Inverse Uncertainty Quantification Research Articles

Uncertainty quantification, propagation and characterization by Bayesian analysis combined with global sensitivity analysis applied to dynamical intracellular pathway models.

MotivationDynamical models describing intracellular phenomena are increasing in size and complexity as more information is obtained from experiments. These models are often over-parameterized with respect to the quantitative data used for parameter estimation, resulting in uncertainty in the individual parameter estimates as well as in the predictions made from the model. Here we combine Bayesian analysis with global sensitivity analysis (GSA) in order to give better informed predictions; to point out weaker parts of the model that are important targets for further experiments, as well as to give guidance on parameters that are essential in distinguishing different qualitative output behaviours.ResultsWe used approximate Bayesian computation (ABC) to estimate the model parameters from experimental data, as well as to quantify the uncertainty in this estimation (inverse uncertainty quantification), resulting in a posterior distribution for the parameters. This parameter uncertainty was next propagated to a corresponding uncertainty in the predictions (forward uncertainty propagation), and a GSA was performed on the predictions using the posterior distribution as the possible values for the parameters. This methodology was applied on a relatively large model relevant for synaptic plasticity, using experimental data from several sources. We could hereby point out those parameters that by themselves have the largest contribution to the uncertainty of the prediction as well as identify parameters important to separate between qualitatively different predictions. This approach is useful both for experimental design as well as model building.Availability and implementationSource code is freely available at https://github.com/alexjau/uqsa.Supplementary information Supplementary data are available at Bioinformatics online.

Inverse uncertainty quantification using the modular Bayesian approach based on Gaussian process, Part 1: Theory

In nuclear reactor system design and safety analysis, the Best Estimate plus Uncertainty (BEPU) methodology requires that computer model output uncertainties must be quantified in order to prove that the investigated design stays within acceptance criteria. "Expert opinion" and "user self-evaluation" have been widely used to specify computer model input uncertainties in previous uncertainty, sensitivity and validation studies. Inverse Uncertainty Quantification (UQ) is the process to inversely quantify input uncertainties based on experimental data in order to more precisely quantify such ad-hoc specifications of the input uncertainty information. In this paper, we used Bayesian analysis to establish the inverse UQ formulation, with systematic and rigorously derived metamodels constructed by Gaussian Process (GP). Due to incomplete or inaccurate underlying physics, as well as numerical approximation errors, computer models always have discrepancy/bias in representing the realities, which can cause over-fitting if neglected in the inverse UQ process. The model discrepancy term is accounted for in our formulation through the "model updating equation". We provided a detailed introduction and comparison of the full and modular Bayesian approaches for inverse UQ, as well as pointed out their limitations when extrapolated to the validation/prediction domain. Finally, we proposed an improved modular Bayesian approach that can avoid extrapolating the model discrepancy that is learnt from the inverse UQ domain to the validation/prediction domain.

Inverse uncertainty quantification using the modular Bayesian approach based on Gaussian Process, Part 2: Application to TRACE

Inverse Uncertainty Quantification (UQ) is a process to quantify the uncertainties in random input parameters while achieving consistency between code simulations and physical observations. In this paper, we performed inverse UQ using an improved modular Bayesian approach based on Gaussian Process (GP) for TRACE physical model parameters using the BWR Full-size Fine-Mesh Bundle Tests (BFBT) benchmark steady-state void fraction data. The model discrepancy is described with a GP emulator. Numerical tests have demonstrated that such treatment of model discrepancy can avoid over-fitting. Furthermore, we constructed a fast-running and accurate GP emulator to replace TRACE full model during Markov Chain Monte Carlo (MCMC) sampling. The computational cost was demonstrated to be reduced by several orders of magnitude. A sequential approach was also developed for efficient test source allocation (TSA) for inverse UQ and validation. This sequential TSA methodology first selects experimental tests for validation that has a full coverage of the test domain to avoid extrapolation of model discrepancy term when evaluated at input setting of tests for inverse UQ. Then it selects tests that tend to reside in the unfilled zones of the test domain for inverse UQ, so that one can extract the most information for posterior probability distributions of calibration parameters using only a relatively small number of tests. This research addresses the "lack of input uncertainty information" issue for TRACE physical input parameters, which was usually ignored or described using expert opinion or user self-assessment in previous work. The resulting posterior probability distributions of TRACE parameters can be used in future uncertainty, sensitivity and validation studies of TRACE code for nuclear reactor system design and safety analysis.

Consistency Analysis for Massively Inconsistent Datasets in Bound-to-Bound Data Collaboration

Bound-to-Bound Data Collaboration (B2BDC) provides a natural framework for addressing both forward and inverse uncertainty quantification problems. In this approach, QOI (quantity of interest) models are constrained by related experimental observations with interval uncertainty. A collection of such models and observations is termed a dataset and carves out a feasible region in the parameter space. If a dataset has a nonempty feasible set, it is said to be consistent. In real-world applications, it is often the case that collections of experiments and observations are inconsistent. Revealing the source of this inconsistency, i.e., identifying which models and/or observations are problematic, is essential before a dataset can be used for prediction. To address this issue, we introduce a constraint relaxation-based approach, entitled the vector consistency measure, for investigating datasets with numerous sources of inconsistency. The benefits of this vector consistency measure over a previous method of consistency analysis are demonstrated in two realistic gas combustion examples.

Kriging-based inverse uncertainty quantification of nuclear fuel performance code BISON fission gas release model using time series measurement data

In nuclear reactor fuel performance simulation, fission gas release (FGR) and swelling involve treatment of several complicated and interrelated physical processes, which inevitably depend on uncertain input parameters. However, the uncertainties associated with these input parameters are only known by “expert judgment”. In this paper, inverse Uncertainty Quantification (UQ) under the Bayesian framework is applied to BISON code FGR model based on Risø-AN3 time series experimental data. Inverse UQ seeks statistical descriptions of the uncertain input parameters that are consistent with the available measurement data. It always captures the uncertainties in its estimates rather than merely determining the best-fit values. Kriging metamodel is applied to greatly reduce the computational cost during Markov Chain Monte Carlo sampling.We performed a dimension reduction for the FGR time series data using Principal Component Analysis. We also projected the original FGR time series measurement data onto the PC subspace as “transformed experiment data”. A forward uncertainty propagation based on the posterior distributions shows that the agreement between BISON simulation and Risø-AN3 time series measurement data is greatly improved. The posterior distributions for the uncertain input factors can be used to replace the expert specifications for future uncertainty/sensitivity analysis.

Electromagnetic wave scattering by random surfaces: Shape holomorphy

For time-harmonic electromagnetic waves scattered by either perfectly conducting or dielectric bounded obstacles, we show that the fields depend holomorphically on the shape of the scatterer. In the presence of random geometrical perturbations, our results imply strong measurability of the fields, in weighted spaces in the exterior of the scatterer. These findings are key to prove dimension-independent convergence rates of sparse approximation techniques of polynomial chaos type for forward and inverse computational uncertainty quantification. Also, our shape-holomorphy results imply parsimonious approximate representations of the corresponding parametric solution families, which are produced, for example, by greedy strategies such as model order reduction or reduced basis approximations. Finally, the presently proved shape holomorphy results imply convergence of shape Taylor expansions of far-field patterns for fixed amplitude domain perturbations in a vicinity of the nominal domain, thereby extending the widely used asymptotic linearizations employed in first-order, second moment domain uncertainty quantification.

Inverse uncertainty quantification of TRACE physical model parameters using sparse gird stochastic collocation surrogate model

Within the BEPU (Best Estimate plus Uncertainty) methodology uncertainties must be quantified in order to prove that the investigated design remains within acceptance criteria. For best-estimate system thermal-hydraulics codes like TRACE and RELAP5, significant uncertainties come from the closure laws which are used to describe transfer terms in the balance equations. The accuracy and uncertainty information of these correlations are usually unknown to the code users, which results in the user simply ignoring or describing them using expert opinion or personal judgment during uncertainty and sensitivity analysis. The purpose of this paper is to replace such ad-hoc expert judgment of the uncertainty information of TRACE physical model parameters with inverse Uncertainty Quantification (UQ) based on OECD/NRC BWR Full-size Fine-Mesh Bundle Tests (BFBT) benchmark steady-state void fraction data. Inverse UQ seeks statistical descriptions of the physical model random input parameters that are consistent with the experimental data. Inverse UQ always captures the uncertainty of its estimates rather than merely determining point estimates of the best-fit input parameters. Bayesian analysis is used to establish the inverse UQ problems based on experimental data, with systematic and rigorously derived surrogate models based on Sparse Gird Stochastic Collocation (SGSC). Global sensitivity analysis including Sobol' indices and correlation coefficients are used to identify the important TRACE input parameters. Several adaptive Markov Chain Monte Carlo (MCMC) sampling techniques are investigated and implemented to explore the posterior probability density functions. This research solves the problem of lack of uncertainty information for TRACE physical model parameters for the closure relations. The quantified uncertainties are necessary for future uncertainty and sensitivity study of TRACE code in nuclear reactor system design and safety analysis.

Inverse uncertainty quantification of reactor simulations under the Bayesian framework using surrogate models constructed by polynomial chaos expansion

Modeling and simulations are naturally augmented by extensive Uncertainty Quantification (UQ) and sensitivity analysis requirements in the nuclear reactor system design, in which uncertainties must be quantified in order to prove that the investigated design stays within acceptance criteria. Historically, expert judgment has been used to specify the nominal values, probability density functions and upper and lower bounds of the simulation code random input parameters for the forward UQ process. The purpose of this paper is to replace such ad-hoc expert judgment of the statistical properties of input model parameters with inverse UQ process. Inverse UQ seeks statistical descriptions of the model random input parameters that are consistent with the experimental data. Bayesian analysis is used to establish the inverse UQ problems based on experimental data, with systematic and rigorously derived surrogate models based on Polynomial Chaos Expansion (PCE).The methods developed here are demonstrated with the Point Reactor Kinetics Equation (PRKE) coupled with lumped parameter thermal-hydraulics feedback model. Three input parameters, external reactivity, Doppler reactivity coefficient and coolant temperature coefficient are modeled as uncertain input parameters. Their uncertainties are inversely quantified based on synthetic experimental data. Compared with the direct numerical simulation, surrogate model by PC expansion shows high efficiency and accuracy. In addition, inverse UQ with Bayesian analysis can calibrate the random input parameters such that the simulation results are in a better agreement with the experimental data.

Evaluating Mixture Rules for Multi-Component Pressure Dependence: H + O2 (+M) = HO2 (+M)

A substantial fraction of the reactions that take place in combustion are complex-forming (i.e. pressure-dependent) reactions, which depend on the collisional energy transfer characteristics of the surrounding mixture (“bath gas”). In contrast to the single-component bath gases used in nearly all fundamental investigations of these reactions, significant fractions of multiple species of disparate collisional energy transfer characteristics are usually present in combustion environments. Calculations of the rate constant for multi-component mixtures thereafter inevitably relies on a “mixture rule” based on rate constants for the individual single-component bath gases. Failure to implement a mixture rule, which is actually quite common in many implementations of kinetic models within combustion codes, can yield errors up to a factor equal to the number of separate expressions for each component. Even still, as demonstrated herein, rate constants evaluated through various proposed mixture rules can differ by over 60% for exactly the mixture compositions of interest to combustion modeling. Therefore, evaluating the suitability of various mixture rules for accurately representing multi-component pressure dependence for combustion modeling purposes appears particularly worthwhile, given that these differences now exceed achievable accuracies for single-component bath gases and, additionally, given that inverse uncertainty quantification procedures almost always rely on starting with an accurate model structure, including mixture rules. The present study evaluates various existing and new mixture rules for representations of multi-component pressure dependence for H+O2(+M)=HO2(+M) through comparison with master equation calculations. The results indicate that mixture rules based on the reduced pressure provide a considerably more accurate description than those based on pressure (reaching accuracies of 2% compared to ∼60%). Recommendations are made for incorporation of these new mixture rules within combustion codes. Until then, potential errors in the full condition-dependence for H+O2(+M)=HO2(+M) of up to ∼90% are essentially unavoidable merely due to available representation options within combustion codes.

Inverse uncertainty quantification of trace physical model parameters using BFBT benchmark data

Forward quantification of simulation (code) response uncertainties requires knowledge of physical model parameter uncertainties. Nuclear thermal-hydraulics codes, such as RELAP5 and TRACE, do not provide any information on uncertainties of physical model parameters. A framework is developed to quantify uncertainties of physical model parameters using Maximum Likelihood Estimation (MLE), Bayesian Maximum A Priori (MAP), and Markov Chain Monte Carlo (MCMC) algorithms.The objective of the present work is to perform the sensitivity analysis of the physical model parameters in code TRACE and calculate their uncertainties using MLE, MAP, and MCMC algorithms. The OECD/NEA BWR Full-size fine-mesh Bundle Test (BFBT) data is used to quantify uncertainty of selected physical models of TRACE code. The BFBT is based on a multi-rod assembly with measured data available for single or two-phase pressure drop, axial and radial void fraction distributions, and critical power for a wide range of system conditions. In this work, the steady-state cross-sectional averaged void fraction distribution is used as the input data for inverse uncertainty quantification (IUQ) algorithms, and the selected physical model’s probability distribution function (PDF) is the desired output quantity.

Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice

Many uncertainty quantification (UQ) approaches suffer from the curse of dimensionality, that is, their computational costs become intractable for problems involving a large number of uncertainty parameters. In these situations, the classic Monte Carlo often remains the preferred method of choice because its convergence rate O(n−1/2), where n is the required number of model simulations, does not depend on the dimension of the problem. However, many high-dimensional UQ problems are intrinsically low-dimensional, because the variation of the quantity of interest (QoI) is often caused by only a few latent parameters varying within a low-dimensional subspace, known as the sufficient dimension reduction (SDR) subspace in the statistics literature. Motivated by this observation, we propose two inverse regression-based UQ algorithms (IRUQ) for high-dimensional problems. Both algorithms use inverse regression to convert the original high-dimensional problem to a low-dimensional one, which is then efficiently solved by building a response surface for the reduced model, for example via the polynomial chaos expansion. The first algorithm, which is for the situations where an exact SDR subspace exists, is proved to converge at rate O(n−1), hence much faster than MC. The second algorithm, which doesn't require an exact SDR, employs the reduced model as a control variate to reduce the error of the MC estimate. The accuracy gain could still be significant, depending on how well the reduced model approximates the original high-dimensional one. IRUQ also provides several additional practical advantages: it is non-intrusive; it does not require computing the high-dimensional gradient of the QoI; and it reports an error bar so the user knows how reliable the result is.

Probabilistic fatigue damage prognosis using surrogate models trained via three-dimensional finite element analysis

Utilizing inverse uncertainty quantification techniques, structural health monitoring (SHM) can be integrated with damage progression models to form a probabilistic prediction of a structure’s remaining useful life (RUL). However, damage evolution in realistic structures is physically complex. Accurately representing this behavior requires high-fidelity models which are typically computationally prohibitive. In this paper, high-fidelity fatigue crack growth simulation times are reduced by three orders of magnitude using a model based on a set of surrogate models trained via three-dimensional finite element analysis. The developed crack growth modeling approach is experimentally validated using SHM-based damage diagnosis data. A probabilistic prediction of RUL is formed for a metallic, single-edge notch tension specimen with a fatigue crack growing under mixed-mode conditions.

Accurate Solution of Bayesian Inverse Uncertainty Quantification Problems Combining Reduced Basis Methods and Reduction Error Models

Computational inverse problems related to partial differential equations (PDEs) often contain nuisance parameters that cannot be effectively identified but still need to be considered as part of the problem. The objective of this work is to show how to take advantage of a reduced order framework to speed up Bayesian inversion on the identifiable parameters of the system, while marginalizing away the (potentially large number of) nuisance parameters. The key ingredients are twofold. On the one hand, we rely on a reduced basis (RB) method, equipped with computable a posteriori error bounds, to speed up the solution of the forward problem. On the other hand, we develop suitable reduction error models (REMs) to quantify in an inexpensive way the error between the full-order and the reduced-order approximation of the forward problem, in order to gauge the effect of this error on the posterior distribution of the identifiable parameters. Numerical results dealing with inverse problems governed by elliptic PDEs in the case of both scalar parameters and parametric fields highlight the combined role played by RB accuracy and REM effectivity.

A NEW INVERSE METHOD FOR THE UNCERTAINTY QUANTIFICATION OF SPATIALLY VARYING RANDOM MATERIAL PROPERTIES

In this paper, a new inverse uncertainty quantification method was proposed to identify statistical parameters associated with spatially varying material properties and to reconstruct their heterogeneous distributions from limited experimental measurements. The proposed method parameterizes statistical models of random fields with analytic co-variance functions and spectral decomposition into Karhunen-Loeve random variables. The statistical model parameters are identified by an experimental-numerical inverse analysis method, which is expected to significantly reduce time and cost required for quantification of material uncertainties.

Stochastic model updating utilizing Bayesian approach and Gaussian process model

Stochastic model updating (SMU) has been increasingly applied in quantifying structural parameter uncertainty from responses variability. SMU for parameter uncertainty quantification refers to the problem of inverse uncertainty quantification (IUQ), which is a nontrivial task. Inverse problem solved with optimization usually brings about the issues of gradient computation, ill-conditionedness, and non-uniqueness. Moreover, the uncertainty present in response makes the inverse problem more complicated. In this study, Bayesian approach is adopted in SMU for parameter uncertainty quantification. The prominent strength of Bayesian approach for IUQ problem is that it solves IUQ problem in a straightforward manner, which enables it to avoid the previous issues. However, when applied to engineering structures that are modeled with a high-resolution finite element model (FEM), Bayesian approach is still computationally expensive since the commonly used Markov chain Monte Carlo (MCMC) method for Bayesian inference requires a large number of model runs to guarantee the convergence. Herein we reduce computational cost in two aspects. On the one hand, the fast-running Gaussian process model (GPM) is utilized to approximate the time-consuming high-resolution FEM. On the other hand, the advanced MCMC method using delayed rejection adaptive Metropolis (DRAM) algorithm that incorporates local adaptive strategy with global adaptive strategy is employed for Bayesian inference. In addition, we propose the use of the powerful variance-based global sensitivity analysis (GSA) in parameter selection to exclude non-influential parameters from calibration parameters, which yields a reduced-order model and thus further alleviates the computational burden. A simulated aluminum plate and a real-world complex cable-stayed pedestrian bridge are presented to illustrate the proposed framework and verify its feasibility.

Subspace-based Inverse Uncertainty Quantification for Nuclear Data Assessment

Safety analysis and design optimization depend on the accurate prediction of various reactor attributes. Predictions can be enhanced by reducing the uncertainty associated with the attributes of interest. An inverse problem can be defined and solved to assess the sources of uncertainty, and experimental effort can be subsequently directed to further improve the uncertainty associated with these sources. In this work a subspace-based algorithm for inverse sensitivity/uncertainty quantification (IS/UQ) has been developed to enable analysts account for all sources of nuclear data uncertainties in support of target accuracy assessment-type analysis. An approximate analytical solution of the optimization problem is used to guide the search for the dominant uncertainty subspace. By limiting the search to a subspace, the degrees of freedom available for the optimization search are significantly reduced. A quarter PWR fuel assembly is modeled and the accuracy of the multiplication factor and the fission reaction rate are used as reactor attributes whose uncertainties are to be reduced. Numerical experiments are used to demonstrate the computational efficiency of the proposed algorithm. Our ongoing work is focusing on extending the proposed algorithm to account for various forms of feedback, e.g., thermal-hydraulics and depletion effects.

Parallel Inverse Modeling and Uncertainty Quantification for Computationally Demanding Groundwater-Flow Models Using Covariance Matrix Adaptation

This study investigates the performance of the covariance matrix adaptation-evolution strategy (CMA-ES), a stochastic optimization method, in solving groundwater inverse problems. The objectives of the study are to evaluate the computational efficiency of the parallel CMA-ES and to investigate the use of the empirically estimated covariance matrix in quantifying model prediction uncertainty due to parameter estimation uncertainty. First, the parallel scaling with increasing number of processors up to a certain limit is discussed for synthetic and real-world groundwater inverse problems. Second, through the use of the empirically estimated covariance matrix of parameters from the CMA-ES, the study adopts the Monte Carlo simulation technique to quantify model prediction uncertainty. The study shows that the parallel CMA-ES is an efficient and powerful method for solving the groundwater inverse problem for computationally demanding groundwater flow models and for deriving covariances of estimated parameters for uncertainty analysis.

Reduced order modeling in iTOUGH2

The inverse modeling and uncertainty quantification capabilities of iTOUGH2 are augmented with reduced order models (ROMs) that act as efficient surrogates for computationally expensive high fidelity models (HFMs). The implementation of the ROM capabilities involves integration of three main computational components. The first component is the ROM itself. Two response surface approximations are currently implemented: Gaussian process regression (GPR) and radial basis function (RBF) interpolation. The second component is a multi-output adaptive sampling procedure that determines the sample points used to construct the ROMs. The third component involves defining appropriate error measures for the adaptive sampling procedure, allowing ROMs to be constructed efficiently with limited user intervention. Details in all three components must complement one another to obtain an accurate approximation. The new capability and its integration with other analysis tools within iTOUGH2 are demonstrated in two examples. The results from using the ROMs in an uncertainty quantification analysis and a global sensitivity analysis compare favorably with the results obtained using the HFMs. GPR is more accurate than RBF, but the difference can be small and similar conclusion can be deduced from the analyses. In the second example involving a realistic numerical model for a hypothetical industrial-scale carbon storage project in the Southern San Joaquin Basin, California, USA, significant reduction in computational effort can be achieved when ROMs are used to perform a rigorous global sensitivity analysis.