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Inverse Uncertainty Quantification (IUQ) method has been widely used to quantify the uncertainty of Physical Model Parameters (PMPs) in nuclear Thermal Hydraulics (TH) systems. This paper introduces a novel hierarchical Bayesian model which aims to mitigate two existing challenges in IUQ: the high variability of PMPs under varying experimental conditions, and unknown model discrepancies or outliers causing over-fitting issues. The proposed hierarchical model is compared with the conventional single-level Bayesian model using TRACE code and the measured void fraction data in the BFBT benchmark. A Hamiltonian Monte Carlo Method - No U-Turn Sampler (NUTS) is used for posterior sampling. The results demonstrate the effectiveness of the proposed hierarchical model in providing better estimates of the posterior distributions of PMPs and being less prone to over-fitting. The proposed method also demonstrates a promising approach for generalizing IUQ to larger databases with broad ranges of experimental conditions.

Uncertainty Quantification (UQ) is an essential step in computational model validation because assessment of the model accuracy requires a concrete, quantifiable measure of uncertainty in the model predictions. The concept of UQ in the nuclear community generally means forward UQ (FUQ), in which the information flow is from the inputs to the outputs. Inverse UQ (IUQ), in which the information flow is from the model outputs and experimental data to the inputs, is an equally important component of UQ but has been significantly underrated until recently. FUQ requires knowledge in the input uncertainties which has been specified by expert opinion or user self-evaluation. IUQ is defined as the process to inversely quantify the input uncertainties based on experimental data. This review paper aims to provide a comprehensive and comparative discussion of the major aspects of the IUQ methodologies that have been used on the physical models in system thermal-hydraulics codes. IUQ methods can be categorized by three main groups: frequentist (deterministic), Bayesian (probabilistic), and empirical (design-of-experiments). We used eight metrics to evaluate an IUQ method, including solidity, complexity, accessibility, independence, flexibility, comprehensiveness, transparency, and tractability. Twelve IUQ methods are reviewed, compared, and evaluated based on these eight metrics. Such comparative evaluation will provide a good guidance for users to select a proper IUQ method based on the IUQ problem under investigation., Comment: 76 pages, 10 figures

The efficient method for computing the sensitivities is the adjoint method. The cost of solving an adjoint equation is comparable to the cost of solving the governing equation. Once the adjoint solution is obtained, the sensitivities to any number of parameters can be obtained with little effort. There are two methods to develop the adjoint equations: continuous method and discrete method. In the continuous method, the control theory is applied to the forward governing equation and produces an analytical partial differential equation for solving the adjoint variable; in the discrete method, the control theory is applied to the discrete form of the forward governing equation and produces a linear system of equations for solving the adjoint variable. In this article, an adjoint sensitivity analysis framework is developed using both the continuous and discrete methods. These two methods are assessed with the faucet flow for steady-state problem and one transient test case based on the BFBT benchmark for transient problem. Adjoint sensitivities from both methods are verified by sensitivities given by the perturbation method. Adjoint sensitivities from both methods are physically reasonable and match each. The sensitivities obtained with discrete method is found to be more accurate than the sensitivities from the continuous method. The continuous method is computationally more efficient than the discrete method because of the analytical coefficient matrices and vectors. However, difficulties are observed in solving the continuous adjoint equation for cases where the adjoint equation contains sharp discontinuities in the source terms; in such cases, the continuous method is not as robust as the discrete adjoint method., Comment: arXiv admin note: substantial text overlap with arXiv:1805.01451

Verification, validation and uncertainty quantification (VVUQ) have become a common practice in thermal-hydraulics analysis. An important step in the uncertainty analysis is the sensitivity analysis of various uncertain input parameters. The common approach for computing the sensitivities, e.g. variance-based and regression-based methods, requires solving the governing equation multiple times, which is expensive in terms of computational cost. An alternative approach to compute the sensitivities is the adjoint method. The cost of solving an adjoint equation is comparable to the cost of solving the governing equation. Once the adjoint solution is available, the sensitivities to any number of parameters can be obtained with little cost. However, successful application of adjoint sensitivity analysis to two-phase flow simulations is rare. In this work, an adjoint sensitivity analysis framework is developed based on the discrete adjoint method and a new implicit forward solver. The framework is tested with the faucet flow problem and the BFBT benchmark. Adjoint sensitivities are shown to match analytical sensitivities very well in the faucet flow problem. The adjoint method is used to propagate uncertainty in input parameters to the void fraction in the BFBT benchmark test. The uncertainty propagation with the adjoint method is verified with the Monte Carlo method and is shown to be efficient.

Inverse Uncertainty Quantification (UQ), or Bayesian calibration, is the process to quantify the uncertainties of random input parameters based on experimental data. The introduction of model discrepancy term is significant because "over-fitting" can theoretically be avoided. But it also poses challenges in the practical applications. One of the mostly concerned and unresolved problem is the "lack of identifiability" issue. With the presence of model discrepancy, inverse UQ becomes "non-identifiable" in the sense that it is difficult to precisely distinguish between the parameter uncertainties and model discrepancy when estimating the calibration parameters. Previous research to alleviate the non-identifiability issue focused on using informative priors for the calibration parameters and the model discrepancy, which is usually not a viable solution because one rarely has such accurate and informative prior knowledge. In this work, we show that identifiability is largely related to the sensitivity of the calibration parameters with regards to the chosen responses. We adopted an improved modular Bayesian approach for inverse UQ that does not require priors for the model discrepancy term. The relationship between sensitivity and identifiability was demonstrated with a practical example in nuclear engineering. It was shown that, in order for a certain calibration parameter to be statistically identifiable, it should be significant to at least one of the responses whose data are used for inverse UQ. Good identifiability cannot be achieved for a certain calibration parameter if it is not significant to any of the responses. It is also demonstrated that "fake identifiability" is possible if model responses are not appropriately chosen, or inaccurate but informative priors are specified.

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.

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., Comment: 27 pages, 10 figures, article