Approach For Uncertainty Propagation Research Articles

Multi-Fidelity Uncertainty Propagation Approach for Multi-Dimensional Correlated Flow Field Responses

ABSTRACT To quantify uncertainties in the multidimensional correlated flow field responses resulting from uncertain model parameters in computational fluid dynamics (CFD), this paper proposes an uncertainty quantification framework that utilises proper orthogonal decomposition (POD) and a co-Kriging model. A mixed sample snapshot set containing both high- and low-fidelity responses is first formed by transferring high-fidelity responses into the space of low-fidelity responses. The use of POD on this snapshot set reduces the multidimensional correlated responses to reduced-dimension responses, which are coefficients of basis functions. A co-Kriging model is then established between the input parameters and reduced-dimension responses. This comprehensive model enables predictions of multidimensional correlated responses under new input parameters. Through an analysis of a NACA0012 airfoil, it is demonstrated that the proposed multi-fidelity approach offers significant advancements in accuracy and efficiency compared to the single-fidelity model, providing an efficient prediction model for uncertainty propagation analysis.

Optimizing resource allocations to improve system reliability via the propagation of statistical moments through fault trees

Fault tree analysis remains a significant alternative for modeling and analyzing reliability and failure modes. Traditionally, the probabilities of the basic events are assumed as point estimates from which the fault tree is used to estimate the probability of the top failure event. This paper instead proposes using this framework for a different purpose: framing the model parameters describing the basic events as random variables, which collectively contribute to the statistics of the estimate of the top failure event. An approach for uncertainty propagation via statistical moments is developed for both static and dynamic gates with specific consideration of the effects of the dependency conditions for different distribution types. Assuming there is a cost to acquire model parameters with reduced uncertainty, this framing can thus be used to identify the best set of model parameters possible within a constrained budget to minimize uncertainty in the resulting estimate of the top failure event. With these models in place, the statistical moments of the top event can be connected to all lower levels, and the statistics in the estimate of the top event can be optimized with a given cost constraint (e.g. allocating resources to reducing uncertainty in the knowledge of basic events to minimize variance in the estimate of the top failure event). This optimization of the uncertainty in the top event is demonstrated with a case study.

Uncertainty-aware automated machine learning toolbox

Abstract Measurement data can be considered complete only with an associated measurement uncertainty to express knowledge about the spread of values reasonably attributed to the measurand. Measurement uncertainty also allows to assess the comparability and the reliability of measurement results as well as to evaluate decisions based on the measurement result. Artificial Intelligence (AI) methods and especially Machine Learning (ML) are often based on measurements, but so far, uncertainty is widely neglected in this field. We propose to apply uncertainty propagation in ML to allow estimating the uncertainty of ML results and, furthermore, an optimization of ML methods to minimize this uncertainty. Here, we present an extension of a previously published automated ML toolbox (AMLT), which performs feature extraction, feature selection and classification in an automated way without any expert knowledge. To this end, we propose to apply the principles described in the “Guide to the Expression of Uncertainty in Measurement” (GUM) and its supplements to carry out uncertainty propagation for every step in the AMLT. In previous publications we have presented the uncertainty propagation for some of the feature extraction methods in the AMLT. In this contribution, we add some more elements to this concept by also including statistical moments as a feature extraction method, add uncertainty propagation to the feature selection methods and extend it to also include the classification method, linear discriminant analysis combined with Mahalanobis distance. For these methods, analytical approaches for uncertainty propagation are derived in detail, and the uncertainty propagation for the other feature extraction and selection methods are briefly revisited. Finally, the use the uncertainty-aware AMLT is demonstrated for a data set consisting of uncorrelated measurement data and associated uncertainties.

Nuclear data uncertainty propagation applied to the versatile test reactor conceptual design

The Versatile Test Reactor (VTR) currently under development is a 300 MWth sodium-cooled fast reactor (SFR) fueled with ternary metal alloy fuel, which aims to accelerate the testing of advanced nuclear fuels, materials, instrumentation, and sensors in high flux environments that are necessary to license the next generation of advanced reactor concepts. To support the VTR design process, uncertainties associated with the nuclear data has been propagated through the reactor core neutronics calculation to global parameters of interest, such as the core multiplication factor, kinetic parameters, and various reactivity feedback coefficients, following the sensitivity based uncertainty propagation approach. By folding the sensitivity coefficients, separately computed by the generalized perturbation theory code PERSENT and Monte Carlo code Serpent 2, with the variance–covariance matrices from COMMARA-2.0, we obtain the reaction-wise, isotope-wise, and overall uncertainties for each response of interest due to nuclear data uncertainty. With Serpent 2, the statistical error of the uncertainty is obtained by propagating the statistical error of the sensitivity coefficients through the same process using a newly developed uncertainty propagation method. From both codes, the overall top uncertainty contributors are found to be the cross section of Fe-56 elastic scattering, Na-23 elastic scattering, and U-238 inelastic scattering. The large contributions of the Fe-56 elastic scattering cross sections to global parameters are due to its relatively large relative uncertainty of 5–10% in nuclear data and the large volume of Fe-containing reflector assemblies in the fairly compact VTR core design. Both codes agreed well for the overall uncertainty estimates of all responses of interest, except the delayed neutron fraction, prompt neutron generation time, and the coolant density feedback coefficient, where Serpent 2 yielded a much larger value than PERSENT due to the large statistical error of sensitivity coefficients. The calculated uncertainties are also compared to those associated with other SFR cores. Another outcome of this study is a variance–covariance matrix of reactivity coefficients, which can be used in the subsequent uncertainty propagation to the system level to investigate the impact of identified uncertainties on system responses in the safety analysis.

Interval uncertainty propagation in metamaterial beams using machine learning based optimization

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.

A Semi-Analytical and Monte Carlo-Based Phase Dynamic Evolution Approach for LEO Mega-Constellations

In recent years, with the reduction of the cost of microsatellites, the development of commercial rockets and the multi-satellite launching technology, the construction of large-scale constellations in low-Earth orbit (Mega-Constellations) has become a development trend. Since the motion of LEO satellites is affected by perturbations such as non-spherical gravitational fields and atmospheric drag, as well as the uncertainty of actuators, measurement systems, and dynamic models, it is easy to cause divergence of constellation configurations. The station-keeping control of the satellites is crucial for the stable operation of the mega-constellation. Aiming at this problem, this paper proposes an uncertainty propagation approach based on semi-analytical and Monte Carlo for LEO Mega-Constellations. Under the assumption that initial uncertainty on the osculating trajectory is Gaussian distribution, through hypothesis testing analysis, the uncertainty propagation simulations of a single satellite suggest that the satellite argument of latitude and the relative phase of co-plane satellites can be both considered as Gaussian distributions with zero means. Multi-group Monte Carlo simulations with product-based least-squares surface fitting establish an approximate mapping between initial and terminal errors. The mapping provides an efficient method for deviation prediction and can be used to design the station-keeping control strategy.

Machine learning based optimization for interval uncertainty propagation

Two non-intrusive uncertainty propagation approaches are proposed for the performance analysis of engineering systems described by expensive-to-evaluate deterministic computer models with parameters defined as interval variables. These approaches employ a machine learning based optimization strategy, the so-called Bayesian optimization, for evaluating the upper and lower bounds of a generic response variable over the set of possible responses obtained when each interval variable varies independently over its range. The lack of knowledge caused by not evaluating the response function for all the possible combinations of the interval variables is accounted for by developing a probabilistic description of the response variable itself by using a Gaussian Process regression model. An iterative procedure is developed for selecting a small number of simulations to be evaluated for updating this statistical model by using well-established acquisition functions and to assess the response bounds. In both approaches, an initial training dataset is defined. While one approach builds iteratively two distinct training datasets for evaluating separately the upper and lower bounds of the response variable, the other one builds iteratively a single training dataset. Consequently, the two approaches will produce different bound estimates at each iteration. The upper and lower response bounds are expressed as point estimates obtained from the mean function of the posterior distribution. Moreover, a confidence interval on each estimate is provided for effectively communicating to engineers when these estimates are obtained at a combination of the interval variables for which no deterministic simulation has been run. Finally, two metrics are proposed to define conditions for assessing if the predicted bound estimates can be considered satisfactory. The applicability of these two approaches is illustrated with two numerical applications, one focusing on vibration and the other on vibro-acoustics.

Uncertainty propagation in risk and resilience analysis of hierarchical systems

A novel formulation is proposed for uncertainty propagation in risk and resilience analysis of hierarchical systems. The main challenges are related to the complexity of hierarchical systems’ computational workflow and high-dimensional probability space. The computational workflow in regional risk and resilience analysis consists of many interconnected sub-models to predict future hazards, the reliability and functionality of physical systems, and the recovery of disrupted services. The complexity of the computational workflow limits the number of model evaluations for uncertainty propagation. In contrast, the computational workflow contains many sources of uncertainty that demand extensive model evaluations to accurately estimate their effects. The proposed formulation in this paper consists of a multi-level uncertainty propagation approach to reduce the problem dimensionality and a variables-grouping approach to reduce the number of model evaluations. The idea of the multi-level uncertainty propagation is to break down the high-dimensional problem into several low-dimensional ones, one for each level of the hierarchy in the computational workflow. The proposed variables-grouping approach provides an adaptive refinement of uncertainty propagation to identify the influential uncertain input data and computational sub-models. The paper illustrates the proposed formulation through a well-known academic problem and regional risk and resilience analysis of a community.

Towards the NASA UQ Challenge 2019: Systematically forward and inverse approaches for uncertainty propagation and quantification

This paper is dedicated to exploring the NASA Langley Challenge on Optimization under Uncertainty by proposing a series of approaches for both forward and inverse treatment of uncertainty propagation and quantification. The primary effort is placed on the categorization of the subproblems as to be forward or inverse procedures, such that dedicated techniques are proposed for the two directions, respectively. The sensitivity analysis and reliability analysis are categorized as forward procedures, while modal calibration & uncertainty reduction, reliability-based optimization, and risk-based design are regarded as inverse procedures. For both directions, the overall approach is based on imprecise probability characterization where both aleatory and epistemic uncertainties are investigated for the inputs, and consequently, the output is described as the probability-box (P-box). Theoretic development is focused on the definition of comprehensive uncertainty quantification criteria from limited and irregular time-domain observations to extract as much as possible uncertainty information, which will be significant for the inverse procedure to refine uncertainty models. Furthermore, a decoupling approach is proposed to investigate the P-box along two directions such that the epistemic and aleatory uncertainties are decoupled, and thus a two-loop procedure is designed to propagate both epistemic and aleatory uncertainties through the systematic model. The key for successfully addressing this challenge is in obtaining on the balance among an appropriate hypothesis of the input uncertainty model, a comprehensive criterion of output uncertainty quantification, and a computational viable approach for both forward and inverse uncertainty treatment.

A non-probabilistic uncertainty analysis method based on ellipsoid possibility model and its applications in multi-field coupling systems

Non-probabilistic methods are usually more appropriate and feasible than probabilistic methods to quantify and propagate uncertainty with small samples in many engineering problems. However, some non-probabilistic methods are easily affected by an outlier in uncertainty quantification and only estimate the bounds of output responses in a feasible region in uncertainty propagation. Moreover, some such methods ignore the situation that its sample points have the characteristic of clustering to the center, which significantly affects the accuracy of uncertainty quantification and propagation. In this study, a novel non-probabilistic uncertainty analysis method is proposed to solve the situation and obtain detailed information within the response interval during the process of uncertainty propagation for some problems with small samples. First, an ellipsoid possibility model (EPM) including multiple ellipsoid domains is established using the estimated covariance matrix and predefined scaling rule, which takes into consideration the distribution characteristic of the sample points. The possibilities of all established ellipsoid domains can be calculated by transforming the oblique ellipsoid space into the standard ellipsoid space. Then, two uncertainty propagation approaches based on the established EPM are presented to obtain the interval of an uncertain response and the possibility on all the possible values of the system response, respectively. Finally, the calculation results based on the proposed uncertainty analysis method are in good agreement with those of the MCS in three numerical examples and two engineering applications. The comparison results with three traditional non-probabilistic methods demonstrate that the proposed non-probabilistic uncertainty analysis method is accurate and practical.

Processing Aleatory and Epistemic Uncertainties in Experimental Data From Sparse Replicate Tests of Stochastic Systems for Real-Space Model Validation

Abstract This paper presents a practical methodology for propagating and processing uncertainties associated with random measurement and estimation errors (that vary from test-to-test) and systematic measurement and estimation errors (uncertain but similar from test-to-test) in inputs and outputs of replicate tests to characterize response variability of stochastically varying test units. Also treated are test condition control variability from test-to-test and sampling uncertainty due to limited numbers of replicate tests. These aleatory variabilities and epistemic uncertainties result in uncertainty on computed statistics of output response quantities. The methodology was developed in the context of processing experimental data for “real-space” (RS) model validation comparisons against model-predicted statistics and uncertainty thereof. The methodology is flexible and sufficient for many types of experimental and data uncertainty, offering the most extensive data uncertainty quantification (UQ) treatment of any model validation method the authors are aware of. It handles both interval and probabilistic uncertainty descriptions and can be performed with relatively little computational cost through use of simple and effective dimension- and order-adaptive polynomial response surfaces in a Monte Carlo (MC) uncertainty propagation approach. A key feature of the progressively upgraded response surfaces is that they enable estimation of propagation error contributed by the surrogate model. Sensitivity analysis of the relative contributions of the various uncertainty sources to the total uncertainty of statistical estimates is also presented. The methodologies are demonstrated on real experimental validation data involving all the mentioned sources and types of error and uncertainty in five replicate tests of pressure vessels heated and pressurized to failure. Simple spreadsheet procedures are used for all processing operations.

Tailored mixture properties for accurate laminar flame speed measurement from spherically expanding flames: Application to H[formula omitted]/O[formula omitted]/N[formula omitted]/He mixtures

The uncertainty on the laminar flame speed extracted from spherically expanding flames can be minimized by using large flame radius data for the extrapolation to zero stretch-rate. However, at large radii, the hydrodynamic and thermo-diffusive instabilities would induce the wrinkling of the flame surface and limit the range of usable data. In the present study, we have employed the flame stability theory of Matalon to tailor the properties of the initial mixture so that onset of cellular flame would occur at a pre-determined, large radius. This approach was employed to measure the laminar flame speeds of H2/O2/N2/He mixtures with equivalence ratios from 0.6 to 2.0, at pressures of 50/80/100 kPa and initial temperature of 300 K. For most experiments we performed, the uncertainty related to the extrapolation to zero stretch-rate (performed with the linear curvature model) was below 2% as shown by the position of the data points in the (Lb/Rf,U, Lb/Rf,L) plan, where Lb is the burned Markstein length; and Rf,L and Rf,U are the flame radii at the lower and upper bounds of the extrapolation range. Unsteady 1-D simulations using four chemical mechanisms were performed to show that unstretched laminar flame speed can be well-characterized with relative error below 10% for most conditions. The flame dynamical response to stretch rate could be captured by the mechanisms only under some conditions. Further analyses on critical flame radius were carried out: (i) the dominant parameters were identified based on a local sensitivity analysis; and (ii) the uncertainty on the predicted critical radius was determined through an uncertainty propagation approach. Uncertainty in critical flame radius calculation comes mostly from the uncertainty on fundamental transport and kinetic data. The present work indicates that although the stability theory of Matalon provides a well defined framework to tailor the mixture properties for improved flame speed measurement, the inaccuracy of some of the required parameters can result in significantly over-estimated critical radius for cellular flame onset which compromises the accuracy of the tailoring procedure.

An Effective Approach to Acquire the Impurity Diffusion Coefficients in Binary Alloys with Quantified Uncertainties

In this paper, we started from the composition-dependent interdiffusion coefficients with quantified uncertainties in binary alloys by integrating the Matano-based method, distribution functions, and uncertainty propagation approach. After carefully defining the numerically stable region for the interdiffusion coefficients, the suitable pre-set functions were screened to achieve the reasonable fit to the D-c and μ-c data according to the Akaike information criterion. With the fitted D-c and μ-c curves, the impurity diffusion coefficients with uncertainties can be directly determined. Benchmark tests in five hypothetical binary systems with different preset D-c relations were then utilized to validate the presently effective approach, followed by practical applications in five real cases, i.e., fcc Ni-Co, fcc Cu-Al, fcc Pt-Ni, hcp Mg-Zn, and bcc Ti-V alloys. The impurity diffusion coefficients with uncertainties derived by the presently effective approach were found to be in excellent agreement with the data by tracer experiments, indicating that this effective approach can serve as a standard one for acquiring the high-quality impurity diffusion coefficients in binary alloys with quantified uncertainties, especially for the noble metals and the cases without suitable radioactive tracer isotopes.

Discrete-Direct Model Calibration and Uncertainty Propagation Method Confirmed on Multi-Parameter Plasticity Model Calibrated to Sparse Random Field Data

Abstract A discrete direct (DD) model calibration and uncertainty propagation approach is explained and demonstrated on a 4-parameter Johnson-Cook (J-C) strain-rate dependent material strength model for an aluminum alloy. The methodology's performance is characterized in many trials involving four random realizations of strain-rate dependent material-test data curves per trial, drawn from a large synthetic population. The J-C model is calibrated to particular combinations of the data curves to obtain calibration parameter sets which are then propagated to “Can Crush” structural model predictions to produce samples of predicted response variability. These are processed with appropriate sparse-sample uncertainty quantification (UQ) methods to estimate various statistics of response with an appropriate level of conservatism. This is tested on 16 output quantities (von Mises stresses and equivalent plastic strains) and it is shown that important statistics of the true variabilities of the 16 quantities are bounded with a high success rate that is reasonably predictable and controllable. The DD approach has several advantages over other calibration-UQ approaches like Bayesian inference for capturing and utilizing the information obtained from typically small numbers of replicate experiments in model calibration situations—especially when sparse replicate functional data are involved like force–displacement curves from material tests. The DD methodology is straightforward and efficient for calibration and propagation problems involving aleatory and epistemic uncertainties in calibration experiments, models, and procedures.

Probabilistic observation model correction using non-Gaussian belief fusion

This paper presents a framework for state estimation which tolerates uncertainty in observation model parameters by (1) incorporating this uncertainty in state observation, and (2) correcting model parameters to improve future state observations. The first objective is met by an uncertainty propagation approach, while the second is achieved by gradient-descent optimization. The novel framework allows state estimates to be represented by non-Gaussian probability distribution functions. By correcting observation model parameters, estimation performance is enhanced since the accuracy of observations is increased. Monte Carlo simulation experiments validate the efficacy of the proposed approach in comparison with conventional estimation techniques, showing that as model parameters converge to ground-truth over time, state estimation correspondingly improves when compared to a static model estimate. Because observation models cannot be known with perfect accuracy and existing approaches do not address parametric uncertainties in non-Gaussian estimation, this work has both novelty and usefulness in most state estimation contexts.

Impact Assessment in the Process of Propagating Climate Change Uncertainties into Building Energy Use

Buildings are subject to significant stresses due to climate change and design strategies for climate resilient buildings are rife with uncertainties which could make interpreting energy use distributions difficult and questionable. This study intends to enhance a robust and credible estimate of the uncertainties and interpretations of building energy performance under climate change. A four-step climate uncertainty propagation approach which propagates downscaled future weather file uncertainties into building energy use is examined. The four-step approach integrates dynamic building simulation, fitting a distribution to average annual weather variables, regression model (between average annual weather variables and energy use) and random sampling. The impact of fitting different distributions to the weather variable (such as Normal, Beta, Weibull, etc.) and regression models (Multiple Linear and Principal Component Regression) of the uncertainty propagation method on cooling and heating energy use distribution for a sample reference office building is evaluated. Results show selecting a full principal component regression model following a best-fit distribution for each principal component of the weather variables can reduce the variation of the output energy distribution compared to simulated data. The results offer a way of understanding compound building energy use distributions and parsing the uncertain nature of climate projections.

Statistical Taylor series expansion: An approach for epistemic uncertainty propagation in Markov reliability models

In this paper we develop a new Taylor series expansion method for computing model output metrics under epistemic uncertainty in the model input parameters. Specifically, we compute the expected value and the variance of the stationary distribution associated with Markov reliability models. In the multi-parameter case, our approach allows to analyze the impact of correlation between the uncertainty on the individual parameters the model output metric. In addition, we also approximate true risk by using the Chebyshev’ inequality. Numerical results are presented and compared to the corresponding Monte Carlo simulations ones.

Gaussian Process Learning-Based Probabilistic Optimal Power Flow

In this letter, we present a novel Gaussian Process Learning-based Probabilistic Optimal Power Flow (GP-POPF) for solving POPF under renewable and load uncertainties of arbitrary distribution. The proposed method relies on a non-parametric Bayesian inference-based uncertainty propagation approach, called Gaussian Process (GP). We also suggest a new type of sensitivity called Subspace-wise Sensitivity, using observations on the interpretability of GP-POPF hyperparameters. The simulation results on 14-bus and 30-bus systems show that the proposed method provides reasonably accurate solutions when compared with Monte-Carlo Simulations (MCS) solutions at different levels of uncertain renewable penetration and load uncertainties. The proposed method requires a lesser number of samples and elapsed time. The non-parametric nature of the proposed method is manifested by testing uncertain injections that follow various distributions in the 118-bus system. The low error value results verify the proposed method's capability of working with different types of input uncertainty distributions.

Selecting Technologies in Aircraft Conceptual Design Using Probabilistic Inversion

Technology forecasting is an essential starting point for conceptual design of any complex engineering system. In fact, many research projects are focused on developing a small set of promising technologies to a suitable readiness level. However, selecting a set of technologies from a larger pool is a nontrivial task, opposed by uncertainty and subjective tradeoffs. This paper proposes a probabilistic method to represent technologies and quantify their effects, while accounting for uncertainty. Using probabilistic inversion, technologies can be selected from a larger set to meet a certain combination of requirements. Several test cases illustrate the method and how it may be used in conceptual design projects. It is concluded that probabilistic inversion enables answering technology development and selection queries, which would be challenging to answer with traditional deterministic approaches, or purely forward uncertainty propagation approaches.

Uncertainty assessment for 3D geologic modeling of fault zones based on geologic inputs and prior knowledge

Abstract. Characterizing the zone of damaged and altered rock surrounding a fault surface is highly relevant to geotechnical and geo-environmental engineering works in the subsurface. Evaluating the uncertainty associated with 3D geologic modeling of these fault zones is made possible using the popular and flexible input-based uncertainty propagation approach to geologic model uncertainty assessment – termed probabilistic geomodeling. To satisfy the automation requirements of probabilistic geomodeling while still preserving the key geometry of fault zones in the subsurface, a clear and straightforward modeling approach is developed based on four geologic inputs used in implicit geologic modeling algorithms (surface trace, structural orientation, vertical termination depth and fault zone thickness). The rationale applied to identifying and characterizing the various sources of uncertainty affecting each input are explored and provided using open-source codes. In considering these sources of uncertainty, a novel model formulation is implemented using prior geologic knowledge (i.e., empirical and theoretical relationships) to parameterize modeling inputs which are typically subjectively interpreted by the modeler (e.g., vertical termination depth of fault zones). Additionally, the application of anisotropic spherical distributions to modeling disparate levels of information available regarding a fault zone's dip azimuth and dip angle is demonstrated, providing improved control over the structural orientation uncertainty envelope. The probabilistic geomodeling approach developed is applied to a simple fault zone geologic model built from historically available geologic mapping data, allowing for a visual comparison of the independent contributions of each modeling input on the combined model uncertainty, revealing that vertical termination depth and structural orientation uncertainty dominate model uncertainty at depth, while surface trace uncertainty dominates model uncertainty near the ground surface. The method is also successfully applied to a more complex fault network model containing intersecting major and minor fault zones. The impacts of the model parameterization choices, the fault zone modeling approach and the effects of fault zone interactions on the final geologic model uncertainty assessment are discussed.