Uncertainty Propagation In Models Research Articles

A Multi-Fidelity Uncertainty Propagation Model for Multi-Dimensional Correlated Flow Field Responses

Given the randomness inherent in fluid dynamics problems and limitations in human cognition, Computational Fluid Dynamics (CFD) modeling and simulation are afflicted with non-negligible uncertainties, casting doubts on the credibility of CFD. Scientifically and rigorously quantifying the uncertainty of CFD is paramount for assessing its credibility and informing engineering decisions. In order to quantify the uncertainty of multidimensional flow field responses stemming from uncertain model parameters, this paper proposes a method based on Gappy Proper Orthogonal Decomposition (POD) for supplementing high-fidelity flow field data within a framework that leverages POD and surrogate models. This approach enables the generation of corresponding high-fidelity flow fields from low-fidelity ones, significantly reducing the cost of high-fidelity flow field computation in uncertainty propagation modeling. Through an analysis of the impact of uncertainty in the coefficients of the Spalart–Allmaras (SA) turbulence model on the distribution of wall friction coefficients for the NACA0012 airfoil and pressure coefficients for the M6 wing, the proposed multi-fidelity modeling approach is demonstrated to offer significant advancements in both accuracy and efficiency compared to single-fidelity methods, providing a robust and efficient prediction model for large-scale random sampling.

Aerodynamic Uncertainty Quantification of a Low-Pressure Turbine Cascade by an Adaptive Gaussian Process

Stochastic variations of the operation conditions and the resultant variations of the aerodynamic performance in Low-Pressure Turbine (LPT) can often be found. This paper studies the aerodynamic performance impact of the uncertain variations of flow parameters, including inlet total pressure, inlet flow angle, and turbulence intensity for an LPT cascade. Flow simulations by solving the Reynolds-averaged Navier–Stokes equations, the SST turbulence model, and γ−Re˜θt transition model equations are first carried out. Then, a Gaussian process (GP) based on an adaptive sampling technique is introduced. The accuracy of adaptive GP (ADGP) is proven to be high through a function experiment. Using ADGP, the uncertainty propagation models between the performance parameters, including total pressure-loss coefficient, outlet flow angle, Zweifel number, and the uncertain inlet flow parameters, are established. Finally, using the propagation models, uncertainty quantifications of the performance changes are conducted. The results demonstrate that the total pressure-loss coefficient and Zweifel number are sensitive to uncertainties, while the outlet flow angle is almost insensitive. Statistical analysis of the flow field by direct Monte Carlo simulation (MCS) shows that flow transition on the suction side and viscous shear stress are rather sensitive to uncertainties. Moreover, Sobol sensitivity analysis is carried out to specify the influence of each uncertainty on the performance changes in the turbine cascade.

Multi-Step Gaussian Regression Prediction in Dynamic Systems: A Case Study in Vehicle Lateral Control

Abstract In this paper, we focus on developing a multi-step uncertainty propagation method for systems with state- and control-dependent uncertainties. System uncertainty creates a mismatch between the actual system and its control-oriented model. Often, these uncertainties are state- and control-dependent, such as modeling error. This uncertainty propagates over time and results in significant errors over a given time horizon, which can disrupt the operation of safety-critical systems. Stochastic predictive control methods can ensure that the system stays within the safe region with a given probability, but requires prediction of the future state distributions of the system over the horizon. Predicting the future state distribution of systems with state- and control-dependent uncertainty is a difficult task. Existing methods only focus on modeling the current or one-step uncertainty, while the uncertainty propagation model over a horizon is generally over-approximated. Hence, we present a multi-step Gaussian process regression method to learn the uncertainty propagation model for systems with state- and control-dependent uncertainties. We also perform a case study on vehicle lateral control problems, where we learn the vehicle’s error propagation model during lane changes. Simulation results show the efficacy of our proposed method.

Microstructure design and optimization of high-sensitivity interdigital capacitive humidity sensor based on uncertainty analysis

The microstructure of the interdigital capacitive humidity sensor plays a critical role in achieving high sensitivity in measurements. Performing a quantitative evaluation to assess the influence of structural parameters on sensor sensitivity is vital for improving the overall performance of the sensor. In this paper, a theoretical mathematical model and a structural uncertainty propagation model are developed for the interdigitated capacitive humidity sensor, considering the uncertain errors that occur during the processing and measurement processes. A quantitative analysis is conducted to investigate the impact of each structural component and its corresponding uncertainty on the output sensitivity. Moreover, by integrating the two constructed models, the paper formulates an optimal equation for sensor performance that mitigates the impact of structural errors on the system while improving sensor sensitivity. Researchers have the flexibility to adjust sensor material, environmental conditions, structural components, and other parameter variables within the model, according to specific application requirements, to achieve the optimal design scheme for a sensor with specific materials and structure. The proposed method and model presented in this article have undergone verification using finite element physical simulation. The sensitivity model achieves an accuracy rate of 94.45%, whereas the uncertainty model achieves an accuracy rate of 81.96%. Additionally, the model reduces the calculation time to 1/550th and the required computational resources to 1/5th of the simulation, thereby greatly improving the efficiency of sensor optimization design.

Orbital Uncertainty Propagation Based on Adaptive Gaussian Mixture Model under Generalized Equinoctial Orbital Elements

The number of resident space objects (RSOs) has been steadily increasing over time, posing significant risks to the safe operation of on-orbit assets. The accurate prediction of potential collision events and implementation of effective and nonredundant avoidance maneuvers require the precise estimation of the orbit positions of objects of interest and propagation of their associated uncertainties. Previous research mainly focuses on striking a balance between accurate propagation and efficient computation. A recently proposed approach that integrates uncertainty propagation with different coordinate representations has the potential to achieve such a balance. This paper proposes combining the generalized equinoctial orbital elements (GEqOE) representation with an adaptive Gaussian mixture model (GMM) for uncertainty propagation. Specifically, we implement a reformulation for the orbital dynamics so that the underlying state and the moment feature of the GMM are propagated under the GEqOE coordinates. Starting from an initial Gaussian probability distribution function (PDF), the algorithm iteratively propagates the uncertainty distribution using a detection-splitting module. A differential entropy-based nonlinear detector and a splitting library are utilized to adjust the number of GMM components dynamically. Component splitting is triggered when a predefined threshold of differential entropy is violated, generating several GMM components. The final probability density function (PDF) is obtained by a weighted summation of the component distributions at the target time. Benefiting from the nonlinearity reduction caused by the GEqOE representation, the number of triggered events largely decreases, causing the necessary number of components to maintain uncertainty realism also to decrease, which enables the proposed approach to achieve good performance with much more efficiency. As demonstrated by the results of propagation in three scenarios with different degrees of complexity, compared with the Cartesian-based approach, the proposed approach achieves comparable accuracy to the Monte Carlo method while largely reducing the number of components generated during propagation. Our results confirm that a judicious choice of coordinate representation can significantly improve the performance of uncertainty propagation methods in terms of accuracy and computational efficiency.

Surrogate modeling for high dimensional uncertainty propagation via deep kernel polynomial chaos expansion

In this paper, deep kernel polynomial chaos expansion (DKPCE) is proposed as a surrogate model for high dimensional uncertainty propagation. Firstly, deep neural network (DNN) and polynomial chaos expansion (PCE) are connected to create a novel network model, the input dimensionality of PCE layer can thus be controlled by restricting the number of neurons in the feature layer. Then, the back-propagation algorithm is employed for computing all the parameters of DKPCE, the dimension reduction and modeling process of DKPCE are thus executed simultaneously. During the modeling process, a data driven method is first implemented for computing the orthogonal polynomial bases within the PCE layer in the forward propagation step, and the partial derivatives for the coefficients of orthogonal polynomial bases are computed first in the back-propagation step. After constructing DKPCE, the coefficients of PCE layer can be utilized to compute the statistical characteristics of system response. Finally, several numerical examples are utilized for validating the effectiveness of DKPCE.

Using UAV Time Series to Estimate Landslides’ Kinematics Uncertainties, Case Study: Chirlești Earthflow, Romania

This paper presents a methodology for evaluating the uncertainties caused by the misalignment between two digital elevation models in estimating landslide kinematics. The study focuses on the earthflow near the town of Chirlești, located in the Bend Subcarpathians, Buzău County, Romania, which poses a high risk of blocking the DN10 national road. Four flights were conducted between 2018 and 2022 using a DJI Phantom 4 UAV using the same flight plan. Monte Carlo simulations were used to model uncertainty propagation of the DEM misalignments in the landslide kinematics analysis. The simulations were applied to the accuracy values of the structure from a motion process used to generate the digital elevation models. The degree of uncertainty was assessed using the displaced material’s total amount in conjunction with the spatial correlation of the displaced material between two consecutive flights. The results revealed that the increase in the RMS values did not determine an increase in the displaced earth between two UAV flights. Instead, combining the RMS values and the correlation coefficient clearly indicated the correspondence between the spatial distribution of the displaced earth material and the overall changes reported between the two UAV flights. An RMS value of up to 1 unit associated with a correlation coefficient of 0.95 could be considered the maximum allowable error for estimating landslide kinematics across space and time. The current methodology is reliable when studying slow-movement landslides and when using short intervals between UAV flights. For rapid movements or significant terrain changes, such as translational and rotational landslides, careful analysis of the correlation coefficient in conjunction with the RMS values is recommended.

A comparison of exposure uncertainty propagation models used in epidemiological studies

Abstract In the field of ecological epidemiological studies, accurate estimation of the long-term health effects of air pollution is crucial. Two-step models that involve exposure assessment and health effects estimation are often used for this purpose. However, the accuracy of exposure assessment is uncertain and may not accurately reflect true exposure. Despite several proposed methods to manage this uncertainty, the impact of different approaches on air pollution inferences remains uncertain. In this study, we conduct a simulation study to compare the inferences of air pollution impact from various exposure uncertainty propagation models while investigating their health effects. The results suggest that the Without-uncertainty model and the Multi-set method are preferable to the prior method and pollution-health jointly model (without cut-off). Moreover, a case study further reinforces the evidence of a link between mortality and PM2.5 concentrations, showing that an increase of 1 μg⋅m−3 in PM2.5 concentration is likely to increase all-cause deaths in Scotland by 4.51% [95% credible interval (CI), 3.42%, 5.49%] to 7.51% (95% CI, 6.28%, 8.80%). These findings have important implications for policymakers and public health officials seeking to mitigate the harmful effects of air pollution.

Collision risk quantification for pairs of recorded aircraft trajectories

AbstractComplex domestic airspace requires collision risk models and monitoring tools suitable for arbitrary aircraft trajectories. This paper presents a new mathematically based collision risk approach that extends the International Civil Aviation Organisation (ICAO) models to full aircraft encounters based on real trajectory data. A new continuous time intervention model is presented, along with a position uncertainty propagation model that better reflects aircraft behaviour and allows generalisation to all trajectories to eliminate degenerate cases. The proposed risk model is computationally efficient compared to the models it is based on and can be applied to large-scale trajectory data. The utility of the model is demonstrated through a series of case studies using real aircraft trajectories.

Lane change for self-driving in highly dense traffic using motion based uncertainty propagation

This paper presents a design of lane change decision and control algorithm in highly dense traffic situation for self-driving vehicles using motion-based adaptive uncertainty propagation and Stochastic Model Predictive Control (SMPC). Essential ideas of the proposed algorithm are introduced; i) an optimal motion in a current situation with multiple criteria decision making (MCDM), ii) four steps to change lane successfully in the dense traffic situation which is modeled as a simple acceleration model based on real driving data, iii) motion-based adaptive uncertainty propagation to consider a model error. The proposed algorithm has been evaluated via simulation studies in MATLAB/Simulink and CARSIM. The simulation results show the effectiveness of the proposed algorithm and its performance for changing lane in the highly-dense traffic situation.

A novel linear uncertainty propagation method for nonlinear dynamics with interval process

Interval process is a preferable model for time-varying uncertainty propagation of dynamic systems when only the range of uncertainties can be obtained. However, for nonlinear systems, except for Monte Carlo (MC) simulation, there are still few efficient uncertainty propagation methods under the interval process model. This paper develops a non-intrusive and semi-analytical uncertainty propagation method, named the “convex model linearization method (CMLM),” by constructing a linearization formulation of a nonlinear system in a non-probabilistic sense. First, the criterion to evaluate the difference between the original system and the linearization formulation is derived, represented by the discrepancy of middle point, radius and correlations of response. By minimizing these three parameters, the coefficients of linear equations will be optimized to obtain the linearization formulation of the original system. Then, analytical equations are built to calculate uncertainty response under the interval process, without time-consuming analysis of the original system. To further improve the efficiency of the linearization process, Chebyshev polynomial is introduced to approximate the nonlinear dynamic analysis. Two numerical examples of duffing oscillators and vehicle rides are set to test the proposed CMLM. Compared to the MC method, with comparable uncertainty response precision, the CMLM just needs 1–10% times of dynamic analyses of the nonlinear system. Furthermore, a practical launch vehicle ascent trajectory problem with black-box dynamics is solved by, respectively, the CMLM and MC method. The results verify the capacity of the CMLM to deal with black-box problems and show that the CMLM performs better in terms of accuracy, efficiency and robustness.

Data-Driven Surrogate Model and Admissible-Region-Based Orbital Uncertainty Propagation

With fast-growing space objects but limited sensor resource, optical-sensor-based short-arc uncertainty propagation is necessary for space object tracking. Recently, the admissible-region-based orbit initialization method has been shown to be effective for uncertainty propagation when the number of observations is very small. However, it is challenging to describe the uncertainty distribution in the admissible region. In this paper, a data-driven kriging method is combined with the admissible region to build an accurate surrogate model for orbital uncertainty propagation instead of using the Monte Carlo method directly. In addition, an active learning strategy is utilized to choose sample data from the admissible region more efficiently for incrementally building the surrogate model. The main advantage of the kriging method is that no a priori probability density function needs to be assumed for the uncertainty distribution. Two numerical examples including a low Earth orbit and a geosynchronous Earth orbit propagation are used to demonstrate the effectiveness of the kriging model. It compares favorably to the polynomial chaos-based uncertainty propagation in terms of accuracy. It is also shown that the proposed method can achieve close performance to the Monte Carlo results but with much less sampling points and higher computational efficiency.

Uncertainty Propagation and Assessment of Five-Axis On-Machine Measurement Error

Abstract As an important technology of adaptive machining, on-machine measurement (OMM) is always used to measure the shape of the to-be-cut surface. Due to the multi-source errors in an OMM system, the confidence of the measurement accuracy becomes particularly important before planning the next cutting procedure. OMM is implemented by multi-components involving machine tools, fixture, and on-machine probe. Each component introduces uncertainty of error into the measurement result. The positioning error, alignment error, and eccentricity error are investigated in terms of the expectation, uncertainty, and probability density function. By the kinematics of the components participated in the measurement, these errors are accumulated to affect the measurement results. Thus, the uncertainty of the final measurement result is determined by the accumulation of uncertainty of error sources. In order to evaluate the uncertainty of measurement error, an uncertainty propagation model involving positioning error of five-axis machine tools, alignment error of workpiece, and eccentricity error of the stylus tip is established in terms of the kinematic chain of OMM. Then, the uncertainty of the measurement result in vector form is obtained. Finally, the simulation and the experiment about the impeller are employed to validate the effectiveness and repeatability of the uncertainty propagation model for five-axis OMM.

Analysis of uncertainty using stochastic flow simulation of LCM at constant thickness

Liquid Composite Moulding (LCM) process design requires that the resin impregnation minimizes both mould filling time, as well as the probability of occurrence of dry, unsaturated zones. This can be challenging, as the degree of variability present in material properties and the manufacturing process itself, can be substantial. The main purpose of this study is to understand and quantify how the different sources of variability, present in LCM, may affect the performance of the mould filling process, creating defects and to quantify how lower scale interactions may affect macro-scale flow. In order to transfer the meso-scale stochastic behaviour towards a complete macro-scale model, an uncertainty propagation model is proposed, whereby local statistical properties are used to predict the upscaled flow behaviour. This is done by means of flow simulation, using stochastic models for the input process variables. Local fabric distortions, combined with fabric permeability, pressure and race-tracking serving as inputs to estimate the joint probability distribution that characterizes the global uncertainty in flow performance. Preliminary uncertainty analyses investigating individual sources of variability are available in the literature, however, a complete framework combining the different variables into a mould filling scenario, was not found. Therefore, this study tries to develop a new multi-variable aggregating concept that is applied to mould filling simulation. The individual contribution of each source of variability used in this work is quantified, providing a better understanding of the reliability of alternative process designs.

Analysis of uncertainty using stochastic flow simulation of LCM at constant thickness

Liquid Composite Moulding (LCM) process design requires that the resin impregnation minimizes both mould filling time, as well as the probability of occurrence of dry, unsaturated zones. This can be challenging, as the degree of variability present in material properties and the manufacturing process itself, can be substantial. The main purpose of this study is to understand and quantify how the different sources of variability, present in LCM, may affect the performance of the mould filling process, creating defects and to quantify how lower scale interactions may affect macro-scale flow. In order to transfer the meso-scale stochastic behaviour towards a complete macro-scale model, an uncertainty propagation model is proposed, whereby local statistical properties are used to predict the upscaled flow behaviour. This is done by means of flow simulation, using stochastic models for the input process variables. Local fabric distortions, combined with fabric permeability, pressure and race-tracking serving as inputs to estimate the joint probability distribution that characterizes the global uncertainty in flow performance. Preliminary uncertainty analyses investigating individual sources of variability are available in the literature, however, a complete framework combining the different variables into a mould filling scenario, was not found. Therefore, this study tries to develop a new multi-variable aggregating concept that is applied to mould filling simulation. The individual contribution of each source of variability used in this work is quantified, providing a better understanding of the reliability of alternative process designs.

Calibration of cellular automata urban growth models from urban genesis onwards - a novel application of Markov chain Monte Carlo approximate Bayesian computation

Cellular Automata (CA) models are widely used to study spatial dynamics of urban growth and evolving patterns of land use. One complication across CA approaches is the relatively short period of data available for calibration, providing sparse information on patterns of change and presenting problematic signal-to-noise ratios. To overcome the problem of short-term calibration, this study investigates a novel approach in which the model is calibrated based on the urban morphological patterns that emerge from a simulation starting from urban genesis, i.e., a land cover map completely void of urban land. The application of the model uses the calibrated parameters to simulate urban growth forward in time from a known urban configuration.This approach to calibration is embedded in a new framework for the calibration and validation of a Constrained Cellular Automata (CCA) model of urban growth. The investigated model uses just four parameters to reflect processes of spatial agglomeration and preservation of scarce non-urban land at multiple spatial scales and makes no use of ancillary layers such as zoning, accessibility, and physical suitability. As there are no anchor points that guide urban growth to specific locations, the parameter estimation uses a goodness-of-fit (GOF) measure that compares the built density distribution inspired by the literature on fractal urban form. The model calibration is a novel application of Markov Chain Monte Carlo Approximate Bayesian Computation (MCMC-ABC). This method provides an empirical distribution of parameter values that reflects model uncertainty. The validation uses multiple samples from the estimated parameters to quantify the propagation of model uncertainty to the validation measures.The framework is applied to two UK towns (Oxford and Swindon). The results, including cross-application of parameters, show that the models effectively capture the different urban growth patterns of both towns. For Oxford, the CCA correctly produces the pattern of scattered growth in the periphery, and for Swindon, the pattern of compact, concentric growth. The ability to identify different modes of growth has both a theoretical and practical significance. Existing land use patterns can be an important indicator of future trajectories. Planners can be provided with insight in alternative future trajectories, available decision space, and the cumulative effect of parcel-by-parcel planning decisions.

Data-driven modal surrogate model for frequency response uncertainty propagation

A method is developed for propagation of model parameter uncertainties into frequency response functions based on a modal representation of the equations of motion. Individual local surrogate models of the eigenfrequencies and residue matrix elements for each mode are trained to build a global surrogate model. The computational cost of the global surrogate model is reduced in three steps. First, modes outside the range of interest, necessary to describe the in-band frequency response, are approximated with few residual modes. Secondly, the dimension of the residue matrices for each mode is reduced using principal component analysis. Lastly, multiple surrogate model structures are employed in a mixture. Cheap second-order multivariate polynomial models and more expensive Gaussian process models with different kernels are used to model the modal data. Leave-one-out cross-validation is used for model selection of the local surrogate models. The approximations introduced allow the method to be used for modally dense models at a small computational cost, without sacrificing the global surrogate model’s ability to capture mode veering and crossing phenomena. The method is compared to a Monte Carlo based approach and verified on one industrial-sized component and on one assembly of two car components.

Multimodal ellipsoid model for non-probabilistic structural uncertainty quantification and propagation

The traditional ellipsoid convex set is a kind of basic non-probabilistic model to measure uncertainties. However, it is difficult or inaccurate to quantify the uncertainties of variables with multimodal distributed samples. In this paper, a more generalized non-probabilistic ellipsoid model named multimodal ellipsoid model is proposed to effectively deal with the multimodal distributed samples. The samples with one or more similar properties are clustered together, and the principal directions of the samples and characteristic matrix are appropriately found through the Gaussian mixture model. Then, the multimodal ellipsoid model can be constructed by using the elliptical contour features of the Gaussian model to measure the uncertainties of variables. The proposed multimodal ellipsoid model can not only establish traditional ellipsoid model, but also establish multi-ellipsoid model for uncertain variables with multimodal samples. Furthermore, combining with the multimodal ellipsoid model and performance measure approach, the uncertain propagation results of system are obtained accurately. Three numerical examples and one engineering application are provided to demonstrate the effectiveness and accuracy of the proposed multimodal ellipsoid model.

Active Sampling Based Polynomial-Chaos–Kriging Model for Orbital Uncertainty Propagation

Propagating uncertainties usually requires repeated evaluation of the subject model for a large number of different input parameters. Surrogate models have been widely used to avoid this computationally intensive uncertainty propagation by replacing the original model by an easy-to-evaluate function model. In this Paper, the polynomial chaos based Kriging is used as such a surrogate model for orbital uncertainty propagation. The polynomial chaos represents the global trend of the uncertainty distribution, while the Kriging describes the local variations. Such a combination can provide a more precise surrogate model than individual polynomial chaos or ordinary Kriging representation. To further enhance the accuracy, a new active sampling strategy is proposed to incrementally build and improve the polynomial chaos based Kriging model. This new modeling scheme only requires a small number of sampling points while achieving close performance to the Monte Carlo based propagation. It is also more accurate than the random sampling based Kriging model. Three orbital uncertainty propagation examples are used to demonstrate the effectiveness of the proposed surrogate model.

Estimation of uncertainty in transmutation rates of LLFPs in a fast reactor transmutation system via an estimation of the cross-section covariances

ABSTRACT A new concept of transmutation system for six long-lived fission products (LLFPs: 79Se, 93Zr, 99Tc, 107Pd, 129I, and 135Cs) consisting of three small-size fast reactor cores was proposed. In order to establish a method to estimate the effects of cross-section uncertainties on the transmutation rates of the LLFPs, cross section covariances for the six LLFPs were estimated using a T6 system which includes resonance and statistical models via propagation of model uncertainties, and the uncertainties of transmutation rates and support factors of the six LLFPs in the three fast reactors were estimated based on a Total Monte Carlo (TMC) method. It was made clear that some of the transmutation rates of these LLFPs have large uncertainties indicating that the accuracy of cross sections for them is not enough, and hence the strategy for efficient transmutation may have to be reconstructed. It is concluded that the T6 system and the TMC method used in this study are quite effective not only to estimate the uncertainties of cross sections but also to assess the effect of error propagation throughout the series of calculations conducting in this study.