High-dimensional Uncertainty Quantification Research Articles

Sensitivity Analysis and Robust Optimization to High-Dimensional Uncertainties of Compressors with Active Subspace method

The performance of compressors often deviates significantly from the design value due to the substantial geometric deviations arising from manufacturing errors and in-service degradation. Robust design optimization (RDO) plays a crucial role in ensuring the high performance and reliability of compressors. The study proposes an active subspace-based dimensionality-reduction RDO method for high-dimensional uncertainties of compressors. By employing the active subspace (AS) method, a dimensionality-reduction artificial neural network (ANN) is raised to solve the "dimension disaster" problem for high-dimensional uncertainty quantification (UQ) analysis. Additionally, to decrease the computation cost of the original AS method, a data-driven method is utilized to calculate the gradient of the quantity of interest (QoI) within the AS dimensionality-reduction framework. Furthermore, Shapley method is applied to study the sensitivity of geometric variables and their variation direction on the mean and variance of the performance, and the robustness improvement method by only modifying key geometric variables based on Shapley results is discussed. The RDO of high dimensional geometrical uncertainties of a transonic compressor stage is studied by using this method. The results demonstrate a reduction from the original 24-dimensional uncertainties to a 4-dimensional space, achieved through this method. Consequently, the required sample size is reduced by 75% while maintaining nearly unchanged model accuracy. Furthermore, the findings reveal that after RDO, the mean efficiency increases by 1.04%, with a 42% reduction in variance. Compared to RDO, which requires simultaneous optimization of all 24 geometric variables, the robustness improvement based on Shapley only necessitates modifying 4 key geometric variables, resulting in a substantial increase in both the mean and variance of efficiency.

Uncertainty quantification with high-dimensional correlated process variations for an in-situ thermal expansion coefficient test structure

The impacts of process variations on the π-shaped thermal expansion coefficient (TEC) in-situ test structure are analyzed. Correlations among high-dimensional design parameters have to be taken into account for this type of structure, while the widely studied polynomial chaos expansion (PCE) technique is incapable of solving such problems. This paper presents a novel method for high-dimensional uncertainty quantification problems with non-Gaussian correlated variants. As a pre-processing method, the Spearman correlation coefficient and generalized lambda distribution are merged to eliminate the correlation. Then a compressed-sensing based Legendre-sparse solver is applied benefiting from the uniformly distributed parameters derived from the pre-processing and thus the cost of rebuilding correlated basis functions is eliminated. This method has demonstrated superior performance over existing methods in the analysis of TEC in-situ test structures. This analysis shows that the measurement accuracy of the test structure demonstrates good performance and is insensitive to process errors, which is consistent with experimental data. The presented technique has the potential to be applied in the design of various microelectromechanical system (MEMS) devices.

An adaptive data-driven subspace polynomial dimensional decomposition for high-dimensional uncertainty quantification based on maximum entropy method and sparse Bayesian learning

Polynomial dimensional decomposition (PDD) is a surrogate method originated from the ANOVA (analysis of variance) decomposition, and has shown powerful performance in uncertainty quantification (UQ) accuracy and convergence recently. However, complex high-dimensional problems result in a large number of polynomial basis functions, leading to heavy computational burden, and the probability distributions of the input random variables are indispensable for PDD modeling and UQ, which may be unavailable in practical engineering. This study establishes an adaptive data-driven subspace PDD (ADDSPDD) for high-dimensional UQ, which employs two types of data for modeling the PDD basis function and the low-dimensional subspace directly, namely, the data of input random variables and the input-response samples. Firstly, we propose a data-driven zero-entropy criterion-based maximum entropy method for reconstructing the probability density functions (PDF) of input variables. Then, with the aid of the established PDFs, a data-driven subspace PDD (DDSPDD) is proposed based on the whitening transformation. To recover the subspace of the function of interest accurately and efficiently, we put forward an approximate active subspace method (AAS) based on the Taylor expansion under some mild premises. Finally, we integrate an adaptive learning algorithm into the DDSPDD framework based on the sparse Bayesian learning theory, obtaining our ADDSPDD; thus, the real subspace and the significant PDD basis functions can be identified with limited computational budget. We validate the proposed method by using four examples, and systematically compare four existing dimension-reduction methods with the AAS. Results show that the proposed framework is effective and the AAS is a good choice when the corresponding assumptions are satisfied.

Classifier-based adaptive polynomial chaos expansion for high-dimensional uncertainty quantification

A novel approach for the construction of polynomial chaos expansion (PCE) is proposed to facilitate high-dimensional uncertainty quantification (UQ). The current PCE techniques are well-known for UQ; however, they are affected by the curse of dimensionality that leads to over-fitting and tractability issues. Although L1-minimization can be used to deal with over-fitting, it is still ineffective for problems with a large number of independent random inputs or requiring a very high-order PCE. Therefore, a classifier-based PCE (CAPCE) is presented to mitigate the factorial growth of basis terms and prevent over-fitting. The adaptive framework includes four basis selection strategies – enrichment, screening, discovery, and recycling – in the inner loop and the enrichment of the training samples in the outer loop. Mainly, an adaptive classifier is trained on the dominant and discarded basis terms obtained from L1-minimization during discovery. It then allows the judicious selection of new higher-order basis candidates for L1-solver in the next iteration, thereby alleviating the effect of the curse of dimensionality. The proposed framework has been tested with analytical problems of varying sizes of independent random inputs and an engineering composite laminate problem. The comparison of CAPCE with the available efficient PCE techniques demonstrated improvements in accuracy using fewer samples due to a faster convergence rate.

Automatic Determination of Set of Multivariate Basis Polynomials Based on Recursion and Application of LSPCR to High-Dimensional Uncertainty Quantification of Multi-Conductor Transmission Lines

Automatic Determination of Set of Multivariate Basis Polynomials Based on Recursion and Application of LSPCR to High-Dimensional Uncertainty Quantification of Multi-Conductor Transmission Lines

Investigation on high-dimensional uncertainty quantification and reliability analysis of aero-engine

Accurate quantification of uncertainties is paramount in the contemporary design of aero-engines, demanding substantial computational resources. This investigation introduces an efficient parallel framework for uncertainty quantification, significantly curtailing sample requirements while upholding computational precision. Furthermore, this research tackles the constraints intrinsic to traditional reliability analyses, which often disregard uncertainty, by conceiving an algorithm tailored to aero-engine reliability analysis and lifespan prediction that deliberately accounts for uncertainties. The findings from the uncertainty quantification disclose that the most pivotal factor contributing to the variability in leakage flow rate is unequivocally the variance index of the tip clearance deviation, demonstrating a substantial variance index of 81.52 %. Concerning cooling performance, a remarkable shift is evident, with inlet total temperature fluctuation eclipsing the tip clearance deviation in terms of significance. This transformation is accentuated by a variance index of 54.52 %. Furthermore, the film cooling effectiveness displays notable variability within the uncertainty framework, underscoring that the blades may be subjected to more severe thermal corrosion and thermal fatigue during operational conditions than initially envisioned. Reliability calculations elucidate that the probability of blade failure experiences the steepest ascent after approximately 15,500 h of operation, necessitating mandatory overhaul at this juncture.

High-Dimensional Uncertainty Quantification of High-Pressure Turbine Vane Based on Multifidelity Deep Neural Networks

Abstract In this work, a new multifidelity (MF) uncertainty quantification (UQ) framework is presented and applied to the LS89 nozzle modified by fouling. Geometrical uncertainties significantly influence the aerodynamic performance of gas turbines. One representative example is given by the airfoil shape modified by fouling deposition, as in turbine nozzle vanes, which generates high-dimensional input uncertainties. However, the traditional UQ approaches suffer from the curse of dimensionality phenomenon in predicting the influence of high-dimensional uncertainties. Thus, a new approach based on multifidelity deep neural networks (MF-DNN) was proposed in this paper to solve the high-dimensional UQ problem. The basic idea of MF-DNN is to ensure the approximation capability of neural networks based on abundant low-fidelity (LF) data and few high-fidelity (HF) data. The prediction accuracy of MF-DNN was first evaluated using a 15-dimensional benchmark function. An affordable turbomachinery UQ platform was then built based on key components including the MF-DNN model, the sampling module, the parameterization module and the statistical processing module. The impact of fouling deposition on LS89 nozzle vane flow was investigated using the proposed UQ framework. In detail, the MF-DNN was fine-tuned based on bi-level numerical simulation results: the 2D Euler flow field as low-fidelity data and the 3D Reynolds-averaged Navier–Stokes (RANS) flow field as high-fidelity data. The UQ results show that the total pressure loss of LS89 vane is increased by at most 17.1% or reduced by at most 4.3%, while the mean value of the loss is increased by 3.4% compared to the baseline. The main reason for relative changes in turbine nozzle performance is that the geometric uncertainties induced by fouling deposition significantly alter the intensity of shock waves near the throat area and trailing edge. The developed UQ platform could provide a useful tool in the design and optimization of advanced turbomachinery considering high-dimensional input uncertainties.

A fast evaluation method of surrogate model based on active subspace dimension reduction

Aimed at the problem of high-dimensional uncertainty quantification commonly existing in engineering complex shapes, traditional methods are often time-consuming and energy-consuming. So the paper proposes an efficient surrogate model method based on active subspace dimension reduction. The method uses active subspace to achieve feature dimension reduction and construct surrogate model, and develops a fast evaluation method of 1-D active subspace based on polynomial least squares decomposition to solve the difficult derivative problems. The detailed implementation process of the method is given in the paper, and the feasibility of the method is verified by examples such as the ablation thermal response and RAE2822 airfoil. It provides a solution for the quantification of multivariate uncertainty in engineering.

Statistical evaluation of stability margin of a multi-stage compressor with geometric variability using adaptive polynomial chaos-Kriging model

Compressed air energy storage systems must promptly adapt to power network demand fluctuations, necessitating a high surge margin in the compression system to ensure safety. It is challenging to completely eliminate blade geometric variations caused by limited machining precision, the important effects of which should be considered during aerodynamic shape design and production inspection. The present paper explores the uncertainty impact of geometric deviations on the stability margin of a multi-stage axial compressor at a low rotational speed. Initially, an adaptive polynomial chaos expansion-based universal Kriging model is introduced, and its superior response performance in addressing high-dimensional uncertainty quantification problems is validated through rigorous analytical and engineering tests. Then, this model is used to statistically evaluate the stability margin improvement (SMI) of the compressor due to the Gaussian and realistic geometric variabilities separately. The results show that the mean and standard deviation of SMI are −0.11% and 0.5% under the Gaussian geometric variability, while those are 0.33% and 0.39% under the realistic variability. For both the geometric variabilities, the stagger angle and maximum thickness deviations of the first-stage rotor are the most influential parameters controlling the uncertainty variations in the stability margin. Finally, the underlying impact mechanism of the influential geometric deviations is investigated. The variation in the stability margin caused by the geometric deviations primarily results from the alteration of inlet incidences, affecting the size of the tip leakage vortex blockage and boundary-layer separation regions near the blade tip of the first-stage rotor.

High-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer based on probability density evolution method

This paper proposed an efficient research method for high-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer. Firstly, the dynamic model of projectile motion is established considering the flexible deformation of the barrel and the interaction between the projectile and the barrel. Subsequently, the accuracy of the dynamic model is verified based on the external ballistic projectile attitude test platform. Furthermore, the probability density evolution method (PDEM) is developed to high-dimensional uncertainty quantification of projectile motion. The engineering example highlights the results of the proposed method are consistent with the results obtained by the Monte Carlo Simulation (MCS). Finally, the influence of parameter uncertainty on the projectile disturbance at muzzle under different working conditions is analyzed. The results show that the disturbance of the pitch angular, pitch angular velocity and pitch angular of velocity decreases with the increase of launching angle, and the random parameter ranges of both the projectile and coupling model have similar influence on the disturbance of projectile angular motion at muzzle.

Gated Linear Model induced U-net for surrogate modeling and uncertainty quantification

We propose a novel deep learning based surrogate model for solving high-dimensional uncertainty quantification and uncertainty propagation problems. The proposed deep learning architecture is developed by integrating the well-known U-net architecture with the Gaussian Gated Linear Network (GGLN) and referred to as the Gated Linear Network induced U-net or GLU-net. The proposed GLU-net treats the uncertainty propagation problem as an image to image regression and hence, is extremely data efficient. Additionally, it also provides estimates of the predictive uncertainty. The network architecture of GLU-net is less complex with 44% fewer parameters than the contemporary works. We illustrate the performance of the proposed GLU-net in solving the Darcy flow problem under uncertainty under the sparse data scenario. We consider the stochastic input dimensionality to be up to 4225. Benchmark results are generated using the vanilla Monte Carlo simulation. We observe the proposed GLU-net to be accurate and extremely efficient even when no information about the structure of the inputs is provided to the network. Case studies are performed by varying the training sample size and stochastic input dimensionality to illustrate the robustness of the proposed approach.

High-Dimensional Uncertainty Quantification in Electrical Impedance Tomography Forward Problem Based on Deep Neural Network

In electrical impedance tomography (EIT), the uncertainty of conductivity distribution may cause the uncertainty in the forward calculation and further affect the inverse problem. In this paper, an improved univariate dimension reduction method based on deep neural network (DNN-UDR) is proposed for the high-dimensional uncertainty quantification in EIT forward problem. Firstly, DNN is studied to build a substitute model for EIT forward problem in order to solve the high-dimensional problem. Three normalized circular finite element models are established with random uniform conductivity distribution. Then UDR is used to analyze and quantify the uncertainty in the simulation with the form of probability. Compared with Monte Carlo simulation (MCS), the probability distribution of voltage is fitted, and the quantification indicators such as mean, variance, variation coefficient and covariance, are also consistent. On the other hand, with the increase of parameter dimensions, DNN-UDR accelerates the computations obviously. This indicates that DNN-UDR is effective and has high structural stability, accurate prediction results and high computational efficiency.

Multi-Fidelity Low-Rank Approximations for Uncertainty Quantification of a Supersonic Aircraft Design

Uncertainty quantification has proven to be an indispensable study for enhancing reliability and robustness of engineering systems in the early design phase. Single and multi-fidelity surrogate modelling methods have been used to replace the expensive high fidelity analyses which must be repeated many times for uncertainty quantification. However, since the number of analyses required to build an accurate surrogate model increases exponentially with the number of random input variables, most surrogate modelling methods suffer from the curse of dimensionality. As an alternative approach, the Low-Rank Approximation method can be applied to high-dimensional uncertainty quantification studies with a low computational cost, where the number of coefficients for building the surrogate model increases only linearly with the number of random input variables. In this study, the Low-Rank Approximation method is implemented for multi-fidelity applications with additive and multiplicative correction approaches to make the high-dimensional uncertainty quantification analysis more efficient and accurate. The developed uncertainty quantification methodology is tested on supersonic aircraft design problems and its predictions are compared with the results of single- and multi-fidelity Polynomial Chaos Expansion and Monte Carlo methods. For the same computational cost, the Low-Rank Approximation method outperformed both in surrogate modeling and uncertainty quantification cases for all the benchmarks and real-world engineering problems addressed in the present study.

A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems

Constructing surrogate models for uncertainty quantification (UQ) on complex partial differential equations (PDEs) having inherently high-dimensional O(10n), n≥2, stochastic inputs (e.g., forcing terms, boundary conditions, initial conditions) poses tremendous challenges. The “curse of dimensionality” can be addressed with suitable unsupervised learning techniques used as a pre-processing tool to encode inputs onto lower-dimensional subspaces while retaining its structural information and meaningful properties. In this work, we review and investigate thirteen dimension reduction methods including linear and nonlinear, spectral, blind source separation, convex and non-convex methods and utilize the resulting embeddings to construct a mapping to quantities of interest via polynomial chaos expansions (PCE). We refer to the general proposed approach as manifold PCE (m-PCE), where manifold corresponds to the latent space resulting from any of the studied dimension reduction methods. To investigate the capabilities and limitations of these methods we conduct numerical tests for three physics-based systems (treated as black-boxes) having high-dimensional stochastic inputs of varying complexity modeled as both Gaussian and non-Gaussian random fields to investigate the effect of the intrinsic dimensionality of input data. We demonstrate both the advantages and limitations of the unsupervised learning methods and we conclude that a suitable m-PCE model provides a cost-effective approach compared to alternative algorithms proposed in the literature, including recently proposed expensive deep neural network-based surrogates and can be readily applied for high-dimensional UQ in stochastic PDEs.

Sparse Polynomial Chaos Expansion for Uncertainty Quantification of Composite Cylindrical Shell with Geometrical and Material Uncertainty

The geometrical dimensions and mechanical properties of composite materials exhibit inherent variation and uncertainty in practical engineering. Uncertainties in geometrical dimensions and mechanical properties propagate to the structural performance of composite cylindrical shells under hydrostatic pressure. However, traditional methods for quantification of uncertainty, such as Monte Carlo simulation and the response surface method, are either time consuming with low convergence rates or unable to deal with high-dimensional problems. In this study, the quantification of the high-dimensional uncertainty of the critical buckling pressure of a composite cylindrical shell with geometrical and material uncertainties was investigated by means of sparse polynomial chaos expansion (PCE). With limited design samples, sparse PCE was built and validated for predictive accuracy. Statistical moments (mean and standard deviation) and global sensitivity analysis results were obtained based on the sparse PCE. The mean and standard deviation of critical buckling pressure were 3.5777 MPa and 0.3149 MPa, with a coefficient of variation of 8.801%. Global sensitivity analysis results from Sobol’ indices and the Morris method showed that the uncertainty of longitudinal modulus has a massive influence on the critical buckling pressure of composite cylindrical shell, whereas the uncertainties of transverse modulus, shear modulus, and Poisson’s ratio have a weak influence. When the coefficient of variation of ply thickness and orientation angle does not surpass 2%, the uncertainties of ply thickness and orientation angle have a weak influence on the critical buckling pressure. The study shows that the sparse PCE is effective at resolving the problem of high-dimensional uncertainty quantification of composite cylindrical shell with geometrical and material uncertainty.

A Survey of Unsupervised Learning Methods for High-Dimensional Uncertainty Quantification in Black-Box-Type Problems

A Survey of Unsupervised Learning Methods for High-Dimensional Uncertainty Quantification in Black-Box-Type Problems

High-Dimensional Uncertainty Quantification via Tensor Regression With Rank Determination and Adaptive Sampling

Fabrication process variations can significantly influence the performance and yield of nanoscale electronic and photonic circuits. Stochastic spectral methods have achieved great success in quantifying the impact of process variations, but they suffer from the curse of dimensionality. Recently, low-rank tensor methods have been developed to mitigate this issue, but two fundamental challenges remain open: how to automatically determine the tensor rank and how to adaptively pick the informative simulation samples. This article proposes a novel tensor-regression method to address these two challenges. We use an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{q}/ \ell _{2}$ </tex-math></inline-formula> group-sparsity regularization to determine the tensor rank. The resulting optimization problem can be efficiently solved via an alternating minimization solver. We also propose a two-stage adaptive sampling method to reduce the simulation cost. Our method considers both exploration and exploitation via the estimated Voronoi cell volume and nonlinearity measurement, respectively. The proposed model is verified with synthetic and some realistic circuit benchmarks, on which our method can well capture the uncertainty caused by 19–100 random variables with only 100–600 simulation samples.

Progress of Tensor-Based High-Dimensional Uncertainty Quantification of Process Variations

Uncertainty quantification under process variations is an important topic in computational electromagnetics (EM) and electronic design automation (EDA). The popular stochastic methods become computationally inefficient when the number of uncertain parameters grows. A promising technique to address this challenge is tensor computation. This short paper reviews some recent tensor methods for high-dimensional uncertainty quantification in the field of EDA and computational EM.

High-Dimensional Uncertainty Quantification in a Hybrid Data + Model-Based Submodeling Method for Refined Response Estimation at Critical Locations

AbstractThis study presents frameworks to build nonintrusive polynomial chaos expansion (PCE) models with regression and Smolyak sparse-grid quadrature to perform high-dimensional uncertainty propa...

A dimension-reduction metamodeling approach to simulation-based uncertainty quantification problems with high dimensionalities

A common strategy to handle simulation-based uncertainty quantification problems is adopting a metamodel to replace time-demanding calculations such as computational fluid dynamics simulation or finite element analysis within Monte Carlo simulation process. However, most of the so far metamodel-assisted uncertainty quantification methods suffer from the ‘curse of dimensionality.’ The required number of evaluations, which determines the computational cost, increases exponentially as the dimensionality of the input uncertainty increases, resulting in unaffordable computational cost for high-dimensional problems. Another challenge emerges when the output uncertainties are a spatially varying field accommodating a huge number of spatial nodes. To solve these issues, here we propose a dimension-reduction metamodeling approach, in which active subspace method is utilized to reduce the input dimensionality and proper orthogonal decomposition method is utilized to reduce the output dimensionality of the spatially varying field. The relationship between the two methods is established by using the support vector regression model. Through uncertainty quantification of seven stochastic analytical functions and one stochastic convection-diffusion equation, the proposed approach was verified to be fairly accurate in propagating high-dimensional input uncertainties to either a scalar value or a spatially varying output. The accuracy and efficiency of the proposed approach in dealing with even more practical simulation-based problems were then validated by uncertainty quantification of a compressor cascade with stochastic protrusions/dents distributed on the blade surface. This work provides an effective and versatile approach for simulation-based high-dimensional uncertainty quantification problems.