Uncertainty Quantification Approach Research Articles

A multivariate interval approach for inverse uncertainty quantification with limited experimental data

This paper introduces an improved version of a novel inverse approach for the quantification of multivariate interval uncertainty for high dimensional models under scarce data availability. Furthermore, a conceptual and practical comparison of the method with the well-established probabilistic framework of Bayesian model updating via Transitional Markov Chain Monte Carlo is presented in the context of the DLR-AIRMOD test structure. First, it is shown that the proposed improvements of the inverse method alleviate the curse of dimensionality of the method with a factor up to 105. Furthermore, the comparison with the Bayesian results revealed that the selection ofthe most appropriate method depends largely on the desired information and availability of data. In case large amounts of data are available, and/or the analyst desires full (joint)-probabilistic descriptors of the model parameter uncertainty, the Bayesian method is shown to be the most performing. On the other hand however, when such descriptors are not needed (e.g., for worst-case analysis), and only scarce data are available, the interval method is shown to deliver more objective and robust bounds on the uncertain parameters. Finally, also suggestions to aid the analyst in selecting the most appropriate method for inverse uncertainty quantification are given.

Glycemic Control of People With Type 1 Diabetes Based on Probabilistic Constraints.

The objective of the paper is to develop an open loop insulin infusion profile, which is capable of controlling the blood glucose level of people with Type 1 diabetes in the presence of broad uncertainties such as inter-patient variability and unknown meal quantity. For illustrative purposes, the Bergman model in conjunction with a gut-dynamics model is chosen to represent the human glucose-insulin dynamics. A recently developed sampling based uncertainty quantification approach is used to determine the statistics (mean and variance) of the evolving states in the model. These statistics are utilized to define chance constraints in an optimization framework. The solution obtained shows that under the assumptions made on the distribution of the model parameters, all possible glucose trajectories over time satisfy the desired glycemic control goals. The solution is also validated on the FDA approved Type 1 Diabetes Metabolic Simulator suggesting that the proposed algorithm is highly suitable for human subjects.

Ensemble-based estimates of eigenvector error for empirical covariance matrices

Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda _i\}$ and eigenvectors $\{\boldsymbol{u}_i\}$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda }_i\}$ and eigenvectors $\{\tilde{\boldsymbol{u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $p\to \infty $ to a spectral density $\rho (\lambda )$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda $ and approximate the distribution of the expected square error $r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$ across the matrix ensemble for all $\boldsymbol{u}_i$ associated with $\lambda _i=\lambda $. We find, for example, that for sufficiently large matrix size $p$ and sample size $n> p$, the probability density of $r$ scales as $1/nr^2$. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if $r$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.

Application of Bound-to-Bound Data Collaboration approach for development and uncertainty quantification of a reduced char combustion model

The development of efficient industrial oxy-coal boilers can be significantly aided by Computational Fluid Dynamics (CFD) tools, as far as fidelity in modeling coal combustion is also complemented by feasible computational costs. Reduced and predictive models are the most suitable for this application scale. Reduced models feature predictivity when they are validated against a broad range of experiments and targeted by Uncertainty Quantification (UQ) procedures. This work proposes a numerical procedure that uses Bound-to-Bound Data Collaboration (B2B-DC) to derive a reduced char combustion model describing transport phenomena and reactions between char carbon and O2, CO2 and H2O, in both conventional and oxy-conditions. The approach determines the consistency between a numerical model and an experimental dataset. The latter is made up of the experiments carried out in an optically accessible laminar entrained flow reactor, operated by Sandia National Laboratories. The procedure follows five steps towards predictive modeling capability, namely: quantification of the uncertainty in the experiments, via instrument verification and modeling; development of a physics model and continuous improvement of its fidelity, via model-form uncertainty; identification of the uncertain and most sensitive parameters and of their prior bounds; sampling of the initial uncertain parameter space and training of a surrogate model; validation of the physics model via inference from the data. The last step, also known as inverse problem, is performed by applying the Bound-to-Bound Data Collaboration approach. A char combustion model is found consistent with the experimental data and its validity stands for conventional and oxy-combustion conditions. It accounts for heterogeneous reactions at the particle surface, mass transport of species in the particle boundary layer, pore diffusion and surface area changes. The consistent reduced model overcomes the differences in mass transport and kinetics observed in the experimental campaign. A reduction of the initial degree of uncertainty in both model and experiments is achieved.

Inverse uncertainty quantification using the modular Bayesian approach based on Gaussian process, Part 1: Theory

In nuclear reactor system design and safety analysis, the Best Estimate plus Uncertainty (BEPU) methodology requires that computer model output uncertainties must be quantified in order to prove that the investigated design stays within acceptance criteria. "Expert opinion" and "user self-evaluation" have been widely used to specify computer model input uncertainties in previous uncertainty, sensitivity and validation studies. Inverse Uncertainty Quantification (UQ) is the process to inversely quantify input uncertainties based on experimental data in order to more precisely quantify such ad-hoc specifications of the input uncertainty information. In this paper, we used Bayesian analysis to establish the inverse UQ formulation, with systematic and rigorously derived metamodels constructed by Gaussian Process (GP). Due to incomplete or inaccurate underlying physics, as well as numerical approximation errors, computer models always have discrepancy/bias in representing the realities, which can cause over-fitting if neglected in the inverse UQ process. The model discrepancy term is accounted for in our formulation through the "model updating equation". We provided a detailed introduction and comparison of the full and modular Bayesian approaches for inverse UQ, as well as pointed out their limitations when extrapolated to the validation/prediction domain. Finally, we proposed an improved modular Bayesian approach that can avoid extrapolating the model discrepancy that is learnt from the inverse UQ domain to the validation/prediction domain.

Configuration-Sampling-Based Surrogate Models for Rapid Parameterization of Non-Bonded Interactions.

In this study, we present an approach for rapid force field parameterization and uncertainty quantification of the non-bonded interaction parameters for classical force fields. The accuracy of most thermophysical properties, and especially vapor-liquid equilibria (VLE), obtained from molecular simulation depends strongly on the non-bonded interactions. Traditionally, non-bonded interactions are parameterized to agree with macroscopic properties by performing large amounts of direct molecular simulation. Due to the computational cost of molecular simulation, surrogate models (i.e., efficient models that approximate direct molecular simulation results) are an essential tool for high-dimensional parameterization and uncertainty quantification of non-bonded interactions. The present study compares two different configuration-sampling-based surrogate models, namely, Multistate Bennett Acceptance Ratio (MBAR) and Pair Correlation Function Rescaling (PCFR). MBAR and PCFR are coupled with the Isothermal Isochoric (ITIC) thermodynamic integration method for estimating vapor-liquid saturation properties. We find that MBAR and PCFR are complementary in their roles. Specifically, PCFR is preferred when exploring distant regions of the parameter space while MBAR is better in the local domain.

A multiobjective Markov chain Monte Carlo approach for history matching and uncertainty quantification

We present a probabilistic approach for integrating multiple data types into subsurface flow models. Our approach is based on a Bayesian framework whereby we exhaustively sample the multi-dimensional posterior distribution to define a Pareto front which represents the trade-off between multiple objectives during history matching. These objectives can be matching of water-cut, GOR, BHP and time-lapse seismic data. For field applications, these objectives do not necessarily move in tandem because of measurement errors and also interpretative nature of the seismic data.Our proposed method is built on a Differential Evolution Markov Chain Monte Carlo (DEMC) algorithm in which multiple Markov Chains are run in parallel. First, a dominance relationship is established amongst multiple models. This is followed by construction of the posterior distribution based on a hypervolume measure. A unique aspect of our method is in the parameter proposal generation which is based on a random walk on two arbitrarily selected chains. This promotes effective mixing of the chains resulting in improved convergence.We illustrate the algorithm using a nine-spot waterflood model whereby we use water-cut and bottomhole flowing pressure data to calibrate the permeability field. The permeability field is re-parameterized using a previously proposed Grid Connectivity Transform (GCT) which is a model order reduction technique defined based only on the decomposition of the grid Laplacian. The compression power of GCT allows us to reconstruct the permeability field with few parameters, thus significantly improving the computational efficiency of the McMC approach. Next, we applied the method to the Brugge benchmark case involving 10 water injectors and 20 producers. For both cases, the algorithm provides an ensemble of models all constrained to the history data and defines a probabilistic Pareto front in the objective space. Several experimental runs were conducted to compare the effectiveness of the algorithm with Non-Dominated Sorting Genetic Algorithms (NSGA-II). Higher hypervolume was constantly measured using our algorithm which indicates that more optimal solutions were sampled. Our method provides a novel approach for subsurface model calibration and uncertainty quantification using McMC in which the communication between parallel Markov chains enhances adequate mixing. This significantly improves the convergence without loss in sampling quality.

Review of Robust Aerodynamic Design Optimization for Air Vehicles

The ever-increasing demands for risk-free, resource-efficient and environment-friendly air vehicles motivate the development of advanced design methodology. As a particularly promising design methodology considering uncertainties, robust aerodynamic design optimization (RADO) is capable of providing robust and reliable aerodynamic configuration and reducing cost under probable uncertainties in the flight envelop and all life cycle of air vehicle. However, the major challenges including high computational cost with increasing dimensionality of uncertainty and complex RADO procedure hinder the wider application of RADO. In this paper, the complete RADO procedure, i.e., uncertainty modeling, establishment of uncertainty quantification approach as well as robust optimization subject to reliability constraints under uncertainty, is elaborated. Systematic reviews of RADO methodology including uncertainty modeling methods, comprehensive uncertainty quantification approaches, and robust optimization methods are provided. Further, this paper presents a brief survey of the main applications of RADO in the aerodynamic design of transonic flow and natural-laminar-flow, and discusses the application prospects of RADO methodology for air vehicles. The detailed statement of the paper indicates the intention, i.e., to present the state of the art in RADO methodology, to highlight the key techniques and primary challenges in RADO, and to provide the beneficial directions for future researches.

An investigation of fuel variability effect on bio-syngas combustion using uncertainty quantification

Fuel variability effects on physicochemical properties such as adiabatic flame temperature and laminar flame speed of premixed bio-syngas combustion are investigated via polynomial chaos expansion (PCE) based uncertainty quantification (UQ) approach at several equivalence ratios. Questions regarding confidence level of using bio-syngas with varying fuel composition are tackled from a statistical point view. Impacts of unburnt gas temperature and different chemical mechanisms (GMI-Mech 3.0 and San Diego Mechanism) on predicted uncertainties of these combustion properties are discussed. It was found that fluctuation of flame temperature at various equivalence ratios is less affected by bio-syngas fuel variabilities, while flame speed is sensitive to uncertainties in fuel composition. For instance, 1.5% fluctuation of bio-syngas constituent can lead to 14% fluctuation of flame speed for rich combustion, and 3% for lean combustion. Less than 0.8% fluctuation of flame temperature due to variability of bio-syngas fuel composition was observed. UQ of bio-syngas combustion showed that hydrogen variability plays a significant role (70–80% at lean condition) in flame speed variation, while methane variability, although thought to be important, has a negligible impact except for fuel-rich combustion. Overall, the current study has provided a fundamental understanding of the effects of fuel variability on physicochemical properties of bio-syngas combustion. Dominating compositions to variations of bio-syngas combustion are provided quantitatively to guide targeted uncertainty reduction from the upstream gasification process.

Kalman filter approach for uncertainty quantification in time-resolved laser-induced incandescence.

Time-resolved laser-induced incandescence (TiRe-LII) data can be used to infer spatially and temporally resolved volume fractions and primary particle size distributions of soot-laden aerosols, but these estimates are corrupted by measurement noise as well as uncertainties in the spectroscopic and heat transfer submodels used to interpret the data. Estimates of the temperature, concentration, and size distribution of soot primary particles within a sample aerosol are typically made by nonlinear regression of modeled spectral incandescence decay, or effective temperature decay, to experimental data. In this work, we employ nonstationary Bayesian estimation techniques to infer aerosol properties from simulated and experimental LII signals, specifically the extended Kalman filter and Schmidt-Kalman filter. These techniques exploit the time-varying nature of both the measurements and the models, and they reveal how uncertainty in the estimates computed from TiRe-LII data evolves over time. Both techniques perform better when compared with standard deterministic estimates; however, we demonstrate that the Schmidt-Kalman filter produces more realistic uncertainty estimates.

A Possibilistic Approach for the Prediction of the Risk of Interference between Power and Signal Lines Onboard Satellites

This work presents a hybrid random/fuzzy approach for uncertainty quantification in electromagnetic modelling, which combines probability and possibility theory in order to properly account for both aleatory and epistemic uncertainty, respectively. In particular, a typical intrasystem electromagnetic-compatibility problem in aerospace applications is considered, where some parameters are affected by fabrication tolerances or other kinds of randomness (aleatory uncertainty) and others are inherently deterministic but unknown due to human’s lack of knowledge (epistemic uncertainty). Namely, a differential-signal line in a satellite is subject to crosstalk due to a nearby dc power line carrying conducted emissions generated by a dc-dc converter in a wide frequency range (up to 100 MHz). The nonideal features of the signal line (e.g., weak unbalance of terminal loads) are treated as random variables (RVs), whereas the mutual position of signal and power line is characterized by possibility theory through suitable fuzzy variables. Such a hybrid approach allows deriving a general and exhaustive description of uncertainty of the target variable of interest, that is, the differential noise voltage induced in the signal line. The obtained results are compared versus a conventional Monte Carlo simulation where all parameters are treated as RVs, and the advantages of the proposed approach (in terms of completeness and richness of information gained about sensitivity of results) are highlighted.

Stochastic Basis Adaptation and Spatial Domain Decomposition for Partial Differential Equations with Random Coefficients

We present a novel uncertainty quantification approach for high-dimensional stochastic partial differential equations that reduces the computational cost of polynomial chaos methods by decomposing ...

Bayesian data analysis to quantify the uncertainty of intact rock strength

One of the main difficulties in the geotechnical design process lies in dealing with uncertainty. Uncertainty is associated with natural variation of properties, and the imprecision and unpredictability caused by insufficient information on parameters or models. Probabilistic methods are normally used to quantify uncertainty. However, the frequentist approach commonly used for this purpose has some drawbacks. First, it lacks a formal framework for incorporating knowledge not represented by data. Second, it has limitations in providing a proper measure of the confidence of parameters inferred from data. The Bayesian approach offers a better framework for treating uncertainty in geotechnical design. The advantages of the Bayesian approach for uncertainty quantification are highlighted in this paper with the Bayesian regression analysis of laboratory test data to infer the intact rock strength parameters σci and mi used in the Hoek–Brown strength criterion. Two case examples are used to illustrate different aspects of the Bayesian methodology and to contrast the approach with a frequentist approach represented by the nonlinear least squares (NLLS) method. The paper discusses the use of a Student's t-distribution versus a normal distribution to handle outliers, the consideration of absolute versus relative residuals, and the comparison of quality of fitting results based on standard errors and Bayes factors. Uncertainty quantification with confidence and prediction intervals of the frequentist approach is compared with that based on scatter plots and bands of fitted envelopes of the Bayesian approach. Finally, the Bayesian method is extended to consider two improvements of the fitting analysis. The first is the case in which the Hoek–Brown parameter, a, is treated as a variable to improve the fitting in the triaxial region. The second is the incorporation of the uncertainty in the estimation of the direct tensile strength from Brazilian test results within the overall evaluation of the intact rock strength.

Probabilistic and ensemble simulation approaches for input uncertainty quantification of artificial neural network hydrological models

ABSTRACTArtificial neural network (ANN) has been demonstrated to be a promising modelling tool for the improved prediction/forecasting of hydrological variables. However, the quantification of uncertainty in ANN is a major issue, as high uncertainty would hinder the reliable application of these models. While several sources have been ascribed, the quantification of input uncertainty in ANN has received little attention. The reason is that each measured input quantity is likely to vary uniquely, which prevents quantification of a reliable prediction uncertainty. In this paper, an optimization method, which integrates probabilistic and ensemble simulation approaches, is proposed for the quantification of input uncertainty of ANN models. The proposed approach is demonstrated through rainfall-runoff modelling for the Leaf River watershed, USA. The results suggest that ignoring explicit quantification of input uncertainty leads to under/over estimation of model prediction uncertainty. It also facilitates identification of appropriate model parameters for better characterizing the hydrological processes.

Efficient Uncertainty Quantification for Unconfined Flow in Heterogeneous Media with the Sparse Polynomial Chaos Expansion

In this study, we explore an efficient stochastic approach for uncertainty quantification of unconfined groundwater flow in heterogeneous media, where a sparse polynomial chaos expansion (PCE) surrogate model is constructed with the aid of the feature selection method. The feature selection method is introduced to construct a sparse PCE surrogate model with a reduced number of basis functions, which is accomplished by the least absolute shrinkage and selection operator-modified least angle regression and cross-validation. The training samples are enriched sequentially with the quasi-optimal samples until the results are satisfactory. In this study, we test the performance of the sparse PCE method for unconfined flow with the presence of random hydraulic conductivity and recharge, as well as pumping well. Numerical experiments reveal that, even with large spatial variability and high random dimensionality, the sparse PCE approach is able to accurately estimate the flow statistics with greatly reduced computational efforts compared to Monte Carlo simulations.

An efficient Bayesian uncertainty quantification approach with application to k-ω-γ transition modeling

An efficient Bayesian uncertainty quantification approach is proposed, which combines the adaptive high dimensional model representation technique (HDMR) and stochastic collocation (SC) method based generalized polynomial chaos (gPC) to construct the surrogate for sampling procedure in Bayesian calibration step. Specifically, the adaptive HDMR technique is used to decompose the original high dimensional problems into several lower-dimensional subproblems, which are subsequently solved with the gPC-based SC method. Then the Bayesian calibration and prediction are carried out with the so-constructed surrogate model. A new indicator based on the variance of the corresponding component function is employed to identify the important components of the HDMR, instead of the original one based on the impact on the output mean, as the input parameters that can be well informed in the inverse problem are the ones that the model output is sensitive to. Further, a rigorous convergence study of the approximate posterior to the true posterior is carried out for the proposed approach. Its applications to both a simple mathematical function and a complex fluid dynamic model, i.e. k-ω-γ transition model, are investigated, demonstrating both its efficiency and accuracy. In the application to k-ω-γ transition model, the results show not only a quantified uncertainty overlapping well with the experimental data, but also a great improvement of the match between the prediction mean and the experimental data, which may be due to the further account of the intermittency through the spread of the model parameters.

A machine learning approach for efficient uncertainty quantification using multiscale methods

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results.

On uncertainty quantification in hydrogeology and hydrogeophysics

Abstract Recent advances in sensor technologies, field methodologies, numerical modeling, and inversion approaches have contributed to unprecedented imaging of hydrogeological properties and detailed predictions at multiple temporal and spatial scales. Nevertheless, imaging results and predictions will always remain imprecise, which calls for appropriate uncertainty quantification (UQ). In this paper, we outline selected methodological developments together with pioneering UQ applications in hydrogeology and hydrogeophysics. The applied mathematics and statistics literature is not easy to penetrate and this review aims at helping hydrogeologists and hydrogeophysicists to identify suitable approaches for UQ that can be applied and further developed to their specific needs. To bypass the tremendous computational costs associated with forward UQ based on full-physics simulations, we discuss proxy-modeling strategies and multi-resolution (Multi-level Monte Carlo) methods. We consider Bayesian inversion for non-linear and non-Gaussian state-space problems and discuss how Sequential Monte Carlo may become a practical alternative. We also describe strategies to account for forward modeling errors in Bayesian inversion. Finally, we consider hydrogeophysical inversion, where petrophysical uncertainty is often ignored leading to overconfident parameter estimation. The high parameter and data dimensions encountered in hydrogeological and geophysical problems make UQ a complicated and important challenge that has only been partially addressed to date.

Optimal estimation and scheduling in aquifer management using the rapid feedback control method

Abstract Management of water resources systems often involves a large number of parameters, as in the case of large, spatially heterogeneous aquifers, and a large number of “noisy” observations, as in the case of pressure observation in wells. Optimizing the operation of such systems requires both searching among many possible solutions and utilizing new information as it becomes available. However, the computational cost of this task increases rapidly with the size of the problem to the extent that textbook optimization methods are practically impossible to apply. In this paper, we present a new computationally efficient technique as a practical alternative for optimally operating large-scale dynamical systems. The proposed method, which we term Rapid Feedback Controller (RFC), provides a practical approach for combined monitoring, parameter estimation, uncertainty quantification, and optimal control for linear and nonlinear systems with a quadratic cost function. For illustration, we consider the case of a weakly nonlinear uncertain dynamical system with a quadratic objective function, specifically a two-dimensional heterogeneous aquifer management problem. To validate our method, we compare our results with the linear quadratic Gaussian (LQG) method, which is the basic approach for feedback control. We show that the computational cost of the RFC scales only linearly with the number of unknowns, a great improvement compared to the basic LQG control with a computational cost that scales quadratically. We demonstrate that the RFC method can obtain the optimal control values at a greatly reduced computational cost compared to the conventional LQG algorithm with small and controllable losses in the accuracy of the state and parameter estimation.

Upscaling Uncertainty with Dynamic Discrepancy for a Multi-Scale Carbon Capture System

ABSTRACTUncertainties from model parameters and model discrepancy from small-scale models impact the accuracy and reliability of predictions of large-scale systems. Inadequate representation of these uncertainties may result in inaccurate and overconfident predictions during scale-up to larger systems. Hence, multiscale modeling efforts must accurately quantify the effect of the propagation of uncertainties during upscaling. Using a Bayesian approach, we calibrate a small-scale solid sorbent model to thermogravimetric (TGA) data on a functional profile using chemistry-based priors. Crucial to this effort is the representation of model discrepancy, which uses a Bayesian smoothing splines (BSS-ANOVA) framework. Our uncertainty quantification (UQ) approach could be considered intrusive as it includes the discrepancy function within the chemical rate expressions; resulting in a set of stochastic differential equations. Such an approach allows for easily propagating uncertainty by propagating the joint model parameter and discrepancy posterior into the larger-scale system of rate expressions. The broad UQ framework presented here could be applicable to virtually all areas of science where multiscale modeling is used. Supplementary materials for this article are available online.