bregman-iterative greedy coordinate descent for global sensitivity analysis

Polynomial chaos expansion (PCE) is widely used in a variety of engineering fields for uncertainty and sensitivity analyses. The computational cost of full PCE is unaffordable due to the ‘curse of dimensionality’ of the expansion coefficients. In this paper, a novel methodology for developing sparse PCE is proposed by making use of the efficiency of greedy coordinate descent (GCD) in sparsity exploitation and the capability of Bregman iteration in accuracy enhancement. By minimizing an objective function composed of the l1 norm (sparsity) of the polynomial chaos (PC) coefficients and regularized l2 norm of the approximation fitness, the proposed algorithm screens the significant basis polynomials and builds an optimal sparse PCE with model evaluations much fewer than unknown coefficients. To validate the effectiveness of the developed algorithm, several benchmark examples are investigated for global sensitivity analysis (GSA). A detailed comparison is made with the well-established orthogonal matching pursuit (OMP), least angle regression (LAR) and two adaptive algorithms. Results show that the proposed method is superior to the benchmark methods in terms of accuracy while maintaining a better balance among accuracy, complexity and computational efficiency.

Efficient uncertainty quantification of CFD problems by combination of proper orthogonal decomposition and compressed sensing

In the current paper, an efficient surrogate model based on combination of Proper Orthogonal Decomposition (POD) and compressed sensing is developed for affordable representation of high dimensional stochastic fields. In the developed method, instead of the full (or classical) Polynomial Chaos Expansion (PCE), the ℓ1-minimization approach is utilized to reduce the computational work-load of the low-fidelity calculations. To assess the model capability in the real engineering problems, two challenging high-dimensional CFD test cases namely; i) turbulent transonic flow around RAE2822 airfoil with 18 geometrical uncertainties and ii) turbulent transonic flow around NASA Rotor 37 with 3 operational and 21 geometrical uncertainties are considered. Results of Uncertainty Quantification (UQ) analysis in both test cases showed that the proposed multi-fidelity approach is able to reproduce the statistics of quantities of interest with much lower computational cost than the classical regression-based PCE method. It is shown that the combination of the POD with the compressed sensing in RAE2822 and Rotor 37 test cases gives respectively computational gains between 1.26–7.72 and 1.79–9.05 times greater than the combination of the POD with the full PCE.

Toward an Efficient Uncertainty Quantification of Streamflow Predictions Using Sparse Polynomial Chaos Expansion

Reliable hydrologic models are essential for planning, designing, and management of water resources. However, predictions by hydrological models are prone to errors due to a variety of sources of uncertainty. More accurate quantification of these uncertainties using a large number of ensembles and model runs is hampered by the high computational burden. In this study, we developed a highly efficient surrogate model constructed by sparse polynomial chaos expansion (SPCE) coupled with the least angle regression method, which enables efficient uncertainty quantifications. Polynomial chaos expansion was employed to surrogate a storage function-based hydrological model (SFM) for nine streamflow events in the Hongcheon watershed of South Korea. The efficiency of SPCE is investigated by comparing it with another surrogate model, full polynomial chaos expansion (FPCE) built by a well-known, ordinary least square regression (OLS) method. This study confirms that (1) the performance of SPCE is superior to that of FPCE because SPCE can build a more accurate surrogate model (i.e., smaller leave-one-out cross-validation error) with one-quarter the size (i.e., 500 versus 2000). (2) SPCE can sufficiently capture the uncertainty of the streamflow, which is comparable to that of SFM. (3) Sensitivity analysis attained through visual inspection and mathematical computation of the Sobol’ index has been of great success for SPCE to capture the parameter sensitivity of SFM, identifying four parameters, α, Kbas, Pbas, and Pchn, that are most sensitive to the likelihood function, Nash-Sutcliffe efficiency. (4) The computational power of SPCE is about 200 times faster than that of SFM and about four times faster than that of FPCE. The SPCE approach builds a surrogate model quickly and robustly with a more compact experimental design compared to FPCE. Ultimately, it will benefit ensemble streamflow forecasting studies, which must provide information and alerts in real time.

Efficient reliability analysis with a CDA-based dimension-reduction model and polynomial chaos expansion

Polynomial chaos expansion (PCE) is a versatile tool for building a meta-model in various engineering fields. Unfortunately, it is largely affected by the curse of dimensionality and its application for reliability analysis is usually hindered unless high truncated degrees are used. To alleviate this problem, this paper presents a new method based on contribution-degree analysis (CDA), an exact dimension-reduction model (DRM) and polynomial chaos expansion for efficient reliability analysis. First, the original performance function is decomposed as a summation of several component functions including one lower-dimensional interacting component function and several one-dimensional non-interacting component functions via a CDA-based DRM. Then, the full PCE or sparse PCE is employed to reconstruct the lower-dimensional interacting component function instead of the original complicated multi-dimensional performance function, whereas one-dimensional full PCEs are utilized for approximating the non-interacting component functions. In this way, we avoid building a PCE meta-model for the original multivariate performance function, thus the number of unknown coefficients is significantly reduced and the computational burden for reliability analysis is eased accordingly. Pertinent examples including both analytical performance functions and finite-element models are investigated, which demonstrates that the proposed method achieves a good trade-off of efficiency and accuracy for reliability analysis.

Efficient uncertainty quantification of turbine blade leading edge film cooling using bi-fidelity combination of compressed sensing and Kriging

In the current paper, efficient uncertainty quantification analysis of the gas turbine blade with film cooling is performed using a bi-fidelity surrogate model. The proposed method is based on the combination of sparse polynomial chaos expansion and Kriging metamodels. The orthogonal matching pursuit technique is employed to retrieve dominant expansion basis functions via low-fidelity calculations and, the high-fidelity computations are estimated based on the most dominant basis. In the developed method, the selected dominant basis functions are used as the trend function of the Kriging method, while the stochastic Gaussian part is estimated via a limited number of high-fidelity calculations. Two challenging analytical test functions are considered to investigate the performance of the metamodel. In the blade leading edge film cooling test case, combination of ten operational and geometrical uncertain parameters are examined. The results show that implementing the multi-fidelity approach in sparse polynomial chaos expansion reduces the computational cost by more than 40% and 80% in comparison to the sparse and full polynomial chaos expansion methods, respectively. In addition, results of sensitivity analysis show that the mainstream angle of attack and the compound angle of coolant hole mostly influence the film cooling adiabatic effectiveness.

Stochastic field representation using bi-fidelity combination of proper orthogonal decomposition and Kriging

In the current paper efficient uncertainty quantification (UQ) of high dimensional stochastic fields is performed via a bi-fidelity surrogate model. The method is based on combination of proper orthogonal decomposition (POD) and Kriging method. In the developed method, the trend function of the Kriging model is estimated via a low cost POD, whilst the stochastic Gaussian contribution is tuned using a limited number of high-fidelity computations. The proposed method is applied to three test cases, including a highly nonlinear test function, thermally driven flow in a cavity with stochastic wall temperatures and turbulent heat transfer in a ribbed channel with stochastic heat flux boundary condition. Implementing both POD and POD–Kriging methods leads to significant reduction of computational cost in comparison to the classical full polynomial chaos expansion (PCE) with least cost saving of more than 60%. Furthermore, in all cases combined POD–Kriging method performs either superior or at least similar to the POD method. More specifically, in surrogate models with lower number of low- and high-fidelity samples greater gains in accuracy are obtained. Consequently, for a specific level of accuracy, POD–Kriging method requires lower computational cost.

An expanded sparse Bayesian learning method for polynomial chaos expansion

Abstract Polynomial chaos expansion (PCE) has been proven to be a powerful tool for developing surrogate models in various engineering fields for uncertainty quantification. The computational cost of full PCE is unaffordable due to the “curse of dimensionality” of the expansion coefficients. In this paper, an expanded sparse Bayesian learning method for sparse PCE is proposed. Firstly, basis polynomials of the full PCE are partitioned into significant terms and complementary non-significant terms. The parameterized priors with distinct variance are assigned to the candidates for the significant terms. Then, the dimensionality of the parameter space is equivalent to the assumed sparsity level of the PCE. Secondly, an approximate Kashyap information criterion (KIC) rule which achieves a balance between model simplicity and goodness of fit is derived for model selection. Finally, an automatic search algorithm is proposed by minimizing the KIC objective function and using the variance contribution of each term to the model output to select significant terms. To assess the performance of the proposed method, a detailed comparison is completed with several well-established techniques. The results show that the proposed method is able to identify the most significant PC contributions with superior efficiency and accuracy.

Multi-level multi-fidelity sparse polynomial chaos expansion based on Gaussian process regression

The polynomial chaos expansion (PCE) approaches have drawn much attention in the field of simulation-based uncertainty quantification (UQ) of stochastic problem. In this paper, we present a multi-level multi-fidelity (MLMF) extension of non-intrusive sparse PCE based on recent work of recursive Gaussian process regression (GPR) methodology. The proposed method firstly builds the full PCE with varying degree of fidelity based on GPR technique using orthogonal polynomial covariance function. Then an autoregressive scheme is used to exploit the cross-correlation of these PCE models of different fidelity level, and this procedure yields a high-fidelity PCE model that encodes the information of all the lower fidelity levels. Furthermore, an iterative scheme is used to detect the important bases of PCE in each fidelity level. Three test examples are investigated d to validate the performance of the proposed method, and the results show that the present method provides an accurate meta-model for UQ of stochastic problem.

Efficient uncertainty quantification of stochastic heat transfer problems by combination of proper orthogonal decomposition and sparse polynomial chaos expansion

To increase the contribution and reliability of computational fluid dynamics efforts in design process of industrial equipments, it is necessary to quantify the effects of uncertainties on the system performance. Due to exponentially increment of the computational cost with number of uncertain variables for uncertainty quantification using classical polynomial chaos expansion methodology, reducing the required number of samples for uncertainty quantification is a real engineering challenge. In this paper, the proper orthogonal decomposition method based on the multifidelity approach is combined with the full and sparse polynomial chaos expansions for efficient uncertainty quantification of complex heat transfer problems with large number of random variables. The conjugate conduction heat transfer in NASA C3X cooled gas turbine blade with geometrical uncertainties and the convective heat transfer in ribbed passage with the stochastic wall heat flux boundary condition are considered as the test cases. Results of uncertainty quantification analysis in both test cases showed that proposed multi-fidelity approaches are able to produce the statistical quantities with much lower computational cost compare to the classical regression-based polynomial chaos method. It is shown that the combination of the proper orthogonal decomposition with the sparse polynomial chaos gives a computational gain at least 2 times greater than combination of the proper orthogonal decomposition with the full polynomial chaos expansion.

Sparse polynomial chaos expansions for global sensitivity analysis with partial least squares and distance correlation

Polynomial chaos expansion (PCE) has been proven to be a powerful tool for developing surrogate models in the field of uncertainty and global sensitivity analysis. The computational cost of classical PCE is unaffordable since the number of terms grows exponentially with the dimensionality of inputs. This considerably restricts the practical use of PCE. An efficient approach to address this problem is to build a sparse PCE. Since some basis polynomials in representation are highly correlated and the number of available training samples is small, the sparse PCE obtained by the original least square (LS) regression using these samples may not be accurate. Meanwhile, correlation between the non-influential basis polynomial and the important basis polynomials may disturb the correct selection of the important terms. To overcome the influence of correlation in the construction of sparse PCE, a full PCE of model response is first developed based on partial least squares technique in the paper. And an adaptive algorithm based on distance correlation is proposed to select influential basis polynomials, where the distance correlation is used to quantify effectively the impact of basis polynomials on model response. The accuracy of the surrogate model is assessed by leave-one-out cross validation. The proposed method is validated by several examples and global sensitivity analysis is performed. The results show that it maintains a balance between model accuracy and complexity.

Sparse polynomial chaos expansion based on D-MORPH regression

Polynomial chaos expansion (PCE) is widely used by engineers and modelers in various engineering fields for uncertainty analysis. The computational cost of full PCE is unaffordable for the “curse of dimensionality” of the expansion coefficients. In this paper, a new method for developing sparse PCE is proposed based on the diffeomorphic modulation under observable response preserving homotopy (D-MORPH) algorithm. D-MORPH is a regression technique, it can construct the full PCE models with model evaluations much less than the unknown coefficients. This technique determines the unknown coefficients by minimizing the least-squared error and an objective function. For the purpose of developing sparse PCE, an iterative reweighted algorithm is proposed to construct the objective function. As a result, the objective in D-MORPH regression is converted to minimize the ℓ1 norm of PCE coefficients, and the sparse PCE is established after the proposed algorithm converges to the optimal value. To validate the performance of the developed methodology, several benchmark examples are investigated. The accuracy and efficiency are compared to the well-established least angle regression (LAR) sparse PCE, and results show that the developed method is superior to the LAR-based sparse PCE in terms of efficiency and accuracy.

Adaptive sparse polynomial chaos expansions for global sensitivity analysis based on support vector regression

In the context of uncertainty analysis, Polynomial chaos expansion (PCE) has been proven to be a powerful tool for developing meta-models in a wide range of applications, especially for sensitivity analysis. But the computational cost of classic PCE grows exponentially with the size of the input variables. An efficient approach to address this problem is to build a sparse PCE. In this paper, a full PCE meta-model is first developed based on support vector regression (SVR) technique using an orthogonal polynomials kernel function. Then an adaptive algorithm is proposed to select the significant basis functions from the kernel function. The selection criterion is based on the variance contribution of each term to the model output. In the adaptive algorithm, an elimination procedure is used to delete the non-significant bases, and a selection procedure is used to select the important bases. Due to the structural risk minimization principle employing by SVR model, the proposed method provides better generalization ability compared to the common least square regression algorithm. The proposed method is examined by several examples and the global sensitivity analysis is performed. The results show that the proposed method establishes accurate meta-model for global sensitivity analysis of complex models.

Effects of operational and geometrical uncertainties on heat transfer and pressure drop of ribbed passages

In this paper uncertainty quantification of turbulent flow and heat transfer through ribbed channels of gas turbine blades is presented and discussed. The uncertainty quantification method is first validated using the Ishigami function and then applied to turbulent flow and heat transfer through a ribbed pipe. Combination of four operational and geometrical uncertain variables (Re, Pr, P/h and D/h) are considered. Results for local Nusselt number, average Nusselt number and non-dimensional pressure drop using a polynomial chaos expansion of order 3 are presented and discussed. Uncertainty quantification results for the local Nusselt number showed that the regions located on top and immediately after the rib are more sensitive to the uncertain input variables. To tackle the curse of dimensionality, the sparse polynomial chaos expansion is further employed in this study. Results for quantities of interest showed that the sparse polynomial chaos method reduces the computational cost by at least 50% in comparison to the full polynomial chaos expansion method. Furthermore, results show that the sparse PCE works better for the non-dimensional pressure drop than the average Nusselt number. Results of sensitivity analysis for non-dimensional pressure-drop showed that, in agreement with the empirical correlations, this quantity is independent of Reynolds number. Also, results of sensitivity analysis for local Nusselt number showed that generally the major variability of Nusselt number is due to the blockage ratio.