Polynomial Chaos Expansion Research Articles

Global sensitivity analysis for multivariate outputs using generalized RBF-PCE metamodel enhanced by variance-based sequential sampling

The mathematical and computational models in engineering applications commonly have multiple outputs, so it is critical to develop global sensitivity analysis (GSA) for multivariate outputs, which can be used to explore the effect of input parameters on output responses. Amongst the existing sensitivity analysis, the covariance-based method is one of the most widely used methods due to its understandability and validity. However, traditional GSA is calculated by Monte Carlo Simulation (MCS), leading to huge computational cost and large sample demand. In this study, the generalized hybrid metamodel using radial basis function (RBF) and sparse polynomial chaos expansion (PCE) is applied for covariance-based GSA. To improve the efficiency and performance of RBF-PCE in mult-models, an efficient sequential sampling method is proposed based on the local variance density of sample points. Three analytical functions and two engineering problems are employed to demonstrate the accuracy and validity of the proposed method. The results indicate that the proposed method has a significant improvement in accuracy and efficiency compared with the existing GSA methods, and the sample demand is reduced by three orders of magnitude compared with MCS-based Sobol' indices.

Uncertainty-oriented optimal impedance control for EPS-human system with reliability evaluation

This study proposes an interval optimal impedance control method for the electric power steering- human (EPS-H) system with reliability constraints. Considering the relationship between the lateral control using electric power steering of the vehicle and the human, a position-based impedance control model for the integrated EPS-H system is first developed. Polynomial chaos expansion in the surrogate model is employed to establish the interval state-space equation to obtain the bounded responses of the impedance control system using the interval dimensional analysis method. Moreover, the influence of uncertain parameters on the overall control system is obtained by deriving the interval optimal control law consisting of the interval linear quadratic tracking (LQT) inner loop and the interval impedance control outer loop through the interval uncertainty analysis. Furthermore, a non-probabilistic time-varying reliability method is proposed to evaluate the safety of the whole impedance system using the partial derivatives with respect to the uncertain parameters derived from the interval state-space equation. Regarding the nominal cost function and steady-state error of nominal responses as two optimization objectives, the multi-objective reliability-based design optimization is built and solved. Finally, a numerical example of the impedance control model for the EPS-H system is applied to verify the effectiveness of the proposed method.

Acoustic Performance of a Metamaterial Coating with Uncertainty in Design Parameters

The effect of uncertainty in design parameters on the acoustic performance of a viscoelastic metamaterial coating is investigated. The coating is composed of a layer of hard inclusions embedded in a soft host medium. The layer of inclusions in the soft medium is modelled as a homogenised layer using effective medium approximation theory. Stochastic models based on the polynomial chaos expansion (PCE) method and Monte Carlo (MC) simulations are developed. The effect of uncertainty in geometric and material parameters are quantified by examining the mean and envelopes of the absorption coefficient. Uncertainty in geometric and material parameters produce similar levels of variation in the absorption coefficient of the coating, particularly around the peaks of the absorption coefficient spectrum.

Probabilistic-based burst failure mechanism analysis and risk assessment of pipelines with random non-uniform corrosion defects, considering the interacting effects

Corrosion defects were primary causes for pipeline burst failures. Traditional methodologies simplified the realistic irregular-shaped defects as uniform ones, ignoring the effects of random morphologies on failure behaviours, which caused deviations in remaining strength pb estimation and reliability analysis. Addressing this issue, through coupling random field, non-linear finite element analysis and Monte-Carlo Simulation, an integrated methodology was developed to describe failure behaviours of pipelines with random defects. The failure mechanism and reliability analysis were performed. The pbs for random defects exhibited significant variabilities, of which the probability distribution patterns altered with different corrosion depths attributed to multiple failure modes. The main value of pb for random defects was lower than that of uniform defects, especially for deeper corrosion and interacting defects. At maximum, this reduction attained over 28%; The spatial variation gradient of remaining wall thickness dominated failure developments, based on which three failure modes were revealed; The wall thickness, corrosion degree and interacting effects were influential on failure probabilities Pfs. An average failure risk of Pf over 0.6 would be suffered for employing traditional methodologies. Polynomial chaos expansion models were fitted to estimate pb from corrosion morphologies, showing reasonable accuracies. Besides, the performance variations with different cases were discussed.

Advanced golden jackal optimization for solving the constrained integer stochastic optimization problems

Constrained integer stochastic optimization problems (CISOP) are optimization problems where the cost function is stochastic and the constraints are deterministic. Solving the CISOP using conventional optimization techniques is time-consuming when the design space is huge. Ordinal optimization offers an efficient technique suitable for solving the CISOP. In this paper, an approach that uses golden jackal optimization assisted by ordinal optimization (GJOO) is developed for resolving the CISOP in a relatively short time. The GJOO is composed of three phases: metamodel, global search, and ranking and selection. At first, the polynomial chaos expansion is used as a metamodel to rapidly assess a solution. Next, advanced golden jackal optimization is adopted to determine N excellent solutions from the entire design space. At last, the modified optimal computing budget allocation is adopted to determine a distinguished solution from the N excellent solutions. The GJOO is utilized to the shipping/receiving docks optimization problem of a container freight station in air cargo. The performances of the GJOO are compared with those of five meta-heuristic methods. Empirical results demonstrate the efficiency and robustness of the GJOO algorithm.

Interval-Probabilistic Electricity-Heat-Gas Flow Calculation by Dual-Level Surrogate Structure

The study of highly efficient methods for performing multiple energy flow (MEF) computation with epistemic and aleatory uncertainties, known as interval-probabilistic energy flow (IPEF) calculation, is a fast-growing area of research. However, existing approaches are easily caught in a quandary among accuracy, efficiency, and scope of applicability. This paper presents a dual-level surrogate structure (DLSS) for dealing with the IPEF calculation of an integrated energy system (IES) in an accurate and efficient manner. The structure accommodates electricity, heat, and gas, attaining a high-level balance among precision, speed, and range of applicability. At the lower level of DLSS, a sparse polynomial chaos expansion (sPCE) model is employed to directly map random input variables to output boundary values, saving massive repetition of deterministic calculation. At the upper level of DLSS, a heterogeneous-learning-based multiple energy flow (HL-MEF) model is proposed to accelerate the indispensable deterministic MEF calculation for online sampling, thereby easing the computational burden of sPCE training. Finally, numerical results on a real-world electricity-heat-gas system demonstrate that the proposed DLSS can solve the IPEF problem at nearly 400 times the speed of the standard double-layer Monte Carlo simulation (DLMCS) with guaranteed accuracy.

Uncertainty quantification in low-probability response estimation using sliced inverse regression and polynomial chaos expansion

For wave energy converters (WECs), wind turbines, etc., estimation of response extremes over a selected exposure time is important during design. Sources of uncertainty arising from background slowly-varying environmental conditions and from shorter time-scale fluctuations in ocean winds, turbulence, etc. must all be considered. Together, these different sources can comprise a high-dimensional vector of stochastic variables (often on the order of hundreds or thousands). To accurately propagate the influence of these uncertainty sources to model outputs, conventional surrogate model building approaches such as polynomial chaos expansion (PCE), stochastic collocation, low-rank tensor approximations, etc. must consider dimension reduction. We explore the use of sliced inverse regression (SIR) combined with polynomial chaos expansion. SIR first reduces the original high-dimensional problem to a low-dimensional one; then, an optimal polynomial PCE model is proposed and applied on “effective” components in the low-dimensional space. SIR-PCE can mitigate the curse of dimensionality. It is employed here in the prediction of the long-term extreme response of offshore structures; it is demonstrated using classical benchmark analytical functions as well as offshore applications including extreme waves and the response of a wave energy converter. Efficiency and accuracy gains over Monte Carlo simulation and other methods in literature are found.

Radial Basis Function Surrogates for Uncertainty Quantification and Aerodynamic Shape Optimization under Uncertainties

This paper investigates the adequacy of radial basis function (RBF)-based models as surrogates in uncertainty quantification (UQ) and CFD shape optimization; for the latter, problems with and without uncertainties are considered. In UQ, these are used to support the Monte Carlo, as well as, the non-intrusive, Gauss Quadrature and regression-based polynomial chaos expansion methods. They are applied to the flow around an isolated airfoil and a wing to quantify uncertainties associated with the constants of the γ−R˜eθt transition model and the surface roughness (in the 3D case); it is demonstrated that the use of the RBF-based surrogates leads to an up to 50% reduction in computational cost, compared with the same UQ method that uses CFD computations. In shape optimization under uncertainties, solved by stochastic search methods, RBF-based surrogates are used to compute statistical moments of the objective function. In applications with geometric uncertainties which are modeled through the Karhunen–Loève technique, the use on an RBF-based surrogate reduces the turnaround time of an evolutionary algorithm by orders of magnitude. In this type of applications, RBF networks are also used to perform mesh displacement for the perturbed geometries.

Robust Constrained Multi-Objective Guidance of Supersonic Transport Landing Using Evolutionary Algorithm and Polynomial Chaos

Landing of supersonic transport (SST) suffers from a large uncertainty due to its highly sensitive aerodynamic properties in the subsonic domain, as well as the wind gusts around runways. At the vehicle design stage, a landing trajectory optimization under wind uncertainty in a multi-objective solution space is desired to explore the possible trade-off in its key flight performance metrics. The proposed algorithm solves this robust constrained multi-objective optimal control problem by integrating non-intrusive polynomial chaos expansion into a constrained evolutionary algorithm. The computationally tractable optimization is made possible through the conversion of a probabilistic problem into an equivalent deterministic representation while maintaining a form of the multi-objective problem. The generated guidance trajectories achieve a significant reduction of the uncertainty in their terminal states with a marginal modification in the control history of the deterministic solutions, validating the importance of the consideration of robustness in trajectory optimization.

Multi-dimensional population balance modeling of sonocrystallization of pyrazinamide with systematic estimation of kinetic parameters based on uncertainty and sensitivity analyses

Application of ultrasound has several beneficial effects on both crystallization process and product crystal properties. This work focuses on experimental study and population balance modeling of ultrasound-assisted unseeded batch cooling crystallization of an important polymorphic pharmaceutical. The δ-polymorph of pyrazinamide, a drug for mycobacterium tuberculosis, exhibits plate-like morphology, and such crystals are best described by atleast two dimensions: length and width. A multi-dimensional population balance model for batch cooling sonocrystallization of pyrazinamide from its 1,4-dioxane solution is developed and validated. A series of experiments are performed with different ultrasound amplitudes (silent, 10%, and 50%) to study the effect of ultrasound on nucleation rate, crystal size, and polymorphism. The ultrasound is applied till the point of nucleation and an enhanced nucleation rate is introduced in the model in the presence of ultrasound irradiation. The various kinetic parameters are estimated following a systematic methodology based on uncertainty quantification and sensitivity analysis using experimental observations related to solute concentration and crystal size distribution. The Polynomial Chaos Expansion (PCE) is used to quantify the uncertainties in model prediction due to kinetic parameter uncertainties and sensitivity analysis is performed using total order Sobol’s sensitivity indices derived from PCE model. The kinetic parameters are estimated in two steps. First, a global search technique is used to simultaneously estimate all the kinetic parameters. Next, the less sensitive parameters are kept fixed and the more significant parameters are fine-tuned using a gradient-based local optimizer. Out of total nine kinetic parameters, only four parameters appeared to be significant for both silent and sonicated cases. The high-resolution finite volume scheme is used to solve the multi-dimensional population balance model and the simulation results agree very well with experimental data related to both concentration and crystal size distributions. The results revealed that compared to silent cases, sonication increased the nucleation rate by 102 times and 104 times for 10% and 50% amplitude, respectively. The growth rate was also enhanced significantly with increase in ultrasonic amplitude. All the experiments yielded the desired δ-polymorph of pyrazinamide.

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.

Uncertainty Quantification for the Noah‐MP Land Surface Model: A Case Study in a Grassland and Sandy Soil Region

AbstractLand surface models rely on a multitude of parameters to simulate land‐atmosphere interactions, but the parameter uncertainty can limit the reliability of model predictions. This study utilizes a parameter uncertainty quantification (UQ) framework to quantify and reduce the parameter uncertainty of the Noah‐MP land surface model in a grassland and sandy soil region in the Midwest of the USA. First, the sparse polynomial chaos expansion method which can quantify the interaction effect of parameters, is employed. A relatively small parameter sample size (i.e., 20 times of the number of parameters) was sufficient to identify the sensitive parameters; an additional sensitive parameter, the saturated soil hydraulic conductivity, was screened out compared to previous study. Then, based on the selected sensitive parameters, the weighted multi‐objective adaptive surrogate modeling‐based optimization algorithm is used as the parameter optimization method. The optimization results showed that the root mean square error of flux of latent heat (FLH) on about 82% of the total grids was reduced, and the number was about 57% for gross primary production (GPP) compared to the results using the original parameter settings, indicating that the Pareto parameter set by the UQ framework improved the Noah‐MP model in simulating FLH and GPP in a grassland and sandy soil region in the Midwest of the USA.

Surrogate modeling in irreversible electroporation towards real-time treatment planning

In this paper, we develop surrogate models that can replace expensive predictive models and account for uncertainties in real-time treatment planning for irreversible electroporation of liver tumors. Standard non-intrusive surrogate modeling techniques that account for the model uncertainty and reduce the computational cost, such as polynomial chaos expansion and Gaussian process regression with conventional kernels, often do not capture the true physical behavior of the treatment outcome as required in the context of treatment planning. We improve the Gaussian process regression model by modifying the kernel function to a non-stationary Gibbs kernel with a support vector machine-based classifier in its length scale definition. This proposed model is compared with the standard surrogates in terms of their performance and accuracy. Our model is able to accurately replicate the behavior of the biophysics-based predictive model. There is a decrease of at least 81% in the overall root-mean-square error for treatment outcome when compared to the Gaussian process regression model with conventional kernels. Furthermore, we illustrate the application of the proposed surrogate model in treatment planning to address a voltage optimization problem for complete tumor ablation. Surrogate-assisted treatment planning exhibited good performance while maintaining similar levels of accuracy in comparison to treatment planning based on biophysical models. Finally, the effect of uncertainty in tissue electrical conductivities on the optimal voltage value is discussed.

Exploring global uncertainty quantification and sensitivity analysis methodologies: CO2 capture absorber model with MEA solvent as a test case

A set of global metamodeling uncertainty quantification (UQ) techniques belonging to non-intrusive and forward propagation categories; Polynomial Chaos Expansion (PCE), Kriging, Canonical Low Rank Approximation (LRA), and Polynomial Chaos Kriging (PC-Kriging) and global sensitivity analysis (SA) techniques, Sobol’ indices, Borgonovo, and Morris were compared to a benchmark methodology of direct Monte Carlo Simulation (MCS). The comparative analysis was demonstrated on a CO2 capture absorber model with MEA solvent. Our analysis concluded as follows; (1) although significant variation in the CO2 capture ratio in the prediction profile is shown, there is no apparent advantage of using a larger sample size via direct MCS except achieving a higher clarity in the output distribution, (2) while a very little effect on the convergence was observed which confirms the number of sample size, the fraction of computational time is enhanced by using Sobol sampling method, (3) the benefit of using metamodeling techniques compared to direct MCS was proven as they provide equal predictions in the output statistical measures with fewer evaluations of the original model and are therefore computationally cheaper, and (4) the global SA results showed Sobol indices is computationally more efficient while sharing similar ranking for the most influential parameter with Borgonovo and Morris.

Bayesian interface based calibration of a novel rockfall protection structure modelled in the non-smooth contact dynamics framework

This article presents the development and calibration of a numerical model simulating the response of a novel rockfall protection structure subjected to localized dynamic loading. This structure is made of piled-up concrete blocks interconnected via metallic components whose dynamics response under projectile impact is examined via real-scale experiments. The corresponding numerical model is developed in a python based open source software Siconos which implements the Non-Smooth Contact Dynamics (NSCD) method. The geometrical features and mechanical properties are incorporated in the model via specific developments pertinent to the modelling requirements in Siconos. Some parameters peculiar to the numerical model are calibrated against the spatial–temporal measurements from two full-scale impact experiments. The Bayesian interface statistical learning method aided by the polynomial chaos expansion based meta-model of the NSCD model is deployed for the calibration. The additional understanding of the model dynamics through the byproducts of the meta-model is highlighted. In the end, the NSCD model is successfully calibrated against the spatial–temporal response of the experimental structure with more than 90% accuracy for impact energies up to 1 MJ.

Analyzing the sensitivity of wave frequency responses of floating vessels to uncertain system variables utilizing Polynomial Chaos Expansion

Abstract From a mathematical viewpoint, the frequency domain analysis of vessel motion responses due to wave actions is based on integration of system dynamics idealized in terms of response amplitude operators (RAOs) for 6 DOF rigid body motions and an input wave spectrum in order to obtain the response spectrum. Various quantities of interest can be deduced from the response spectrum which are then used for deriving response-based operational limits for marine operations, also including extreme value and fatigue analysis. The variation of such quantities, owing to the uncertainties associated with the vessel system parameters, can be quantified by performing uncertainty propagation (UP) and consequent sensitivity analysis (SA). This study emphasizes and proposes a computational-efficient way of assessing the sensitivity of the system model output with respect to the uncertainties residing in the input parameters by operating on a surrogate model representation. In this respect, the global sensitivity analysis was effectively carried out by deploying an efficient non-intrusive polynomial chaos expansion (PCE) surrogate model built using a point collocation strategy. Successively, the Sobol' indices were obtained from the analytical decomposition of the polynomial coefficients. The indices, eventually, are employed to quantitatively measure the effects of input uncertainties on the output 6 DOF vessel Root Mean Square responses.

Uncertainty Quantification and Sensitivity Analysis for the Self-Excited Vibration of a Spline-Shafting System

Abstract Self-excited vibrations can occur in the spline-shafting system due to internal friction of the tooth surface. However, due to manufacturing errors, design tolerances, and time-varying factors, the parameters that induce self-excited vibrations are always uncertain. This study provides new insights into the uncertainty quantification and sensitivity analysis of a spline-shaft system suffering from self-excited vibrations. The nonintrusive generalized polynomial chaos expansion (gPCE) with unknown deterministic coefficients is used to represent the propagation of uncertainties in the rotor dynamics, which allows rapid estimation of the statistics of the nonlinear responses. Furthermore, the global sensitivity analysis of the stochastic self-excited vibration response of the rotor system with probabilistic uncertain parameters is evaluated by Sobol indices. The relative influence of different random parameters on the vibration behavior and initial displacement conditions for the occurrence of self-excited vibration is investigated. The accuracy of the adopted method based on the gPCE metamodel is validated by conventional Monte Carlo simulation (MCS). Finally, the effects of parameter uncertainties considering random distribution characteristics on the stochastic vibration characteristics of the rotor system are discussed, which demonstrates the need to consider input uncertainties in analysis and design to ensure robust system performance.

Uncertainty quantification of the effect of cardiac position variability in the inverse problem of electrocardiographic imaging

Objective. Electrocardiographic imaging (ECGI) is a functional imaging modality that consists of two related problems, the forward problem of reconstructing body surface electrical signals given cardiac bioelectric activity, and the inverse problem of reconstructing cardiac bioelectric activity given measured body surface signals. ECGI relies on a model for how the heart generates bioelectric signals which is subject to variability in inputs. The study of how uncertainty in model inputs affects the model output is known as uncertainty quantification (UQ). This study establishes develops, and characterizes the application of UQ to ECGI. Approach. We establish two formulations for applying UQ to ECGI: a polynomial chaos expansion (PCE) based parametric UQ formulation (PCE-UQ formulation), and a novel UQ-aware inverse formulation which leverages our previously established ‘joint-inverse’ formulation (UQ joint-inverse formulation). We apply these to evaluate the effect of uncertainty in the heart position on the ECGI solutions across a range of ECGI datasets. Main results. We demonstrated the ability of our UQ-ECGI formulations to characterize the effect of parameter uncertainty on the ECGI inverse problem. We found that while the PCE-UQ inverse solution provided more complex outputs such as sensitivities and standard deviation, the UQ joint-inverse solution provided a more interpretable output in the form of a single ECGI solution. We find that between these two methods we are able to assess a wide range of effects that heart position variability has on the ECGI solution. Significance. This study, for the first time, characterizes in detail the application of UQ to the ECGI inverse problem. We demonstrated how UQ can provide insight into the behavior of ECGI using variability in cardiac position as a test case. This study lays the groundwork for future development of UQ-ECGI studies, as well as future development of ECGI formulations which are robust to input parameter variability.

On a Stochastic Fundamental Lemma and Its Use for Data-Driven Optimal Control

Data-driven control based on the fundamental lemma by Willems et al. is frequently considered for deterministic LTI systems subject to measurement noise. However, besides measurement noise, stochastic disturbances might also directly affect the dynamics. In this paper, we leverage Polynomial Chaos Expansions (PCE) to extend the deterministic fundamental lemma towards stochastic systems. This extension allows to predict future statistical distributions of the inputs and outputs for stochastic LTI systems in data-driven fashion, i.e., based on the knowledge of previously recorded input-output-disturbance data and of the disturbance distribution we perform data-driven uncertainty propagation. Finally, we analyze data-driven stochastic optimal control problems and we propose a conceptual framework for data-driven stochastic predictive control. Numerical examples illustrate the efficacy of the proposed concepts.

Fast Uncertainty Quantification by Sparse Data Learning From Multiphysics Systems

Data learning is a burgeoning area; here, we refer to it as a gorgeous blend of scientific computing and high-dimensional statistics. In this article, we propose a generalized polynomial chaos expansion (gPCE)-based surrogate model, with a sparse dataset, for multiphysics problems with associated uncertainty. This data-driven method avoids the dilemma of traditional intrusive methods, which usually require tremendous revisions over the backend source code. To circumvent the curse of dimensionality in uncertainty quantification, we perform sparse data learning for the relationship between the system output and the gPCE bases. Numerical experiments showcase that the surrogate model exhibits superior fast performance in system-output prediction, statistical-feature extraction, and sensitivity analysis of the 3-D multiphysics problems.