Use Of Polynomial Chaos Expansions Research Articles

Polynomial chaos expansions for modelling the statistics of acoustic propagation in random waveguides

The use of Polynomial Chaos expansions to model the transfer of the statistical properties of the sound speed in ocean waveguides to those of acoustic waves passing through them is described. A perturbational framework approximating the interaction of acoustic normal modes with fluctuations around a background sound speed is used, with the fluctuations being represented vertically by empirical orthogonal functions and horizontally by correlation functions. The Polynomial Chaos expansions are derived to second order in the uncorrelated random variable representing the sound speed fluctuations for the generalized phase of the complex modal amplitudes, and to third order for the modal amplitudes themselves. The ability of the second order polynomial chaos expansion for the generalized model phase to accurately model acoustic propagation statistics is closely coupled to the accuracy of the adiabatic approximation in light scattering regimes. The weights of the Polynomial Chaos expansions are obtained using both direct (theoretical) and indirect (least-squares fit) methods. Divergence between these two sets of weights increases with combinations of increasing strength of sound speed fluctuations, increasing frequency, or increasing range, indicating where a higher order expansion may be necessary. Modelling results for the first and second moments of the acoustic intensity and the Scintillation Index in random waveguides are presented and compared to Monte Carlo results.

Global sensitivity analysis for phosphate slurry flow in pipelines using generalized polynomial chaos

Slurry transportation via pipelines has garnered growing attention across various industries worldwide, thanks to its efficiency and environmental friendliness. It has emerged as a vital tool for conveying significant volumes of raw phosphate material from extraction points to industrial plants, where it is processed into fertilizers. Yet, optimal and secure pipeline operations necessitate the careful calibration of several physical parameters and their interplay to minimize energy losses. A thorough exploration of the flow pressure drop and the various factors that influence it constitutes a crucial step in attaining this goal. The computational fluid dynamics techniques required to simulate three-dimensional slurry pipe flows pose formidable challenges, primarily due to their high computational costs. Furthermore, numerical solutions for slurry flows are frequently subject to uncertainties arising from the initial and boundary conditions in the mathematical models employed. In this study, we propose the use of polynomial chaos expansions to estimate the uncertainty inherent in the desired slurry flow and perform a sensitivity analysis of flow energy efficiency. In this framework, five parameters are considered as random variables with a given probability distribution over a prescribed range of investigation. The uncertainty is then propagated through the two-phase flow model to statistically quantify their effect on the results. Our findings reveal that variations in slurry velocity and particle size play a pivotal role in determining energy efficiency. Therefore, controlling these factors represents a critical step in ensuring the efficient and safe transportation of slurry through pipelines.

Uncertainty propagation in pore water chemical composition calculation using surrogate models

Performance assessment in deep geological nuclear waste repository systems necessitates an extended knowledge of the pore water chemical conditions prevailing in host-rock formations. In the last two decades, important progress has been made in the experimental characterization and thermodynamic modeling of pore water speciation, but the influence of experimental artifacts and uncertainties of thermodynamic input parameters are seldom evaluated. In this respect, we conducted an uncertainty propagation study in a reference geochemical model describing the pore water chemistry of the Callovian-Oxfordian clay formation. Nineteen model input parameters were perturbed, including those associated to experimental characterization (leached anions, exchanged cations, cation exchange selectivity coefficients) and those associated to generic thermodynamic databases (solubilities). A set of 13 quantities of interest were studied by the use of polynomial chaos expansions built non-intrusively with a least-squares forward stepwise regression approach. Training and validation sets of simulations were carried out using the geochemical speciation code PHREEQC. The statistical results explored the marginal distribution of each quantity of interest, their bivariate correlations as well as their global sensitivity indices. The influence of the assumed distributions for input parameters uncertainties was evaluated by considering two parametric domain sizes.

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

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

Propagation of parameter uncertainty in SWAT: A probabilistic forecasting method based on polynomial chaos expansion and machine learning

Soil and Water Assessment Tool (SWAT) is one of the most widely used semi-distributed hydrological models. Assessment of the uncertainties in SWAT outputs is a popular but challenging topic due to the significant number of parameters. The purpose of this study is to investigate the use of Polynomial Chaos Expansion (PCE) in assessing uncertainty propagation in SWAT under the impact of significant parameter sensitivity. Furthermore, for the first time, a machine learning technique (i.e., artificial neural network, ANN) is integrated with PCE to expand its capability in generating probabilistic forecasts of daily flow. The traditional PCE and the proposed PCE-ANN methods are applied to a case study in the Guadalupe watershed in Texas, USA to assess the uncertainty propagation in SWAT for flow prediction during the historical and forecasting periods. The results show that PCE provides similar results as the traditional Monte-Carlo (MC) method, with a coefficient of determination (R2) value of 0.99 for the mean flow, during the historical period; while the proposed PCE-ANN method reproduces MC output with a R2 value of 0.84 for mean flow during the forecasting period. It is also indicated that PCE and PCE-ANN are as reliable as but much more efficient than MC. PCE takes about 1% of the computational time required by MC; PCE-ANN only takes a few minutes to produce probabilistic forecasting, while MC requires running the model for dozens or hundreds, even thousands, of times. Notably, the development of the PCE-ANN framework is the first attempt to explore PCE’s probabilistic forecasting capability using machine learning. PCE-ANN is a promising uncertainty assessment and probabilistic forecasting technique, as it is more efficient in terms of computation time, and it does not cause loss of essential uncertainty information.

A Novel Modeling Framework for Computationally Efficient and Accurate Real‐Time Ensemble Flood Forecasting With Uncertainty Quantification

AbstractA novel modeling framework that simultaneously improves accuracy, predictability, and computational efficiency is presented. It embraces the benefits of three modeling techniques integrated together for the first time: surrogate modeling, parameter inference, and data assimilation. The use of polynomial chaos expansion (PCE) surrogates significantly decreases computational time. Parameter inference allows for model faster convergence, reduced uncertainty, and superior accuracy of simulated results. Ensemble Kalman filters assimilate errors that occur during forecasting. To examine the applicability and effectiveness of the integrated framework, we developed 18 approaches according to how surrogate models are constructed, what type of parameter distributions are used as model inputs, and whether model parameters are updated during the data assimilation procedure. We conclude that (1) PCE must be built over various forcing and flow conditions, and in contrast to previous studies, it does not need to be rebuilt at each time step; (2) model parameter specification that relies on constrained, posterior information of parameters (so‐called Selected specification) can significantly improve forecasting performance and reduce uncertainty bounds compared to Random specification using prior information of parameters; and (3) no substantial differences in results exist between single and dual ensemble Kalman filters, but the latter better simulates flood peaks. The use of PCE effectively compensates for the computational load added by the parameter inference and data assimilation (up to ~80 times faster). Therefore, the presented approach contributes to a shift in modeling paradigm arguing that complex, high‐fidelity hydrologic and hydraulic models should be increasingly adopted for real‐time and ensemble flood forecasting.

Efficient Model-Assisted Probability of Detection and Sensitivity Analysis for Ultrasonic Testing Simulations Using Stochastic Metamodeling

Abstract Model-assisted probability of detection (MAPOD) and sensitivity analysis (SA) are important for quantifying the inspection capability of nondestructive testing (NDT) systems. To improve the computational efficiency, this work proposes the use of polynomial chaos expansions (PCEs), integrated with least-angle regression (LARS), a basis-adaptive technique, and a hyperbolic truncation scheme, in lieu of the direct use of the physics-based measurement model in the MAPOD and SA calculations. The proposed method is demonstrated on three ultrasonic testing cases and compared with Monte Carlo sampling (MCS) of the physics model, MCS-based kriging, and the ordinary least-squares (OLS)-based PCE method. The results show that the probability of detection (POD) metrics of interests can be controlled within 1% accuracy relative to using the physics model directly. Comparison with metamodels shows that the LARS-based PCE method can provide up to an order of magnitude improvement in the computational efficiency.

Stochastic back-off-based robust process design for continuous crystallization of ibuprofen

Robust model-based process design in continuous pharmaceutical manufacturing aims to implement quality by design principles under uncertainty. Notably, various studies have discussed the back-off concept to solve the underlying robust optimization problem; however, for the concept to have practical value, its efficiency and convergence must be improved. In this work, we introduce a novel, highly efficient stochastic back-off strategy. Instead of using statistical moments of limited validity, we incorporate the full statistical information of the constraints to solve the robust process design problem. To ensure manageable computational costs, we make use of polynomial chaos expansion for uncertainty quantification and propagation. The proposed concept is demonstrated with the design of a tubular crystallizer for ibuprofen crystallization. The results show that the novel stochastic back-off strategy is considerably faster compared with the standard back-off concept and provides more reliable quality by design results in general.

Analysis of spacecraft disposal solutions from LPO to the Moon with high order polynomial expansions

This paper presents the analysis of disposal trajectories from libration point orbits to the Moon under uncertainty. The paper proposes the use of polynomial chaos expansions to quantify the uncertainty in the final conditions given an uncertainty in initial conditions and disposal manoeuvre. The paper will compare the use of polynomial chaos expansions against high order Taylor expansions computed with point-wise integration of the partials of the dynamics, the use of the covariance matrix propagated using a unscented transformation and a standard Monte Carlo simulation. It will be shown that the use of the ellipsoid of uncertainty, that corresponds to the propagation of the covariance matrix with a first order Taylor expansions, is not adequate to correctly capture the dispersion of the trajectories that can intersect the Moon. Furthermore, it will be shown that polynomial chaos expansions better represent the distribution of the final states compared to Taylor expansions of equal order and are comparable to a full scale Monte Carlo simulations but at a fraction of the computational cost.

Using sparse polynomial chaos expansions for the global sensitivity analysis of groundwater lifetime expectancy in a multi-layered hydrogeological model

The study makes use of polynomial chaos expansions to compute Sobol׳ indices within the frame of a global sensitivity analysis of hydro-dispersive parameters in a simplified vertical cross-section of a segment of the subsurface of the Paris Basin. Applying conservative ranges, the uncertainty in 78 input variables is propagated upon the mean lifetime expectancy of water molecules departing from a specific location within a highly confining layer situated in the middle of the model domain. Lifetime expectancy is a hydrogeological performance measure pertinent to safety analysis with respect to subsurface contaminants, such as radionuclides. The sensitivity analysis indicates that the variability in the mean lifetime expectancy can be sufficiently explained by the uncertainty in the petrofacies, i.e. the sets of porosity and hydraulic conductivity, of only a few layers of the model. The obtained results provide guidance regarding the uncertainty modeling in future investigations employing detailed numerical models of the subsurface of the Paris Basin. Moreover, the study demonstrates the high efficiency of sparse polynomial chaos expansions in computing Sobol׳ indices for high-dimensional models.

The Use of the Least-Squares Probabilistic-Collocation Method in Decision Making in the Presence of Uncertainty for Chemical-Enhanced-Oil-Recovery Processes

SummaryThe least-squares Monte Carlo method (LSM) is a decision-evaluation method that can capture the value of flexibility of a process. This method was shown to provide us with some insight into the effect of uncertainty on decision making and to help us capture the upside potential or mitigate the downside effects for a chemical enhanced-oil-recovery (EOR) process. The method is a stochastic approximate dynamic programming approach to decision making. It is modeled after a forward simulation coupled with a recursive algorithm, which produces the near-optimal policy. It relies on Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir-engineering problems because this requires running many reservoir simulations.The objective of this study was to enhance the performance of the LSM by improving the sampling method used to generate the technical uncertainties used in producing the production profiles and to extend its application to different chemical EOR processes. The probabilistic-collocation method has been proved to be a robust and efficient uncertainty-quantification method. It approximates the random input distributions by use of polynomial-chaos expansions and produces a proxy polynomial for the output parameter requiring a limited number of model responses that is conditional on the number of random inputs and the order of the approximation desired. The resulting proxy can then be used to generate the different statistical moments with negligible computational requirement. By use of the sampling methods of the probabilistic-collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision-evaluation method. This is known as the least-squares probabilistic-collocation method (LSPCM).Both methods were demonstrated on a chemical EOR problem by use of a number of stylized reservoir models. The technical uncertainties considered were the residual oil saturation to chemical flooding, surfactant and polymer adsorption, and the viscosity multiplier of the polymer. The economic uncertainties considered were the oil price and the surfactant and polymer price. Both methods were applied by use of three reservoir case studies: a simple homogeneous model, the PUNQ-S3 (2010) model, and a modified portion of the SPE10 model (Christie and Blunt 2001). The results show that the use of the sampling techniques of the probabilistic-collocation method produced relatively accurate responses compared with the original method. For instance, it was possible to produce the same output for the modified SPE10 model by use of the second-order quadrature nodes—81 realizations—rather than the 103 realizations used for the LSM, thus achieving an order-of-magnitude reduction in computational time.The scalability limits of both methods and the different possible enhancements to practically adapt the LSPCM to more-realistic and more-complex reservoir models were discussed. The application was extended to other chemical EOR processes, such as alkaline/surfactant/polymer flooding and polymer flooding, and to more-complex decision problems. The results show that the use of the LSPCM produced accurate results compared with the LSM.

MEMS stochastic model order reduction method based on polynomial chaos expansion

Modeling and simulation of MEMS devices is a very complex task which involve the electrical, mechanical, fluidic and thermal domains, and there are still some uncertainties need to be accounted because of uncertain material and/or geometric parameters factors. According to these problems, we put forward to stochastic model order reduction method under random input conditions to facilitate fast time and frequency domain analyses, the method firstly process model order reduction by Structure Preserving Reduced-order Interconnect Macro Modeling method, then makes use of polynomial chaos expansions in terms of the random input and output variables for the matrices of a finite element model of the system; at last we give the expected values and standard deviations computing method to MEMS stochastic model. The simulation results verify the method is effective in large scale MEMS design process.

On the use of polynomial chaos expansions and generalized moments within the frame of gas radiation in non-uniform media

In many applications involving gaseous media at high temperature, accurate but computationally efficient models are required for the radiative properties of gases. The aim of the present work is to show how, by combining results from Polynomial Chaos framework and Devyatov׳s Method Of Moments, one can provide simple estimates of the transmission functions of gases both in uniform and non-uniform media. The proposed model only involves polynomials. It is probably one of the simplest in terms of mathematical formulation but also one of the most sophisticated, considering the concepts that it involves. It can be applied in uniform and non-uniform media, where it is shown to be equivalent, in terms of accuracy, to usual C–k models.

Postmaneuver Collision Probability Estimation Using Sparse Polynomial Chaos Expansions

This paper describes the use of polynomial chaos expansions to approximate the probability of a collision between two satellites after at least one performs a translation maneuver. Polynomial chaos provides a computationally efficient means to generate an approximate solution to a stochastic differential equation without introducing any assumptions on the a posteriori distribution. The stochastic solution then allows for orbit state uncertainty propagation. For the maneuvering spacecraft in the presented scenarios, the polynomial chaos expansion is sparse, allowing for the use of compressive sampling methods to improve solution tractability. This paper first demonstrates the use of these techniques for possible intraformation collisions for the Magnetospheric Multi-Scale mission. The techniques are then applied to a potential collision with debris in low Earth orbit. Results demonstrate that these polynomial chaos-based methods provide a Monte Carlo-like estimate of the collision probability, including adjustments for a spacecraft shape model, with only minutes of computation cost required for scenarios with a probability of collision as low as . A graphics processing unit implementation of the polynomial chaos expansion analysis further reduces the computation time for the scenarios presented.

A New MEMS Stochastic Model Order Reduction Method: Research and Application

Modeling and simulation of MEMS devices is a very complex tasks which involve the electrical, mechanical, fluidic, and thermal domains, and there are still some uncertainties that need to be accounted for during the robust design of MEMS actuators caused by uncertain material and/or geometric parameters. According to these problems, we put forward stochastic model order reduction method under random input conditions to facilitate fast time and frequency domain analyses; the method makes use of polynomial chaos expansions in terms of the random input variables for the matrices of a finite element model of the system and then uses its transformation matrix to reduce the model; the method is independent of the MOR algorithm, so it is seamlessly compatible with MOR method used in popular finite element solvers. The simulation results verify the method is effective in large scale MEMS design process.

Structural Identification and Monitoring based on Uncertain/Limited Information

The goal of the present study is to propose a structural identification framework able to exploit both vibrational response and operational condition information in extracting structural models, able to represent the systemspecific structural behavior in its complete operational spectrum. In doing so, a scheme need be derived for the extraction of salient features, which are indicative of structural condition. Such a scheme should account for variations attributed to operational effects, such as environmental and operational load variations, and which likely lie within regular structural condition bounds, versus variations which indicate short- or long-term damage effects. The latter may be achieved via coupling of sparse, yet diverse, monitoring information with appropriate stochastic tools, able to infer the underlying dependences between the monitored input and output data. This in turn allows for extraction of quantities, or features, relating to structural condition, which may further be utilized as performance indicators. The computational tool developed herein for realizing such a framework, termed the PCE-ICA scheme, is based on the use of Polynomial Chaos Expansion (PCE) tool, along with an Independent Component Analysis (ICA) algorithm. The benefits of additionally fusing a data-driven system model will further be discussed for the case of complex structural response. The method is assessed via implementation on field data acquired from diverse structural systems, namely a benchmark bridge case study and a wind turbine tower structure, revealing a robust condition assessment tool.

Probabilistic Assessment of Lifetime of Low-Earth-Orbit Spacecraft: Uncertainty Propagation and Sensitivity Analysis

This paper is devoted to the probabilistic uncertainty quantification of orbital lifetime estimation of low-altitude satellites. Specifically, given a detailed characterization of the dominant sources of uncertainty, we map this input into a probabilistic characterization of the orbital lifetime through orbital propagation. Standard Monte Carlo propagation is first considered. The concept of drag correction is then introduced to facilitate the use of polynomial chaos expansions and to make uncertainty propagation computationally effective. Finally, the obtained probabilistic model is exploited to carry out stochastic sensitivity analyses, which in turn allow gaining insight into the impact uncertainties have on orbital lifetime. The proposed developments are illustrated using one CubeSat of the QB50 constellation.

Computing derivative-based global sensitivity measures using polynomial chaos expansions

In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol׳ indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol׳ indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSMs) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. Closed-form expressions for Hermite, Legendre and Laguerre polynomial expansions are given. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis.

Influence of Polynomial Chaos expansion order on an uncertain asymmetric rotor system response

A stochastic harmonic balance method with a recursive procedure is developed to evaluate the steady-state response of a rotor system with uncertain stiffness and asymmetric coupling that involves time-dependent terms. The Polynomial Chaos (PC) expansion is proposed to evaluate the mean and the standard deviation of the responses and the harmonic amplitudes of orders 1–4 involving 288 degrees-of-freedom. To determine the coefficients of the expansion requires to solve a large system of equations. To achieve this and to avoid numerical problems related to the size of the system, the first contribution of the study consists of a recursive evaluation of the Polynomial Chaos coefficients to be able to estimate the stochastic response for a high order of the Polynomial Chaos expansion.From the methodology implemented, the steady-state responses and n× harmonic components (for n=1, 2, 3 and 4 in the present study) of the asymmetric rotor with uncertainty are evaluated for several PC orders. Then, the second main contribution focuses on a clarification and analysis of the use of Polynomial Chaos expansion around the critical speeds. First, it is observed that the convergence is slow: the response obtained with 30 PCs is twice the one obtained with 200 PCs. Second, it is noted that the parity of the PC order has a strong influence on the response level: whereas two responses obtained with two consecutive even PC orders (or two consecutive odd PC orders) are almost the same for a given rotation speed, the ratio of the responses evaluated with two consecutive PC orders (one even order and one odd order) may be large (e.g. oscillations between two consecutive PC orders greater than 10 are noticeable if the PC order is about 30).

Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos

This paper demonstrates the use of polynomial chaos expansions for the nonlinear, non-Gaussian propagation of orbit state uncertainty. Using linear expansions in tensor products of univariate orthogonal polynomial bases, polynomial chaos expansions approximate the stochastic solution of the ordinary differential equation describing the propagated orbit, and include information on covariance, higher moments, and the spatial density of possible solutions. Results presented in this paper use non-intrusive, i.e., sampling-based, methods in combination with either least-squares regression or pseudospectral collocation to estimate the polynomial chaos expansion coefficients at any future point in time. Such methods allow for the usage of existing orbit propagators. Samples based on sun-synchronous and Molniya orbit scenarios are propagated for up to ten days using two-body and higher-fidelity force models. Tests demonstrate that the presented methods require the propagation of orders of magnitude fewer samples than Monte Carlo techniques, and provide an approximation of the a posteriori probability density function that achieves the desired accuracy. Results also show that Poincaré-based polynomial chaos expansions require fewer samples to achieve a given accuracy than Cartesian-based solutions. In terms of probability density function accuracy, the polynomial chaos expansion-based solutions represent an improvement over the linear propagation and unscented transformation techniques.