Polynomial Chaos Expansion Metamodel Research Articles

Efficient MAPoD via Least Angle Regression based Polynomial Chaos Expansion Metamodel for Eddy Current NDT

In this article, a metamodeling approach based on non-intrusive polynomial chaos expansion (PCE) with least angle regression (LAR) method is used in boundary element analysis for a model-assisted probability of detection (MAPoD) study of eddy current nondestructive testing (NDT) systems. The LAR-PCE metamodel represents the NDT system model responses by a set of coefficients with the polynomial basis functions in lieu of pure kernel degeneration accelerated boundary element method (KD-BEM) based physical model. Both the computational accuracy and efficiency of the LAR-PCE metamodel over the ordinary least squares (OLS) based PCE metamodel are demonstrated by testing the 3D eddy current NDT benchmarks with different system setups, flaw lengths and widths. The simulation results show two digits accuracy of the PoD metrics compared with the ones achieved by the KD-BEM based physical model as the benchmark. The LAR-PCE metamodel has remarkable improvements in computational efficiency over the OLS-PCE metamodel and accelerates the MAPoD study.

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.

Optimum aerodynamic shape design under uncertainty by utility theory and metamodeling

In this work, utility theory is introduced to optimum aerodynamic shape design under uncertainty (also called robust aerodynamic design). Specifically, utility theory is used to formulate the objective function of the optimization problem. The advantage of the proposed approach over the commonly used weighted sum method is that it does not require the use of weighting factors or the addition of constraints on the statistical moments. The polynomial chaos expansion metamodel with the least-angle regression and the hyperbolic truncation scheme are used to accelerate the uncertainty propagation and the optimization process. The proposed approach is demonstrated on the optimum airfoil shape design under uncertainty at steady transonic conditions. Two design cases are considered. In both cases the drag coefficient is to be minimized subject to constraints on the lift coefficient and the airfoil thickness at two chordwise locations. The first case treats the Mach number as uncertainty parameter and the second case treats the Mach number and the lift coefficient at target as uncertain. The proposed method is compared with deterministic single-point optimization and multi-point optimization, and the standard robust design formulations. Results show that the proposed approach is capable of obtaining airfoil design of the lowest mean drag coefficient with the smallest standard deviation as compared to the other approaches. In the first case, the proposed approach yields a design with a mean drag coefficient comparable to the standard robust design (around 97.6 drag counts, cts), but with a significantly lower variance (0.6 cts versus 2.5 cts). In the second case, the proposed approach yielded a design with a higher mean drag coefficient than the standard approach (98.4 cts versus 97.6 cts), but with a lower standard deviation (4.1 cts versus 4.8 cts). In both cases, the deterministic approaches yield designs with comparable or lower drag coefficients but with significantly larger standard deviations.

Data-driven polynomial chaos expansion for machine learning regression

We present a regression technique for data-driven problems based on polynomial chaos expansion (PCE). PCE is a popular technique in the field of uncertainty quantification (UQ), where it is typically used to replace a runnable but expensive computational model subject to random inputs with an inexpensive-to-evaluate polynomial function. The metamodel obtained enables a reliable estimation of the statistics of the output, provided that a suitable probabilistic model of the input is available.Machine learning (ML) regression is a research field that focuses on providing purely data-driven input-output maps, with the focus on pointwise prediction accuracy. We show that a PCE metamodel purely trained on data can yield pointwise predictions whose accuracy is comparable to that of other ML regression models, such as neural networks and support vector machines. The comparisons are performed on benchmark datasets available from the literature. The methodology also enables the quantification of the output uncertainties, and is robust to noise. Furthermore, it enjoys additional desirable properties, such as good performance for small training sets and simplicity of construction, with only little parameter tuning required.

Performance of ensemble Kalman filter and Markov chain Monte Carlo under uncertainty in forecast model

Abstract Ensemble Kalman filter (EnKF) and Markov chain Monte Carlo (MCMC) are popular methods to obtain the posterior distribution of unknown parameters in the reservoir model. However, millions of simulation runs may be required in MCMC for accurate sampling of posterior as subsurface flow problems are highly nonlinear and non-Gaussian. Similarly, EnKF formulated on the basis of linear and Gaussian assumptions may also require a large number of realizations to correctly map the solution space of the unknown model parameters, ultimately resulting in the high computational cost. Data-driven meta/surrogate/proxy models provide an alternative solution to alleviate the issue of high computational cost. Since these models are not as accurate as numerical solutions of partial differential equations (PDE), their implementation may add an uncertainty in the forecast model. In literature, the effect of uncertainty in forecast model on data assimilation is not well studied, especially with field-scale reservoir models. In this work, we take the opportunity to evaluate and compare the performance of EnKF and MCMC using polynomial chaos expansion (PCE) based forecast model. Proposed forecast model relies on reducing parameter space using Karhunen–Loeve (KL) expansion which preserves the two-point statistics of the field. Random variables from KL expansion and orthogonal polynomials corresponding to the prior probability density function (pdf) form the set of input parameters in PCE. Further, non-intrusive probabilistic collocation method (PCM) is used to compute PCE coefficients. PCE forecast model is then used in EnKF and MCMC to calculate the likelihood of the samples in place of high fidelity full physics simulation runs. A case study is performed using a 3D field scale model of a reservoir located near Fort McMurray in northern Alberta, Canada. Performance of EnKF and MCMC are assessed under forecast model uncertainty using rigorous qualitative and quantitative analysis and posterior distribution characterization. Results clearly depict that, although EnKF provided reliable mean and variance estimates of model parameters, MCMC outperformed the former even under the uncertainty associated with PCE metamodel. Inaccurate initial assumptions of model parameters were successfully handled by MCMC, although, with a longer burn-in period. Furthermore, characterization of posterior demonstrated reduced uncertainty in the estimation of model parameters using MCMC as compared to EnKF. Practical implications of the proposed approach and performance assessment under forecast model uncertainty will be consequential in designing accurate and computationally efficient reservoir characterization workflows and hence, improved decision-making in reservoir management.

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.