Uncertainty Quantification Approach Research Articles

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.

A decomposition-based uncertainty quantification approach for environmental impacts of aviation technology and operation

AbstractAs a measure to manage the climate impact of aviation, significant enhancements to aviation technologies and operations are necessary. When assessing these enhancements and their respective impacts on the climate, it is important that we also quantify the associated uncertainties. This is important to support an effective decision and policymaking process. However, such quantification of uncertainty is challenging, especially in a complex system that comprises multiple interacting components. The uncertainty quantification task can quickly become computationally intractable and cumbersome for one individual or group to manage. Recognizing the challenge of quantifying uncertainty in multicomponent systems, we utilize a divide-and-conquer approach, inspired by the decomposition-based approaches used in multidisciplinary analysis and optimization. Specifically, we perform uncertainty analysis and global sensitivity analysis of our multicomponent aviation system in a decomposition-based manner. In this work, we demonstrate how to handle a high-dimensional multicomponent interface using sensitivity-based dimension reduction and a novel importance sampling method. Our results demonstrate that the decomposition-based uncertainty quantification approach can effectively quantify the uncertainty of a feed-forward multicomponent system for which the component models are housed in different locations and owned by different groups.

Ensemble-based uncertainty quantification for coordination and control of thermostatically controlled loads

This work investigates an uncertainty quantification (UQ) framework that analyses the uncertainty involved in modelling control systems to improve control strategy performance. The framework involves solving four problems: identifying uncertain parameters, propagating uncertainty to the quantity of interest, data assimilation and making decisions under quantified uncertainties. A specific group of UQ approaches, known as the ensemble-based methods, are adopted to solve these problems. This UQ framework is applied to coordinating a group of thermostatically controlled loads, which relies on simulating a second-order equivalent thermal parameter model with some uncertain parameters. How this uncertainty affects the prediction and the control of total power is examined. The study shows that uncertainty can be effectively reduced using the measurement of air temperatures. Also, the control objective is achieved fairly accurately with a quantification of the uncertainty.

Asynchronous space–time domain decomposition method with localized uncertainty quantification

The computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest. In this case, a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response. We present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions. In particular, we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions. As a result, computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required. To verify the numerical implementation, we consider elastic wave propagation in an axially loaded beam. Moreover, we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain. A projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method.

Multifidelity Sparse-Grid-Based Uncertainty Quantification for the Hokkaido Nansei-oki Tsunami

With uncertainty quantification, we aim to efficiently propagate the uncertainties in the input parameters of a computer simulation, in order to obtain a probability distribution of its output. In this work, we use multi-fidelity sparse grid interpolation to propagate the uncertainty in the shape of the incoming wave for the Okushiri test-case, which is a wave tank model of a part of the 1993 Hokkaido Nansei-oki tsunami. An important issue with many uncertainty quantification approaches is the ‘curse of dimensionality’: the overall computational cost of the uncertainty propagation increases rapidly when we increase the number of uncertain input parameters. We aim to mitigate the curse of dimensionality by using a multifidelity approach. In the multifidelity approach, we combine results from a small number of accurate and expensive high-fidelity simulations with a large number of less accurate but also less expensive low-fidelity simulations. For the Okushiri test-case, we find an improved scaling when we increase the number of uncertain input parameters. This results in a significant reduction of the overall computational cost. For example, for four uncertain input parameters, accurate uncertainty quantification based on only high-fidelity simulations comes at a normalised cost of 219 high-fidelity simulations; when we use a multifidelity approach, this is reduced to a normalised cost of only 10 high-fidelity simulations.

Investigation of Biotransport in a Tumor With Uncertain Material Properties Using a Nonintrusive Spectral Uncertainty Quantification Method.

In this study, statistical models are developed for modeling uncertain heterogeneous permeability and porosity in tumors, and the resulting uncertainties in pressure and velocity fields during an intratumoral injection are quantified using a nonintrusive spectral uncertainty quantification (UQ) method. Specifically, the uncertain permeability is modeled as a log-Gaussian random field, represented using a truncated Karhunen-Lòeve (KL) expansion, and the uncertain porosity is modeled as a log-normal random variable. The efficacy of the developed statistical models is validated by simulating the concentration fields with permeability and porosity of different uncertainty levels. The irregularity in the concentration field bears reasonable visual agreement with that in MicroCT images from experiments. The pressure and velocity fields are represented using polynomial chaos (PC) expansions to enable efficient computation of their statistical properties. The coefficients in the PC expansion are computed using a nonintrusive spectral projection method with the Smolyak sparse quadrature. The developed UQ approach is then used to quantify the uncertainties in the random pressure and velocity fields. A global sensitivity analysis is also performed to assess the contribution of individual KL modes of the log-permeability field to the total variance of the pressure field. It is demonstrated that the developed UQ approach can effectively quantify the flow uncertainties induced by uncertain material properties of the tumor.

Uncertainty quantification and management in additive manufacturing: current status, needs, and opportunities

One of the major barriers that hinder the realization of significant potential of metal-based additive manufacturing (AM) techniques is the variation in the quality of the manufactured parts. Uncertainty quantification (UQ) and uncertainty management (UM) can resolve this challenge based on the modeling and simulation of the AM process. This paper reviews the research state of the art and discusses needs and opportunities in the UQ/UM of the AM processes, with a focus on laser powder bed fusion AM. The major methods and models of laser powder bed fusion AM process are summarized first. The current research work in UQ of AM processes is then reviewed. Based on the review of AM process models and current UQ approaches for the AM process, this paper presents insights into how the current state of the art UQ and UM techniques can be applied to AM to improve the product quality. Future research needs in UQ and UM of AM are also discussed. Laser sintering of metal nanoparticles, which is part of the micro-AM process, is used as an example to illustrate the application of UQ and UM in the AM.

A two-stage approach of multiplicative dimensional reduction and polynomial chaos for global sensitivity analysis and uncertainty quantification with a large number of process uncertainties

Uncertainties associated with estimates of model parameters are inevitable when simulating and modeling chemical processes and significantly affect safety, consistency, and decision making. Quantifying those uncertainties is essential for emulating the actual system behaviors because they can change the management recommendations that are drawn from the model. The use of conventional approaches for uncertainty quantification (e.g., Monte-Carlo and standard polynomial chaos methods) is computationally expensive for complex systems with a large/moderate number of uncertainties. This paper develops a two-stage approach to quantify the uncertainty of complex chemical processes with a moderate/large number of uncertainties (greater than 5). The first stage applies a multiplicative dimensional reduction method to approximate the variance-based global sensitivity measures (Sobol's method), and to simplify the model for the uncertainty quantification stage. The second stage uses the generalized polynomial chaos approach to quantify uncertainty of the simplified model from the first stage. A rigorous simulation illustrates the proposed approach using an interface between MATLAB and HYSYS for three complex chemical processes. The proposed method was compared with conventional approaches, such as the Quasi Monte-Carlo sampling-based method and standard polynomial chaos-based method. The results revealed the clear advantage of the proposed approach in terms of the computational efforts.

Uncertainty quantification in risk assessment - Representation, propagation and treatment approaches: Application to atmospheric dispersion modeling

Quantitative risk analysis (QRA) is a fundamental part of the decision-making process when it comes to the safety of people and the environment. However, due to the uncertainty involved, the credibility of risk assessment results is still a major issue. This paper aims to explore the most commonly used approaches to quantify uncertainty in risk analysis: interval analysis, fuzzy theory, probability theory, evidence theory, and the mixed probabilistic-fuzzy approach. These approaches are used to characterize uncertainty in model inputs obtained from different sources, such as statistical data and expert judgments, and to which different types of uncertainty can be attached. These uncertainty characterizations are then propagated through the model to obtain the corresponding representation of uncertainty for the model outputs. The paper presents the application of these quantification approaches to a loss of containment scenario (LOC), representing one of the most likely situations to occur in industry. The overall aim is to study the effects of uncertainty and compare the different approaches. Indeed, the uncertainty quantification approaches presented can lead to different representations of uncertainty in the outputs and hence to different decisions. The use of an inappropriate approach in an inappropriate place may lead to under or overestimation of risk and subsequently to a bad decision.

BayesWHAM: A Bayesian approach for free energy estimation, reweighting, and uncertainty quantification in the weighted histogram analysis method.

The weighted histogram analysis method (WHAM) is a powerful approach to estimate molecular free energy surfaces (FES) from biased simulation data. Bayesian reformulations of WHAM are valuable in proving statistically optimal use of the data and providing a transparent means to incorporate regularizing priors and estimate statistical uncertainties. In this work, we develop a fully Bayesian treatment of WHAM to generate statistically optimal FES estimates in any number of biasing dimensions under arbitrary choices of the Bayes prior. Rigorous uncertainty estimates are generated by Metropolis-Hastings sampling from the Bayes posterior. We also report a means to project the FES and its uncertainties into arbitrary auxiliary order parameters beyond those in which biased sampling was conducted. We demonstrate the approaches in applications of alanine dipeptide and the unthreading of a synthetic mimic of the astexin-3 lasso peptide. Open-source MATLAB and Python implementations of our codes are available for free public download. © 2017 Wiley Periodicals, Inc.

Epistemic uncertainty quantification in metal fatigue crack growth analysis using evidence theory

Uncertainties originate from physical variability, data uncertainty, and modeling errors in the fatigue crack growth prediction analysis. This study presents an evidential uncertainty quantification (UQ) approach for determining uncertainties involved in revealing the material constants of the metal fatigue crack growth model with imprecise uncertainty information (i.e., epistemic uncertainty). The parameters in fatigue crack growth model are obtained by fitting the available sparse experimental data, and then the uncertainties in these parameters are considered. To alleviate the computational difficulties in the UQ analysis based on evidence theory, an interval optimization method based on differential evolution is used in finding the propagated belief structure. The overall procedure is demonstrated using the results of several replicated experiments on aluminum alloy CCT specimens.

Multiscale Validation and Uncertainty Quantification for Problems With Sparse Data

Quantification of uncertainty in the simulation results becomes difficult for complex real-world systems with little or no experimental data. This paper describes a validation and uncertainty quantification (VUQ) approach that integrates computational and experimental data through a range of experimental scales and a hierarchy of complexity levels. This global approach links dissimilar experimental datasets at different scales, in a hierarchy, to reduce quantified error bars on case with sparse data, without running additional experiments. This approach was demonstrated by applying on a real-world problem, greenhouse gas (GHG) emissions from wind tunnel flares. The two-tier validation hierarchy links, a buoyancy-driven helium plume and a wind tunnel flare, to increase the confidence in the estimation of GHG emissions from wind tunnel flares from simulations.

Quantifying uncertainty of a reacting multiphase flow in a bench-scale fluidized bed gasifier: A Bayesian approach

Adequate assessment of the uncertainties is becoming an integral part of the engineering design. This study is based on several efforts that are underway to investigate the uncertainty quantification (UQ) methods that are applicable to multiphase flows. Although particular emphasis has been given on identifying, characterizing and managing uncertainties in computational fluid dynamics (CFD) simulations, experimental data used in the validation of the CFD model is as critical. Hence, the goal of this paper is to demonstrate application of non-intrusive Bayesian uncertainty quantification methodology in multiphase (gas-solid) flows with experimental data as part of our research efforts to determine the most suited approach for UQ. For this purpose, a bench scale fluidized bed gasifier experiment from a prior study is used to demonstrate the applicability of Bayesian techniques on sparse data. Global sensitivity analysis performed as part of the UQ study shows that among the three operating factors, steam to oxygen ratio has the most influence on syngas composition in the gasifier. An analysis for forward propagation of uncertainties shows that an increase in steam to oxygen ratio leads to an increase in H2 mole fraction and a decrease in CO mole fraction. These findings are in agreement with the reference experimental study. Another contribution in addition to the UQ analysis is the optimization-based approach employed to identify next best set of additional experimental samples. Hence, the surrogate models constructed as part of the UQ analysis is employed to improve the information gain and make incremental recommendation, should the possibility to add more experiments arise. The insight gained from this study has been extensively used in the follow-up UQ study with the CFD modeling of the same gasifier configuration.

Efficient inversion and uncertainty quantification of a tephra fallout model

AbstractAn efficient and effective inversion and uncertainty quantification approach is proposed for estimating eruption parameters given a data set collected from a tephra deposit. The approach is model independent and here is applied using Tephra2, a code that simulates advective and dispersive tephra transport and deposition. The Levenburg‐Marquardt algorithm is combined with formal Tikhonov and subspace regularization to invert eruption parameters; a linear equation for conditional uncertainty propagation is used to estimate posterior parameter uncertainty. Both the inversion and uncertainty analysis support simultaneous analysis of the full eruption and wind field parameterization. The combined inversion/uncertainty quantification approach is applied to the 1992 eruption of Cerro Negro and the 2011 Kirishima‐Shinmoedake eruption. While eruption mass uncertainty is reduced by inversion against tephra isomass data, considerable uncertainty remains for many eruption and wind field parameters, such as plume height. Supplementing the inversion data set with tephra granulometry data is shown to further reduce the uncertainty of most eruption and wind field parameters. The eruption mass of the 2011 Kirishima‐Shinmoedake eruption is 0.82 × 1010 kg to 2.6 × 1010 kg, with 95% confidence; total eruption mass for the 1992 Cerro Negro eruption is 4.2 × 1010 kg to 7.3 × 1010 kg, with 95% confidence. These results indicate that eruption classification and characterization of eruption parameters can be significantly improved through this uncertainty quantification approach.

Comparison of Uncertainty Quantification Approaches in a Supersonic Biplane Airfoil Problem

Comparison of Uncertainty Quantification Approaches in a Supersonic Biplane Airfoil Problem

Error correction in multi-fidelity molecular dynamics simulations using functional uncertainty quantification

We use functional, Fréchet, derivatives to quantify how thermodynamic outputs of a molecular dynamics (MD) simulation depend on the potential used to compute atomic interactions. Our approach quantifies the sensitivity of the quantities of interest with respect to the input functions as opposed to its parameters as is done in typical uncertainty quantification methods. We show that the functional sensitivity of the average potential energy and pressure in isothermal, isochoric MD simulations using Lennard–Jones two-body interactions can be used to accurately predict those properties for other interatomic potentials (with different functional forms) without re-running the simulations. This is demonstrated under three different thermodynamic conditions, namely a crystal at room temperature, a liquid at ambient pressure, and a high pressure liquid. The method provides accurate predictions as long as the change in potential can be reasonably described to first order and does not significantly affect the region in phase space explored by the simulation. The functional uncertainty quantification approach can be used to estimate the uncertainties associated with constitutive models used in the simulation and to correct predictions if a more accurate representation becomes available.

Application of an Adaptive Polynomial Chaos Expansion on Computationally Expensive Three-Dimensional Cardiovascular Models for Uncertainty Quantification and Sensitivity Analysis.

When applying models to patient-specific situations, the impact of model input uncertainty on the model output uncertainty has to be assessed. Proper uncertainty quantification (UQ) and sensitivity analysis (SA) techniques are indispensable for this purpose. An efficient approach for UQ and SA is the generalized polynomial chaos expansion (gPCE) method, where model response is expanded into a finite series of polynomials that depend on the model input (i.e., a meta-model). However, because of the intrinsic high computational cost of three-dimensional (3D) cardiovascular models, performing the number of model evaluations required for the gPCE is often computationally prohibitively expensive. Recently, Blatman and Sudret (2010, "An Adaptive Algorithm to Build Up Sparse Polynomial Chaos Expansions for Stochastic Finite Element Analysis," Probab. Eng. Mech., 25(2), pp. 183-197) introduced the adaptive sparse gPCE (agPCE) in the field of structural engineering. This approach reduces the computational cost with respect to the gPCE, by only including polynomials that significantly increase the meta-model's quality. In this study, we demonstrate the agPCE by applying it to a 3D abdominal aortic aneurysm (AAA) wall mechanics model and a 3D model of flow through an arteriovenous fistula (AVF). The agPCE method was indeed able to perform UQ and SA at a significantly lower computational cost than the gPCE, while still retaining accurate results. Cost reductions ranged between 70-80% and 50-90% for the AAA and AVF model, respectively.

A random matrix approach for quantifying model-form uncertainties in turbulence modeling

With the ever-increasing use of Reynolds-Averaged Navier–Stokes (RANS) simulations in mission-critical applications, the quantification of model-form uncertainty in RANS models has attracted attention in the turbulence modeling community. Recently, a physics-based nonparametric approach for quantifying model-form uncertainty in RANS simulations has been proposed, where Reynolds stresses are projected to physically meaningful dimensions and perturbations are introduced only in the physically realizable limits (Xiao et al., 2016). However, a challenge associated with this approach is to assess the amount of information introduced in the prior distribution and to avoid imposing unwarranted constraints. In this work we propose a random matrix approach for quantifying model-form uncertainties in RANS simulations with the realizability of the Reynolds stress guaranteed, which is achieved by construction from the Cholesky factorization of the normalized Reynolds stress tensor. Furthermore, the maximum entropy principle is used to identify the probability distribution that satisfies the constraints from available information but without introducing artificial constraints. We demonstrate that the proposed approach is able to ensure the realizability of the Reynolds stress, albeit in a different manner from the physics-based approach. Monte Carlo sampling of the obtained probability distribution is achieved by using polynomial chaos expansion to map independent Gaussian random fields to the Reynolds stress random field with the marginal distributions and correlation structures as specified. Numerical simulations on a typical flow with separation have shown physically reasonable results, which verify the proposed approach. Therefore, the proposed method is a promising alternative to the physics-based approach for model-form uncertainty quantification of RANS simulations. The method explored in this work is general and can be extended to other complex physical systems in applied mechanics and engineering.