Input Uncertainty Attribution by Uncertainty Propagation

Joint kinematics are typically described using Cardan angles or the attitude vector and its projection on the joint axes. Whichever the notation used, the uncertainties present in gait measurements affect the computed kinematics, especially for the knee joint. One notation – the attitude vector – enables the derivation of an analytical model of the propagation of uncertainty. Thus, the objective of this study was to derive this analytical model and assess the propagation of uncertainty in knee joint angle computation. Multi-session gait data acquired from one asymptomatic adult participant was used as reference data (experimental mean curve and standard deviations). Findings showed that an input uncertainty of 5° in the attitude vector and joint axes parameters matched experimental standard deviations. Taking each uncertainty independently, the cross-talk effect could result from uncertainty in the orientation of either the attitude vector (intrinsic variability) or the first joint axis (extrinsic variability). We concluded that the model successfully estimated the propagation of input uncertainties on joint angles and enabled an investigation of how that propagation occurred. The analytical model could be used to a priori estimate the standard deviations of experimental kinematics curves based on expected intrinsic and extrinsic uncertainties.

In recent years, the demand to provide best-estimate predictions with confident bounds is increasing for the nuclear reactor performance and safety analysis. The Organisation for Economic Co-operation and Development Nuclear Energy Agency has been developing an international benchmark of the light water reactor (LWR) uncertainty analysis in modeling (UAM) for the examination of uncertainty quantification and propagation methodologies with various modeling and simulation code systems. The objective of the present work is to develop an uncertainty propagation mechanism based on the stochastic sampling method by taking into account the uncertainties of both basic nuclear data and fuel modeling parameters in the simulation of pressurized water reactors (PWRs) that can be incorporated in the conventional LWR simulation approach. More specifically, the Three Mile Island Unit 1–related exercises from the LWR-UAM benchmark have been modeled using the coupled TRACE/PARCS code system in the three-dimensional core representation. The input uncertainties of the neutronics simulation include few-group cross sections and kinetics parameters generated using the Sampler/Polaris sequence of SCALE 6.2.1. Several heat transfer–related variables for the fuel modeling were considered as sources of input uncertainty of the thermal-hydraulics simulations, including the thermal conductivity of fuel and cladding, fuel heat capacity, and the gap conductance. Dakota was used to sample input parameters of the coupled code system and to perform the uncertainty analysis. Two types of simulations were conducted: steady-state calculation at hot full-power condition and transient scenario initiated by the spatially asymmetric rod ejection accident. Quantities of interest for the steady-state calculation, including core multiplication factor and power peaking factors, were calculated with associated uncertainties. For transient calculations, best-estimate plus uncertainty results of the time evolution of core reactivity, core power, and peak fuel temperature were generated and analyzed. The Wilks’ formula was used to determine the necessary sample size to achieve a 95% confidence of the 95% limit of output quantities of interest. Although the uncertainty propagation and quantification method presented in this paper was developed for PWRs, it could be in general applicable to the multiphysics uncertainty quantification of other types of LWR cores.

Uncertainty propagation (UP) is crucial for assessing the impact of input uncertainty on structural responses, holding significant importance in engineering applications. However, achieving accurate and efficient UP remains challenging, especially for highly nonlinear structures. Bayesian neural networks (BNN) have gained attention for addressing UP issues, yet current BNN models only utilize input samples and corresponding structural responses for training. However, incorporating gradients of structural responses with respect to input samples provides valuable information. This study proposes a novel approach called gradient-enhanced Bayesian neural networks (GEBNN) to tackle UP tasks. In the GEBNN, a modified evidence lower bound (MELBO) loss is developed to consider both structural responses and gradient information simultaneously. This includes disparities between actual and predicted responses, as well as disparities between actual and predicted derivatives. Additionally, a gradient screening strategy based on the marginal probability density functions (PDFs) of input samples is established to identify significant derivative data for GEBNN training. Once the GEBNN is configured, it is utilized to replace the computationally intensive finite element model to efficiently provide UP results. Various applications, including nonlinear numerical examples, and mechanical, civil, and aeronautical structures, are presented to demonstrate the effectiveness of the GEBNN. The results show that the GEBNN significantly enhances the computational accuracy of the BNN in solving UP tasks.

Propagation of uncertainty in outputs of a standard integrated model of transportation and land use has been examined. Austin-calibrated DRAM-EMPAL predictions of residence and work locations were used as inputs to a UTPP-type four-step travel demand model, and the resulting travel times were fed forward into the future period’s land use models. Covariance in inputs (including model parameters and demographic variables) was accommodated through multivariate Monte Carlo sampling of 200 scenarios. Variances in land use and travel predictions were analyzed over time and as a function of input values. Results indicate that output variations were most sensitive to the exponent of the link performance function, the split of trips between peak and off-peak, and several trip generation and attraction rates. Twenty years in the future, final uncertainty levels (as measured by coefficients of variation) solely due to input and parameter estimation errors are on the order of 38% for total regional peak-period vehicle miles traveled (VMT), 45% for peak-period flows, and 50% and 37% for residential and employment densities, respectively. This means central point estimates of key model outputs are very likely (>30%) to fall 38% to 50% below or above the mean value. In the Austin example, 15% of the region’s 200 simulated peak-period VMT estimates fell below 3.7 million mi per day and 15% exceeded 8.4 million mi. Such substantial variation is solely due to standard model parameter and input uncertainties. Other uncertainty about the future and about human behavior exists and will add more variation.

Equilibrium modeling of U(VI) speciation in high carbonate groundwaters: Model error and propagation of uncertainty

A review of uncertainty quantification techniques is provided for a variety of situations involving uncertainties in model inputs (independent variables). The situations of interest are divided into three categories: (i) when model prediction uncertainties are quantified based on uncertainties in uncertain inputs, (ii) when parameter estimate uncertainties are calculated by propagation of uncertainties from measured inputs and outputs, and (iii) when model prediction uncertainties are quantified based on corresponding uncertainties in measured inputs and uncertain parameter estimates. For all three situations, linearization‐based and Monte Carlo‐based techniques are reviewed and details for their corresponding algorithms are presented. Recommendations are provided on which uncertainty quantification techniques are best for different types of chemical engineering models based on the amount of input uncertainty and nonlinearity over the range of plausible input and parameter values.

Uncertainty propagation in coupled hydrological models using winding stairs and null-space Monte Carlo methods

A generalized uncertainty propagation criterion from benchmark studies of microstructured material systems

In modern engineering, physical processes are modelled using advanced computer simulation tools (e.g. finite element models) to assess and optimize their performance. Moreover, awareness is growing on concepts like structural reliability and robust design, hence making the efficient quantification and propagation of uncertainties a key challenge. A major part of those analyses is the characterization of uncertainty in the input, which is typically done with probabilistic variables. In the case of sparse data sets, however, probability theory is often not the optimal choice. Imprecise probabilities provide a more general framework accounting for both aleatory and epistemic uncertainties. The uncertainty propagation of imprecise probabilities leads to imprecise responses. In this context, an algorithm for solving imprecise structural reliability problems is presented. The algorithm transforms the imprecise problem into two precise structural reliability problems, which opens up possibilities for using traditional structural reliability analyses techniques. An adaptive experimental design algorithm based on Polynomial-Chaos-Kriging is used to efficiently estimate imprecise failure probabilities. The capabilities of the framework are illustrated through an application using an analytical function and a realistic engineering problem setting.

Generalized likelihood uncertainty estimation (GLUE) using adaptive Markov Chain Monte Carlo sampling

The objective of the study described in this paper was to develop an efficient uncertainty quantification framework capable of analyzing uncertainty in integrated spacecraft system models. Specifically, this paper discusses the capabilities of the developed framework and the res ults when applied to the multidisciplinary analysis of a reusable launch veh icle (RLV). This particular framework is capable of efficiently propagating mixed (inherent and epistemic) uncertainties through complex simulation codes. The Second-Order Probability Theory utilizing a stochastic response surface obta ined with Point-Collocation NonIntrusive Polynomial Chaos was used for the propagation of the mixed uncertainties. This particular methodology was applied to the RLV analysis, and the uncertainty in the output parameters of interested was obtained in terms of intervals at various probability levels. This study has also demonstrated the feasibility of the developed uncertainty quantifica tion framework for efficient propagation of mixed uncertainties in the analysis of complex aerospace systems. I. Introduction ncertainties are generally ubiquitous in the analys is and design of highly complex engineering systems. Uncertainties can arise from the lack of k nowledge in physical modeling (epistemic uncertainty), inherent variations in the systems (a leatory uncertainty), and numerical errors in the computational procedures used for analysis. It is i mportant to account for all of these uncertainties in applications such as robust and reliable design of multi-disciplinary aerospace systems. A reusable la unch vehicle (RLV) is a highly complex aerospace system, which represents a cost viable option for access t o space missions due to its reusability aspect. Since an RLV system is composed of various subsystems which must work together to serve an over-arching p urpose or mission objective/constraint, a multidisciplinary approach for the analysis and des ign of RLV systems is required. Uncertainties are generally present in the models used in each discipline of an RLV multidisciplinary analysis framework. It is important to account for all of these uncertainties for accurate and reliabl e estimates of the RLV system performance. The primary purpose of this paper will be to demonstrate an efficient approach for uncertainty quantification in the multidiscipl inary analysis of a RLV, which has a mixture of aleatory (inherent) and epistemic input uncertainties. Uncertainty quantification will be performed on various output variables of interest f or an RLV system such as the weight and vehicle crossrange. For the propagation of mixed (aleatory-epist emic) uncertainty, Second-Order Probability Theory utilizing Point-Collocation Non-Intrusive Polynomia l Chaos (NIPC) 1 will be used. 2-3 In general, the NIPC methods, which are based on the spectral representa tion of uncertainty, are computationally more effic ient than traditional Monte Carlo methods for moderate number of uncertain variables and can give highly 1 Aerospace Engineer, 4020 Long Beach Blvd, Long Beach, CA, 90807, Member AIAA

This paper outlines an optimization-based method to estimate the reach set of a system while acknowledging unknown-but-bounded input and parametric uncertainty in the underlying dynamical model. The approach is grounded in a second-order Taylor-series expansion of the system's state variables along the solution trajectories as a function of the uncertain elements. Subsequently, over the time horizon of interest, Quadratically Constrained Quadratic Programs (QCQPs) are formulated to estimate maximum and minimum bounds on the state variables to recover the reach set. To contend with the nonconvexity of the QCQPs, Lagrangian relaxations are leveraged to formulate Semidefinite Programs (SDPs) that provide guaranteed bounds to the solutions of the QCQPs. Applications of the method to quantify the impact of uncertain power injections in power-system dynamic models are demonstrated with numerical examples.

The oil and gas industry is faced with uncertainty in many activities. Whereas in many of its areas, such as finance, geology and reservoirs these uncertainties are already incorporated in the modelling and studies. This is not the reality when it comes to modelling artificial lift and multiphase flow. Studies related with uncertainties in oil and gas production are limited. Therefore, this paper aims to develop a methodology to identify and quantify uncertainties, obtaining thus more accurate data to be used in production modelling. In addition, the methodology intends to evaluate oil production forecasting considering uncertainty propagation in oil flow simulation software. This study was divided in statistical analysis and production forecasting. An algorithm using R software analyzed and treated production data. It was applied statistical modelling techniques to data series. Deviations from these data were adjusted to a continuous distribution that provides the parameters to be used by Monte Carlo simulation method to generate random values to be input uncertainties of the MARLIM multiphase flow simulator. Oil flow rate, as output simulator, was adjusted to a new distribution and finally the intervals of occurrence probabilities of oil flow rate forecasts. This methodology was applied to BSW (Basic Sediments and Water) data from a representative field, showing the importance of including uncertainty analysis in order to generate greater reliability and accuracy in the production flow modeling. In conclusion, the method presented excellent results when applied to BSW.

Metamaterials have recently emerged in the search for lightweight noise and vibration solutions. One of their appealing properties for noise control engineering is the ability to create stop bands, which are frequency ranges without free wave propagation. These stop bands arise from the sub-wavelength addition of identically tuned resonators in or on a host structure. However, when manufacturing metamaterials, variability in material properties and geometry is inevitably introduced. On the one hand, the metamaterial attenuation performance can deteriorate due to variability, while on the other hand, variability can even broaden their typically narrowband performance. During the early phases in the design, when often little information concerning the inherent variability is available, it would be desirable to be able to assess the input uncertainty effects on the performance of metamaterials. This work focuses on the numerical assessment of the impact of uncertainties in resonator properties on the vibration attenuation performance of metamaterial beams. To this end, a machine learning based optimization strategy, the so-called Bayesian optimization, is employed in a recently developed non-intrusive uncertainty propagation approach. This enables a very efficient evaluation of the upper and lower bounds on the metamaterial performance, with the uncertain resonator parameters defined as interval variables.

PurposeFor fast calculation of complex structure in engineering, correlations among input variables are often ignored in uncertainty propagation, even though the effect of ignoring these correlations on the output uncertainty is unclear. This paper aims to quantify the inputs uncertainty and estimate the correlations among them acorrding to the collected observed data instead of questionable assumptions. Moreover, the small size of the experimental data should also be considered, as it is such a common engineering problem.Design/methodology/approachIn this paper, a novel method of combining p-box with copula function for both uncertainty quantification and correlation estimation is explored. Copula function is utilized to estimate correlations among uncertain inputs based upon the observed data. The p-box method is employed to quantify the input uncertainty as well as the epistemic uncertainty associated with the limited amount of the observed data. Nested Monte Carlo sampling technique is adopted herein to ensure that the propagation is always feasible. In addition, a Kriging model is built up to reduce the computational cost of uncertainty propagation.FindingsTo illustrate the application of this method, an engineering example of structural reliability assessment is performed. The results indicate that it may significantly affect output uncertainty whether to quantify the correlation among input variables. Furthermore, an additional advantage for risk management is obtained in this approach due to the separation of aleatory and epistemic uncertainties.Originality/valueThe proposed method takes advantage of p-box and copula function to deal with the correlations and limited amount of the observed data, which are two important issues of uncertainty quantification in engineering. Thus, it is practical and has the ability to predict accurate response uncertainty or system state.