Inverse Uncertainty Quantification Research Articles

Development of higher order deterministic approaches of forward and inverse uncertainty quantification for nonlinear engineering systems

This paper introduces an innovative numerical scheme that may accurately quantify the parameter and response distributions with minimal computational costs, with specific applications to the numerical computation of multiphase flows under various fluid conditions. The probabilistic approach of uncertainty quantification, e.g., the Monte Carlo simulation, although known to be effective in estimating the parameter/response distribution for highly nonlinear problems, is usually not applicable to multi-physics and multi-scale integrated systems due to its high computational cost. The objective of this study is first, to reduce the computing demand for complex physical system calculations and, at the same time, to accurately predict actual probability distributions by developing a higher order deterministic approach to uncertainty quantification. To achieve this, instead of executing simulation codes multiple times to estimate the probability distribution, the first to fourth moments of the probability density function are derived directly by employing a second order Taylor expansion form of the responses along with Bayesian statistics. This characterizes the parameter and response distributions computed by inverse and forward uncertainty quantification based on their mean value, mode, covariance, skewness, and kurtosis. The parameter distribution is estimated by the numerically calculated higher moments of the parameters with a posteriori probability density function being derived by Bayesian inference. The parameter distribution is then mapped into the second order approximated response function to compute the characteristics of the response distribution through both analytical calculation and numerical approximation of the moments of the responses. The proposed approach is numerically demonstrated through simulations of two benchmark problems: MIT pressurizer and FEBA tests. The prediction of the responses is improved by inverse uncertainty quantification as numerical calculations with the a posteriori model parameters are in better agreement with the measured data over a wide range of transient periods. Additionally, the response distribution estimated by the proposed approach of forward uncertainty quantification accurately predicts the distribution computed by the Monte Carlo simulation.

Global Sensitivity Analysis for Segmented Inverse Uncertainty Quantification in the Safety Analysis of Nuclear Power Plants

Within the Best Estimate Plus Uncertainty framework for the safety analysis of Nuclear Power Plants, the quantification of the uncertainties affecting the Thermal-Hydraulics (T-H) codes used is crucial. For this, Inverse Uncertainty Quantification (IUQ) methodologies are being developed for determining the probability density functions of relevant T-H codes input parameters, based on experimental data from Separate Effect Tests (SETs) experimental facilities. In practice, IUQ is challenged by the large range of variability of the experimental data in terms of Initial and Boundary Conditions (ICs & BCs), because the experimental campaigns are designed to cover the widest possible domain of conditions with the smallest number of experiments, so that same or similar ICs and BCs are seldomly repeated. To address this issue, we propose to use global sensitivity analysis, to tailor the IUQ on specific sub-regions described by segmented ICs & BCs domains. The methodology proposed is exemplified on two SETs, namely Sozzi-Sutherland and Super Moby Dick, whose experimental databases have been made available in the ATRIUM (Application Tests for Realization of Inverse Uncertainty quantification and validation Methodologies in thermal hydraulics) project promoted by the OECD/NEA/CSNI. The results obtained are superior to those of traditional IUQ methodologies for models highly sensitive to ICs & BCs.

Functional PCA and deep neural networks-based Bayesian inverse uncertainty quantification with transient experimental data

This work focuses on developing an inverse uncertainty quantification (IUQ) process for time-dependent responses, using dimensionality reduction by functional principal component analysis (PCA) and deep neural network (DNN)-based surrogate models. The demonstration is based on the IUQ of TRACE physical model parameters using the FEBA benchmark transient experimental data on peak cladding temperatures during reflooding. The quantity-of-interest (QoI) is time-dependent peak cladding temperature (PCT) profiles. Conventional PCA can hardly represent the data precisely due to the sudden temperature drop at the time of quenching. As a result, a functional alignment method is used to separate the phase and amplitude information in the PCT profiles before conventional PCA is applied for dimensionality reduction. The resulting PC scores are then used to build DNN-based surrogate models to significantly reduce the computational cost in Markov Chain Monte Carlo sampling, while the code/interpolation uncertainty is accounted for using Bayesian neural networks. We compared four IUQ processes with different dimensionality reduction methods and surrogate models. The proposed approach has demonstrated the best performance in reducing the dimensionality of the transient PCT profiles. This approach also produces the posterior distributions of the physical model parameters with which the model predictions have the best agreement with the experimental data.

Simultaneous Inverse Design and Uncertainty Quantification for Frequency Selective Rasorber with Tunable and Switchable Abilities by Bayesian Deep Learning

Simultaneous Inverse Design and Uncertainty Quantification for Frequency Selective Rasorber with Tunable and Switchable Abilities by Bayesian Deep Learning

Scalable Inverse Uncertainty Quantification by Hierarchical Bayesian Modeling and Variational Inference

Inverse Uncertainty Quantification (IUQ) has gained increasing attention in the field of nuclear engineering, especially nuclear thermal-hydraulics (TH), where it serves as an important tool for quantifying the uncertainties in the physical model parameters (PMPs) while making the model predictions consistent with the experimental data. In this paper, we present an extension to an existing Bayesian inference-based IUQ methodology by employing a hierarchical Bayesian model and variational inference (VI), and apply this novel framework to a real-world nuclear TH scenario. The proposed approach leverages a hierarchical model to encapsulate group-level behaviors inherent to the PMPs, thereby mitigating existing challenges posed by the high variability of PMPs under diverse experimental conditions and the potential overfitting issues due to unknown model discrepancies or outliers. To accommodate computational scalability and efficiency, we utilize VI to enable the framework to be used in applications with a large number of variables or datasets. The efficacy of the proposed method is evaluated against a previous study where a No-U-Turn-Sampler was used in a Bayesian hierarchical model. We illustrate the performance comparisons of the proposed framework through a synthetic data example and an applied case in nuclear TH. Our findings reveal that the presented approach not only delivers accurate and efficient IUQ without the need for manual tuning, but also offers a promising way for scaling to larger, more complex nuclear TH experimental datasets.

Sparse-grids uncertainty quantification of part-scale additive manufacturing processes

The present paper aims at applying uncertainty quantification methodologies to process simulations of powder bed fusion of metal. In particular, for a part-scale thermomechanical model of an Inconel 625 super-alloy beam, we study the uncertainties of three process parameters, namely the activation temperature, the powder convection coefficient and the gas convection coefficient. First, we perform a variance-based global sensitivity analysis to study how each uncertain parameter contributes to the variability of the beam displacements. The results allow us to conclude that the gas convection coefficient has little impact and can therefore be fixed to a constant value for subsequent studies. Then, we conduct an inverse uncertainty quantification analysis, based on a Bayesian approach on synthetic displacement data, to quantify the uncertainties of the two remaining parameters, namely the activation temperature and the powder convection coefficient. Finally, we use the results of the inverse uncertainty quantification analysis to perform a data-informed forward uncertainty quantification analysis of the residual strains. Crucially, we make use of surrogate models based on sparse grids to keep to a minimum the computational burden of every step of the uncertainty quantification analysis. The proposed uncertainty quantification workflow allows us to substantially ease the typical trial-and-error approach used to calibrate power bed fusion part-scale models, and to greatly reduce uncertainties on the numerical prediction of the residual strains. In particular, we demonstrate the possibility of using displacement measurements to obtain a data-informed probability density function of the residual strains, a quantity much more complex to measure than displacements.

Inverse uncertainty quantification of a mechanical model of arterial tissue with surrogate modelling

Disorders of coronary arteries lead to severe health problems such as atherosclerosis, angina, heart attack and even death. Considering the clinical significance of coronary arteries, an efficient computational model is a vital step towards tissue engineering, enhancing the research of coronary diseases and developing medical treatment and interventional tools. In this work, we applied inverse uncertainty quantification to a microscale agent-based arterial tissue model, a component of a three-dimensional multiscale in-stent restenosis model. Inverse uncertainty quantification was performed to calibrate the arterial tissue model to achieve a mechanical response in line with tissue experimental data. Bayesian calibration with a bias term correction was applied to reduce the uncertainty of unknown polynomial coefficients of the attractive force function and achieve agreement with the mechanical behaviour of arterial tissue based on the uniaxial strain tests. Due to the high computational costs of the model, a surrogate model based on the Gaussian process was developed to ensure the feasibility of the computations.

Fast uncertainty reduction of chemical kinetic models with complex spaces using hybrid response-surface networks

Response-surface (RS) surrogate approaches permit efficient inverse uncertainty quantification (UQ) of combustion kinetic models, wherein the uncertainty of reaction rates is reduced from observed targets. For kinetic models with parameters characterized by large uncertainty factors, strong nonlinearities, and reaction couplings (e.g., reduced mechanisms of real fuels; such models are referred to be “complex” in this work), the global RS is difficult to approximate, precluding conventional surrogate approaches. This paper proposes a framework that is extendable to such systems, termed Hybrid Response Surface Networks followed by a Stochastic Gradient Descent Ensemble (HRSN-SGDE). This technique focuses on mapping the local RS of just the uncertain spaces in the vicinity of the observed target, referred to as the rate target subspace. Two neural network surrogates are considered: a classifier that predicts the probability of data residing in the rate target subspace and a local RS surrogate which maps the RS of this subspace. A hybrid surrogate loss function is then defined using these surrogates to optimize uncertain rates repeatedly to get an ensemble of solutions representing the constrained rate space. HRSN-SGDE is demonstrated on a complex jet fuel model developed using the hybrid chemistry (HyChem) approach with a low temperature chemistry sub-model using a series of ignition delay times as targets. Results show that the method's local RS objective enables efficient and accurate construction of the surrogates through active learning-based sampling. Also, the unique formulation of the surrogate loss function enables optimization that is robust to suboptimal local minima and faster than evolutionary algorithms by several orders of magnitude. It is shown that HRSN-SGDE method is highly efficacious and suitable to conducting inverse UQ on such complex kinetic models.

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

AbstractThis article introduces an ‐adaptive multi‐element stochastic collocation method, which additionally allows to re‐use existing model evaluations during either ‐ or ‐refinement. The collocation method is based on weighted Leja nodes. After ‐refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub‐element in a hierarchical manner. For ‐refinement, the local polynomial approximations are based on total‐degree or dimension‐adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non‐smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

Multi-Output Gaussian Processes for Inverse Uncertainty Quantification in Neutron Noise Analysis

In a fissile material, the inherent multiplicity of neutrons born through induced fissions leads to correlations in their detection statistics. The correlations between neutrons can be used to trace back some characteristics of the fissile material. This technique, known as neutron noise analysis, has applications in nuclear safeguards or waste identification. It provides a nondestructive examination method for an unknown fissile material. This is an example of an inverse problem where the cause is inferred from observations of the consequences. However, neutron correlation measurements are often noisy because of the stochastic nature of the underlying processes. This makes the resolution of the inverse problem more complex since the measurements are strongly dependent on the material characteristics. A minor change in the material properties can lead to very different outputs. Such an inverse problem is said to be ill posed. For an ill-posed inverse problem, the inverse uncertainty quantification is crucial. Indeed, seemingly low noise in the data can lead to strong uncertainties in the estimation of the material properties. Moreover, the analytical framework commonly used to describe neutron correlations relies on strong physical assumptions, and is thus inherently biased. This paper addresses dual goals. First, surrogate models are used to improve neutron correlation predictions and quantify the errors on those predictions. Then the inverse uncertainty quantification is performed to include the impact of measurement error alongside the residual model bias.

Comparison of inverse uncertainty quantification methods for critical flow test

A problem of epistemic uncertainties introduced by code input parameters that cannot be estimated other than by user expertise has existed in nuclear reactor systems analyses since first uncertainty analyses were conducted. Inverse Uncertainty Quantification (IUQ) methods aim at providing an estimation of the distributions of such parameters. This paper compares and assesses two approaches to quantify the uncertainty of critical flow model parameters available in the TRACE code. First novel approach is based on a machine-learning algorithm, using a Random Forest classifier to assign the results of calculations to one of the defined classes of prediction accuracy with respect to experimental data. The second approach is based on Markov Chain Monte Carlo sampling and Bayesian inference. Both methods allows to assign probability distribution functions to TRACE internal variables, which is a goal for IUQ methods.Both approaches were used to estimate uncertainties for four critical flow parameters available in the TRACE code based on the Marviken critical flow experiment. In case of two parameters both methods produced similar results of the probability density functions. Based on the results of machine learning two additional parameters were deemed noninfluential, while for the second method such definitive assessment could not be made.

Bayesian Hierarchical Modelling for Uncertainty Quantification in Operational Thermal Resistance of LED Systems

Remaining useful life (RUL) prediction is central to prognostics and reliability assessment of light-emitting diode (LED) systems. Their unknown long-term service life remaining when subject to specific operating conditions is affected by various sources of uncertainty stemming from production of individual system components, application of the whole system, measurement and operation. To enhance the reliability of model-based predictions, it is essential to account for all of these uncertainties in a systematic manner. This paper proposes a Bayesian hierarchical modelling framework for inverse uncertainty quantification (UQ) in LED operation under thermal loading. The main focus is on the LED systems’ operational thermal resistances, which are subject to system and application variability. Posterior inference is based on a Markov chain Monte Carlo (MCMC) sampling scheme using the Metropolis–Hastings (MH) algorithm. Performance of the method is investigated for simulated data, which allow to focus on different UQ aspects in applications. Findings from an application scenario in which the impact of disregarded uncertainty on RUL prediction is discussed highlight the need for a comprehensive UQ to allow for reliable predictions.

Extension of the CIRCE methodology to improve the Inverse Uncertainty Quantification of several combined thermal-hydraulic models

The Inverse Uncertainty Quantification (IUQ) of physical correlations used in thermal-hydraulic system codes is a crucial issue in the BEPU (Best Estimate Plus Uncertainty) framework for the licensing and safety analyses of nuclear reactors. The CIRCE methodology is one of the most widely applied IUQ methods. It estimates the (log-)normal probability distribution of a multiplicative factor applied to the reference closure model used in thermal-hydraulic computer codes. CIRCE can jointly estimate the uncertainty of several physical models. However, in some particular cases, a problem of identifiability may occur resulting in less accuracy and precision of the estimated uncertainties.For this reason, in this work, the methodology is improved under the name of CIRCE 2-Steps. This new approach aims at improving the quantification of the model uncertainties when experiments with more than one physical phenomena interacting simultaneously on the output of interest has to be used. In a first step, the uncertainties of the least influential models are evaluated with classic CIRCE methodology using experiments for which only one physical phenomenon is dominant. Then, these uncertainties are properly taken into account when quantifying the remaining uncertainties on the experimental database. This approach is proven to provide more accurate and precise results on several analytical test cases where the uncertainties of two models has to be jointly estimated. The study of the impact of the relative importance of the two models shows that the biggest gain in precision is obtained when the hierarchy of models is such that one model is significantly more influential than the other.Finally, the newly developed methodology is applied to estimate the uncertainties of two condensation heat transfer correlations in the cold leg of a pressurised nuclear reactor.

LaSDI: Parametric Latent Space Dynamics Identification

Enabling fast and accurate physical simulations with data has become an important area of computational physics to aid in inverse problems, design-optimization, uncertainty quantification, and other various decision-making applications. This paper presents a data-driven framework for parametric latent space dynamics identification procedure that enables fast and accurate simulations. The parametric model is achieved by building a set of local latent space model and designing an interaction among them. An individual local latent space dynamics model achieves accurate solution in a trust region. By letting the set of trust region to cover the whole parameter space, our model shows an increase in accuracy with an increase in training data. We introduce two different types of interaction mechanisms, i.e., point-wise and region-based approach. Both linear and nonlinear data compression techniques are used. We illustrate the framework of Latent Space Dynamics Identification (LaSDI) enable a fast and accurate solution process on various partial differential equations, i.e., Burgers’ equations, radial advection problem, and nonlinear heat conduction problem, achieving O(100)x speed-up and O(1)% relative error with respect to the corresponding full order models.

Effect of mesh refinement on the solution of the inverse uncertainty quantification problem for transient physics

In this paper, effect of mesh refinement on the solution of the IUQ problem for transient physics was studied. Two mathematical formulations, the Maximum Likelihood Estimate (MLE) and the Maximum A Posterior (MAP), were adopted to solve the IUQ problem. Global Sensitivity Analysis (GSA) and Local Sensitivity Analysis (LSA) were performed to provide the necessary data for MLE and MAP implementation. TRACE models with three different mesh numbers based on the transient Flooding Experiment with Blocked Array (FEBA) benchmark were developed. Results of this analysis show that statistical expectation of the physical models (statistical mean) tends to shift towards a certain value with mesh refinement. This shifting behavior can be related to the decrease in the truncation error resulting from numerical discretization due to mesh refinement, consequently, compensating for the mismatch between experimental data and code prediction results that may be falsely credited to the uncertainties inherent in the physical models. Moreover, the results show that the relative absolute error between experimental data and code prediction results was decreased upon incorporating the input parameter uncertainties that were determined based on MLE and MAP formulations. In all cases, the error was smallest for the most refined mesh.

Passive safety systems analysis: A novel approach for inverse uncertainty quantification based on Stacked Sparse Autoencoders and Kriging metamodeling

In passive safety system analysis, it is important to provide the uncertainty quantification of the Thermal-Hydraulic (T-H) code output (e.g., the amount of energy exchanged by the passive safety system during an accidental transient). This requires setting proper Probability Density Functions (PDFs) to represent the uncertainty of selected code inputs and the propagation of this uncertainty through the code. One way to obtain the PDF is by Inverse Uncertainty Quantification (IUQ) methods, which rely directly on experimental data and code simulation results. In this work, we present an innovative IUQ method based on: (i) Stacked Sparse Autoencoders (SSAEs) to reduce the problem dimensionality; and (ii) Kriging metamodels to lower the computational burden associated with the sampling of the uncertain input parameters posterior PDF by Markov Chain Monte Carlo (MCMC) (for which many model simulations are typically required). The novelty stands in the use of SSAEs for dimensionality reduction: this allows using directly the raw data available from experimental facilities or computer codes (typically characterized by small signal-to-noise ratios) without having to resort to filtering techniques, whose choice and setting are nontrivial and bias the results. The proposed approach is applied to the power exchanged by the Heat Exchanger (HX) predicted by the RELAP5-3D model of the PERSEO facility, characterized by a small signal-to-noise ratio (SNR) value. Principal Component Analysis (PCA) and SSAE are compared to explain the application of these methodologies in the context of IUQ and highlight the main advantages and drawbacks while also showing the suitability to deal with non-filtered (raw) data.

Sensitivity analysis and inverse uncertainty quantification for the left ventricular passive mechanics

Personalized computational cardiac models are considered to be a unique and powerful tool in modern cardiology, integrating the knowledge of physiology, pathology and fundamental laws of mechanics in one framework. They have the potential to improve risk prediction in cardiac patients and assist in the development of new treatments. However, in order to use these models for clinical decision support, it is important that both the impact of model parameter perturbations on the predicted quantities of interest as well as the uncertainty of parameter estimation are properly quantified, where the first task is a priori in nature (meaning independent of any specific clinical data), while the second task is carried out a posteriori (meaning after specific clinical data have been obtained). The present study addresses these challenges for a widely used constitutive law of passive myocardium (the Holzapfel-Ogden model), using global sensitivity analysis (SA) to address the first challenge, and inverse-uncertainty quantification (I-UQ) for the second challenge. The SA is carried out on a range of different input parameters to a left ventricle (LV) model, making use of computationally efficient Gaussian process (GP) surrogate models in place of the numerical forward simulator. The results of the SA are then used to inform a low-order reparametrization of the constitutive law for passive myocardium under consideration. The quality of this parameterization in the context of an inverse problem having observed noisy experimental data is then quantified with an I-UQ study, which again makes use of GP surrogate models. The I-UQ is carried out in a Bayesian manner using Markov Chain Monte Carlo, which allows for full uncertainty quantification of the material parameter estimates. Our study reveals insights into the relation between SA and I-UQ, elucidates the dependence of parameter sensitivity and estimation uncertainty on external factors, like LV cavity pressure, and sheds new light on cardio-mechanic model formulation, with particular focus on the Holzapfel-Ogden myocardial model.

Bayesian inverse uncertainty quantification of the physical model parameters for the spallation neutron source first target station

The reliability of the mercury spallation target is mission-critical for the neutron science program of the spallation neutron source at the Oak Ridge National Laboratory. We present an inverse uncertainty quantification (UQ) study using the Bayesian framework for the mercury equation of state model parameters, with the assistance of polynomial chaos expansion surrogate models. By leveraging high-fidelity structural mechanics simulations and real measured strain data, the inverse UQ results reveal a tight posterior distribution for the tensile cutoff threshold, the mercury density, and the mercury speed of sound. The updated distributions do not necessarily represent the nominal mercury physical properties, but the ones that fit the strain data and the solid mechanics model we have used, and can be explained by three reasons. First, the limitations of the computer model or what is known as the “model-form uncertainty” that would result from numerical methods and physical approximations. Second, is the biases and errors in the experimental data. Third, is the mercury cavitation damage that also contributes to the change in mercury behavior. Consequently, the mercury equation of state model parameters try to compensate for these effects to improve fitness to the real data. The mercury target simulations using the posterior parametric values result in an excellent agreement with 88% average accuracy compared to experimental data, 6% average increase compared to reference parameters, with some sensors experiencing an increase of more than 25%. With a more accurate strain response predicted by the calibrated simulations, the component fatigue analysis can utilize the comprehensive strain history data to evaluate the target vessel’s lifetime closer to its real limit, saving tremendous target cost and improving the design of future targets as well.

Extension of the Circe Methodology to Improve the Inverse Uncertainty Quantification of Several Combined Thermal-Hydraulic Models

Extension of the Circe Methodology to Improve the Inverse Uncertainty Quantification of Several Combined Thermal-Hydraulic Models

Bayesian Inverse Uncertainty Quantification of the Physical Model Parameters for the Spallation Neutron Source First Target Station

Bayesian Inverse Uncertainty Quantification of the Physical Model Parameters for the Spallation Neutron Source First Target Station