Arbitrary Polynomial Chaos Expansion Research Articles

Polynomial chaos expansion of SAR and temperature increase variability in 3 T MRI due to stochastic input data.

Numerical simulations are largely adopted to estimate dosimetric quantities, e.g., specific absorption rate (SAR) and temperature increase, in tissues to assess the patient exposure to the radiofrequency field generated during magnetic resonance imaging (MRI). Simulations rely on reference anatomical human models and tabulated data of electromagnetic and thermal properties of biological tissues. However, concerns may arise about the applicability of the computed results to any phenotype, introducing a significant degree of freedom in the simulation input data. In addition, simulation input data can be affected by uncertainty in relative positioning of the anatomical model with respect to the radiofrequency coil. The objective of this work is the to estimate the variability of SAR and temperature increase at 3 T head MRI due to different sources of variability in input data, with the final aim to associate a global uncertainty to the dosimetric outcomes. A stochastic approach based on arbitrary Polynomial Chaos Expansion is used to evaluate the effects of several input variabilities (anatomy, tissue properties, body position) on dosimetric outputs, referring to head imaging with a 3 T MRI scanner. It is found that head anatomy is the prevailing source of variability for the considered dosimetric quantities, rather than the variability due to tissue properties and head positioning. From knowledge of the variability of the dosimetric quantities, an uncertainty can be attributed to the results obtained using a generic anatomical head model when SAR and temperature increase values are compared with safety exposure limits. This work associates a global uncertainty to SAR and temperature increase predictions, to be considered when comparing the numerically evaluated dosimetric quantities with reference exposure limits. The adopted methodology can be extended to other exposure scenarios for MRI safety purposes.

Improved hierarchical Bayesian modeling framework with arbitrary polynomial chaos for probabilistic model updating

Bayesian finite element model updating techniques have found widespread application in the structural health monitoring. The conventional Bayesian modeling framework (CBMF) identifies the posterior distribution of structural parameters using data from a single experiment. However, structural parameters exhibit variability due to changes in environmental or experimental conditions, an aspect overlooked by CBMF. In recent years, the hierarchical Bayesian modeling framework (HBMF) has gradually attracted attention. This framework incorporates an additional layer of hyperparameters and utilizes multiple sets of experimental data to describe the variability in structural parameters caused by changes in experimental conditions. Unfortunately, the increase in hyperparameters and data size leads the HBMF to face the dilemma of exceedingly high computational costs. To address this issue, this study proposes an improved hierarchical Bayesian modeling framework (IHBMF). To enhance the computational efficiency of IHBMF, the importance sampling method is employed to simplify the computation of the likelihood function for hyperparameters. Subsequently, a surrogate model based on arbitrary polynomial chaos expansion is introduced to further reduce the computational cost of IHBMF. The framework's accuracy and efficiency were validated through analysis of a simply supported beam and a steel pedestrian bridge. The results demonstrate that IHBMF not only delivers a more accurate quantification of structural parameter uncertainty than CBMF but also exhibits higher computational efficiency compared to HBMF, showcasing its superior capability in structural health monitoring applications.

Sequential simulated annealing for life-cycle optimization of nonlinear stochastic systems via arbitrary polynomial chaos expansion

Quantifying the uncertainties of engineering systems modeled as nonlinear oscillators subject to random excitation is a theoretically complex and computationally demanding task. Consequently, finding an optimal design of such systems considering their life-cycle performance is prohibitive with Monte Carlo simulation methods. In this paper, an efficient performance-based design optimization approach is developed for finding the optimal parameters of engineering systems modeled as nonlinear/hysteretic oscillators subject to stationary and non-stationary excitation. This novel approach utilizes an arbitrary polynomial chaos expansion to estimate the expected cost of failure without performing a computationally expensive integration over the hazard levels. Moreover, we introduce a novel sequential heuristic optimization scheme based on simulated annealing to minimize the total expected cost over the structure life-cycle. Three examples are included in the paper to assess the developed optimization scheme. First, we use the developed framework to optimize a linear single-degree-of-freedom oscillator subject to broadband excitation. Second, a multi-degree-of-freedom oscillator with cubic nonlinearity in damping and stiffness, subject to stationary broadband excitation, is optimized to show the influence of the problem dimensionality in the optimization process. In the last example, a multi-story reinforced concrete shear building modeled as a multi-degree-of-freedom Bouc-Wen oscillator with stiffness and strength degradation and subject to multi-hazards modeled as stationary (wind excitation) and non-stationary (earthquake) stochastic processes, is optimized.

Contaminant source identification in an aquifer using a Bayesian framework with arbitrary polynomial chaos expansion

Contaminant source identification in an aquifer using a Bayesian framework with arbitrary polynomial chaos expansion

Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes–Darcy flow problems

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy’s law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations.

Aerodynamic evaluation of cascade flow with actual geometric uncertainties using an adaptive sparse arbitrary polynomial chaos expansion

In this paper, an adaptive sparse arbitrary polynomial chaos expansion (PCE) is first proposed to quantify the performance impact of realistic multi-dimensional manufacturing uncertainties. The Stieltjes algorithm is employed to generate the PCE basis functions concerning geometric variations with arbitrary distributions. The basis-adaptive Bayesian compressive sensing algorithm is introduced to retain a small number of significant PCE basis functions, requiring fewer model training samples while preserving fitting accuracy. Second, several benchmark tests are used to verify the computational efficiency and accuracy of the proposed method. Eventually, the coexistence effects of six typical machining deviations on the aerodynamic performance and flow fields of a controlled diffusion compressor cascade are investigated. The probability distributions of the machining deviations are approximated by limited measurement data using kernel density estimation. By uncertainty quantification, it can be learned that the mean performance seriously deteriorates with increasing incidences, while the performance at negative incidences is more dispersed. By global sensitivity analysis, the leading-edge profile error should be given high priority when working at negative incidences, and the inlet metal angle error would be carefully inspected first when the cascade works at high positive incidences. Furthermore, controlling the manufacturing accuracy of the suction surface profile error can play a certain role in improving the robustness of aerodynamic performance in off-design conditions. Through flow field analysis, it further proves that actual leading-edge errors are the most important ones to aerodynamics and reveals how the effects of leading-edge errors propagate in the cascade passage, thus affecting the aerodynamic loss.

Real-Time Hybrid Simulation Using Nonlinear Autoregressive with Exogenous Input Model through Data-Driven Arbitrary Polynomial Chaos Surrogate

The current practice of real-time hybrid simulation (RTHS) often requires specialized finite element programs for computational modeling of the analytical substructures. Considering the limited nonlinear modeling capacity or the increasing computation cost for complex modeling, surrogate models of the analytical substructure provide novel alternatives for RTHS to avoid finite element analysis with fast computation. This study explores the use of arbitrary polynomial chaos expansion (APC) and nonlinear autoregressive with exogenous input (NARX) model to emulate the dynamic behavior of analytical substructures in RTHS. The NARX model training can be conducted numerically in an off-line mode using existing general purpose finite element analysis software, and its implementation presents minimum computational demands on the RTHS equipment. RTHS of a single-degree-of-freedom structure with a self-centering viscous damper is conducted as proof of concept to experimentally demonstrate the effectiveness and superiority of the proposed APC-NARX-based approach. The APC is further compared with other metamodeling techniques including polynomial chaos expansion (PCE) and Kriging to surrogate NARX model coefficients to account for ground motion uncertainties in RTHS. It is demonstrated that APC-NARX modeling with optimal order enables better accuracy of RTHS results than those of Kriging- and PCE-NARX.

Arbitrary polynomial chaos expansion for uncertainty analysis of the one-dimensional hindered-compression continuous settling model

Secondary settling tank (SST) models play a significant role in the simulation of a wastewater treatment system. They can estimate effluent and underflow quality and thus help with the design, management, and optimization of wastewater treatment systems. SST modeling consists of an empirical settling velocity function where parameter uncertainty could raise. The performance of an SST model could suffer from parameter uncertainty, which makes parameter uncertainty assessment valuable for SST modeling. Monte Carlo simulation (MCS) is a classical technique for assessing uncertainty, but it requires parameter distribution information and is computationally expensive. To overcome these limitations, arbitrary polynomial chaos expansion (aPCE), a novel approach has been adopted for the first time in this study. The well-recognized Bürger-Diehl SST model is used and the uncertainties originating from five essential model parameters are assessed by the novel aPCE method with the MCS technique being used as a benchmark. Probabilistic estimations of the model output, i.e., sludge blanket height (SBH), are generated by both aPCE and MCS. The comparison results between aPCE and MCS suggest that the aPCE approach can be as effective as MCS in quantifying the uncertainties associated with SST model parameters, while significantly reducing approximately 90 % computational requirements. This study explicitly quantifies the uncertainties associated with SST model parameters in an efficient manner, which can provide robust support for the design, management, and optimization of wastewater treatment systems.

Uncertainty quantification of multi-scale resilience in networked systems with nonlinear dynamics using arbitrary polynomial chaos

Complex systems derive sophisticated behavioral dynamics by connecting individual component dynamics via a complex network. The resilience of complex systems is a critical ability to regain desirable behavior after perturbations. In the past years, our understanding of large-scale networked resilience is largely confined to proprietary agent-based simulations or topological analysis of graphs. However, we know the dynamics and topology both matter and the impact of model uncertainty of the system remains unsolved, especially on individual nodes. In order to quantify the effect of uncertainty on resilience across the network resolutions (from macro-scale network statistics to individual node dynamics), we employ an arbitrary polynomial chaos (aPC) expansion method to identify the probability of a node in losing its resilience and how the different model parameters contribute to this risk on a single node. We test this using both a generic networked bi-stable system and also established ecological and work force commuter network dynamics to demonstrate applicability. This framework will aid practitioners to both understand macro-scale behavior and make micro-scale interventions.

Stochastic analysis of structures under limited observations using kernel density estimation and arbitrary polynomial chaos expansion

Over the past decades there has been considerable interest among the scientific community in modeling random system inputs from limited observations and propagating the system responses. In this paper, we develop a novel method for the reasonable modeling of system inputs and efficient propagation of response. Our method firstly constructs the random model of system inputs from observations by developing a novel kernel density estimator (KDE) for Karhunen–Loeve (KL) variables. By further implementing the arbitrary polynomial chaos (aPC) formulation on dependent non-Gaussian KL variables, the associated aPC-based response propagation is then developed. In our method, the developed random model can accurately represent the input parameters from limited observations as the developed KDE of KL variables can incorporate the inherent relation between marginals of input parameters and KL variables, while empowering the input model to preserve the second-order correlations. Furthermore, since the statistical moments of KL variables can be analytically evaluated, the aPC formulation for achieving optimal convergence of system responses can be accurately determined. In addition, the system response can be accurately propagated in an efficient way by developing an aPC-based regression method using space-filling design, in which the conditional distributions in Rosenblatt transformation of aPC variables can be efficiently determined without evaluations of multi-dimensional integrations. In this way, the current work provides an effective framework for the reasonable stochastic modeling​ and efficient response propagation of real-life engineering systems with limited observations. Two numerical examples are presented to highlight the effectiveness of the developed method.

Deep adaptive arbitrary polynomial chaos expansion: A mini-data-driven semi-supervised method for uncertainty quantification

All kinds of uncertainties influence the reliability of the engineering system. Thus, uncertainty quantification is significant to the system reliability analysis. Polynomial chaos expansion (PCE) is an effective method for uncertainty quantification while it requires sufficient labeled data to quantify uncertainty accurately. To overcome this problem, this paper proposes the adaptive arbitrary polynomial chaos (aPC) and proves two properties of the adaptive expansion coefficients. Sequentially, a semi-supervised deep adaptive arbitrary polynomial chaos expansion (Deep aPCE) method is proposed based on the adaptive aPC and the deep neural network (DNN). The Deep aPCE method uses two properties of the adaptive aPC to assist in training the DNN by a small amount of labeled data and abundant unlabeled data, significantly reducing the training data cost. On the other hand, the Deep aPCE method adopts the DNN to fine-tune the adaptive expansion coefficients dynamically, improving the accuracy of uncertainty quantification. Besides, the Deep aPCE method can directly construct accurate surrogate models of the high dimensional stochastic systems. Five numerical examples are used to verify the effectiveness of the Deep aPCE method. Finally, the Deep aPCE method is applied to the reliability analysis of an axisymmetric conical aircraft.

Probabilistic Power Flow Analysis Based on Partial Least Square and Arbitrary Polynomial Chaos Expansion

This paper presents a new algorithm based on the partial least square (PLS) techniques and the arbitrary polynomial chaos expansion (aPCE) for the probabilistic power flow (PPF) analysis of a power system having many uncertain variables. The proposed method uses the nonlinear PLS to transform a set of random input variables to a smaller number of de-correlated random variables. Then, the aPCE technique is applied to generate the basis polynomial functions and build a surrogate model of the power system response. The algorithm has been implemented and tested with the modified IEEE 118-bus and European 1354-bus systems. The numerical results indicate that, similar to the sparse PCE method and the low-rank approximation technique, the proposed method can be applied to high-dimensional PPF problems. Nonetheless, the proposed approach uses smaller computation time, and estimates statistical characteristics of the response with higher accuracy.

Arbitrary multi-resolution multi-wavelet-based polynomial chaos expansion for data-driven uncertainty quantification

Various real world problems deal with data-driven uncertainty. In particular, in geophysical applications the amount of available data is often limited, posing a challenge in the construction of an appropriate stochastic discretization. Arbitrary polynomial chaos is an alternative to tackle this challenge. Approximating the dependence of model output on the uncertain model parameters by expansion in an orthogonal polynomial basis using data-driven principles. This type of global polynomial representation suffers often from Gibbs’ phenomena, especially if applied in non-linear convection dominated problems that require to deal with discontinuities. The multi-resolution or multi-element framework has been successfully used for reducing Gibbs’ phenomena in intrusive stochastic discretizations. In the present work, we introduce a multi-resolution extension of the arbitrary polynomial chaos expansion which is based on the construction of piecewise polynomial. Gaussian quadrature nodes and weights that are computed using only stochastic (localized) moments provided by the underlying raw data. We enhance our approach by a multi-wavelet based stochastic adaptivity that assures a significant reduction of the computational costs. Numerical experiments of increasing complexity demonstrate the performance of the non-intrusive implementation of the introduced methods in relevant scenarios. The use of a carbon dioxide storage benchmark scenario allows one to compare the presented methodology with other stochastic discretization techniques applied to this benchmark.

Uncertainty Quantification of Multi-Scale Resilience in Nonlinear Complex Networks Using Arbitrary Polynomial Chaos

Resilience characterizes a system's ability to retain its original function when perturbations happen. In the past years our attention mainly focused on small-scale resilience, yet our understanding of resilience in large-scale network considering interactions between components is limited. Even though, recent research in macro and micro resilience pattern has developed analytical tools to analyze the relationship between topology and dynamics across network scales. The effect of uncertainty in a large-scale networked system is not clear, especially when uncertainties cascade between connected nodes. In order to quantify resilience uncertainty across the network resolutions (macro to micro),an arbitrary polynomial chaos (aPC) expansion method is developed in this paper to estimate the resilience subject to parameter uncertainties with arbitrary distributions. For the first time and of particular importance, is our ability to identify the probability of a node in losing its resilience and how the different model parameters contribute to this risk. We test this using a generic networked bi-stable system and this will aid practitioners to both understand macro-scale behaviour and make micro-scale interventions.

Assessing uncertainty propagation in hybrid models for daily streamflow simulation based on arbitrary polynomial chaos expansion

Accurate streamflow prediction is of significant importance in watershed management and has been attracting a lot of research interest. Hybrid hydrological models that employ advantages of physically-based models (PBMs) and data-driven models (DDMs) emerge recently as reliable tools for analyzing hydrological processes. Both PBMs and DDMs have uncertain parameters that could affect the hybrid modeling results. Conventional computational algorithms that rely on repeated sampling, such as Monte Carlo Simulation (MCS), have been recognized as robust tools for assessing the propagation of parameter uncertainties. However, the efficiency of these computational algorithms can decrease exponentially as the number of parameters increases. Meanwhile, although parameter uncertainty of PBMs has been widely studied, that of DDMs in a hybrid modeling framework has not been well investigated. In this study, a hybrid model framework is proposed for streamflow prediction, and an effective arbitrary polynomial chaos expansion (aPCE) method is implemented to assess the propagation of DDM parameter uncertainty within the hybrid modeling framework. The hybrid hydrological model integrating fully-distributed MIKE SHE and four machine-learning techniques increases the coefficient of determination (R2) from 0.70 to 0.82 for two validation periods. The aPCE method generates highly similar probabilistic results as MCS and achieves a 90% efficiency enhancement in uncertainty assessment. The results prove that aPCE is an efficient alternative to MCS for assessing uncertainties of hyperparameters in hybrid models. This study helps improve the performance of the streamflow prediction and also provide an insight into the uncertainty propagation involved in the hybrid models.

A Novel Sparse Polynomial Expansion Method for Interval and Random Response Analysis of Uncertain Vibro-Acoustic System

For the vibro-acoustic system with interval and random uncertainties, polynomial chaos expansions have received broad and persistent attention. Nevertheless, the cost of the computation process increases sharply with the increasing number of uncertain parameters. This study presents a novel interval and random polynomial expansion method, called Sparse Grids’ Sequential Sampling-based Interval and Random Arbitrary Polynomial Chaos (SGS-IRAPC) method, to obtain the response of a vibro-acoustic system with interval and random uncertainties. The proposed SGS-IRAPC retains the accuracy and the simplicity of the traditional arbitrary polynomial chaos method, while avoiding its inefficiency. In the SGS-IRAPC, the response is approximated by the moment-based arbitrary polynomial chaos expansion and the expansion coefficient is determined by the least squares approximation method. A new sparse sampling scheme combined the sparse grids’ scheme with the sequential sampling scheme which is employed to generate the sampling points used to calculate the expansion coefficient to decrease the computational cost. The efficiency of the proposed surrogate method is demonstrated using a typical mathematical problem and an engineering application.

Dynamic graph and polynomial chaos based models for contact tracing data analysis and optimal testing prescription

In this study, we address three important challenges related to disease transmissions such as the COVID-19 pandemic, namely, (a) providing an early warning to likely exposed individuals, (b) identifying individuals who are asymptomatic, and (c) prescription of optimal testing when testing capacity is limited. First, we present a dynamic-graph based SEIR epidemiological model in order to describe the dynamics of the disease propagation. Our model considers a dynamic graph/network that accounts for the interactions between individuals over time, such as the ones obtained by manual or automated contact tracing, and uses a diffusion–reaction mechanism to describe the state dynamics. This dynamic graph model helps identify likely exposed/infected individuals to whom we can provide early warnings, even before they display any symptoms and/or are asymptomatic. Moreover, when the testing capacity is limited compared to the population size, reliable estimation of individual’s health state and disease transmissibility using epidemiological models is extremely challenging. Thus, estimation of state uncertainty is paramount for both eminent risk assessment, as well as for closing the tracing-testing loop by optimal testing prescription. Therefore, we propose the use of arbitrary Polynomial Chaos Expansion, a popular technique used for uncertainty quantification, to represent the states, and quantify the uncertainties in the dynamic model. This design enables us to assign uncertainty of the state of each individual, and consequently optimize the testing as to reduce the overall uncertainty given a constrained testing budget. These tools can also be used to optimize vaccine distribution to curb the disease spread when limited vaccines are available. We present a few simulation results that illustrate the performance of the proposed framework, and estimate the impact of incomplete contact tracing data.

Surrogate-based Bayesian comparison of computationally expensive models: application to microbially induced calcite precipitation

Geochemical processes in subsurface reservoirs affected by microbial activity change the material properties of porous media. This is a complex biogeochemical process in subsurface reservoirs that currently contains strong conceptual uncertainty. This means, several modeling approaches describing the biogeochemical process are plausible and modelers face the uncertainty of choosing the most appropriate one. The considered models differ in the underlying hypotheses about the process structure. Once observation data become available, a rigorous Bayesian model selection accompanied by a Bayesian model justifiability analysis could be employed to choose the most appropriate model, i.e. the one that describes the underlying physical processes best in the light of the available data. However, biogeochemical modeling is computationally very demanding because it conceptualizes different phases, biomass dynamics, geochemistry, precipitation and dissolution in porous media. Therefore, the Bayesian framework cannot be based directly on the full computational models as this would require too many expensive model evaluations. To circumvent this problem, we suggest to perform both Bayesian model selection and justifiability analysis after constructing surrogates for the competing biogeochemical models. Here, we will use the arbitrary polynomial chaos expansion. Considering that surrogate representations are only approximations of the analyzed original models, we account for the approximation error in the Bayesian analysis by introducing novel correction factors for the resulting model weights. Thereby, we extend the Bayesian model justifiability analysis and assess model similarities for computationally expensive models. We demonstrate the method on a representative scenario for microbially induced calcite precipitation in a porous medium. Our extension of the justifiability analysis provides a suitable approach for the comparison of computationally demanding models and gives an insight on the necessary amount of data for a reliable model performance.

Arbitrary polynomial chaos expansion and its application to power flow analysis-Fast approximation of probability distribution by arbitrary polynomial expansion

This paper introduces an arbitrary polynomial chaos expansion method for performing probabilistic power flow analysis in power systems. The proposed method is used for uncertainty analysis, expressing the uncertainty of a system as random variables with an arbitrary output distribution based on orthogonal polynomial expansion. This method is advantageous because of its calculation speed and accuracy. This study expresses probabilistic power flow in a power system with many uncertain power sources using linear combination polynomial expansion. The orthogonal polynomial system employed is generated by moment analysis from renewable energy output data, with the polynomial coefficients derived from a collocation method. Simulation of probabilistic power flow using the proposed method is applied to a 29-bus transmission network model including three renewable energies, and the calculation speed and accuracy are evaluated by changing the expansion order of the polynomial. In addition, the influence on the polynomial coefficient is assessed when the system topology is changed due to a line fault. Therefore, since the arbitrary polynomial chaos expansion method can represent complex networks by linear combination of orthogonal polynomial sets, calculation based on it is several hundred times faster than the conventional Monte Carlo method. The results demonstrate that the proposed method is very useful for analyzing the probabilistic power distribution and that third-order expansion is practically appropriate.

Global sensitivity analysis of a CaO/Ca(OH)2 thermochemical energy storage model for parametric effect analysis

Simulation models have been widely used for thermochemical energy storage to better understand its behavior and consequently to improve operational control of the device. However, incomplete knowledge of system properties leads to a significant number of uncertain parameters in the simulation models, which in turn cause uncertainties in system predictions. In this work, we perform global sensitivity analysis to identify the effect of uncertain parameters on the outputs of a thermochemical energy storage model, so that we can better understand the predictive uncertainties, proceed with targeted data acquisition or even simplify the corresponding uncertainty quantification. To get reliable sensitivity analysis results, we use both variance-based and regional sensitivity analysis since they focus on different probabilistic features of model outputs. Since the simulation model is computationally expensive, we use model reduction via the (arbitrary) polynomial chaos expansion. Then, to further confirm the results, the regional sensitivity index is also estimated based on the original model with the same given sample set. Based on the results of both sensitivity analysis methods, we can find that there are 8 unimportant parameters among 16 analyzed parameters. Thus, we can focus resources on investigating the important uncertain parameters. Also, we can ignore the uncertainty of unimportant parameters to simplify the corresponding uncertainty quantification.