Uncertainty quantification

In view of the often considerable limitations of available data supporting exposure assessment, which sometimes limit the extent of uncertainty quantification and the need to explicitly identify sources of uncertainty prior to their quantification, this section provides an overview of existing concepts and proposes a harmonized approach for the qualitative analysis of uncertainty in exposure assessment. 

There are various levels of pursuit that may be taken simultaneously to address the challenges that arise in multiscale problems. One is to select one or more challenging exemplar problems, selected for their importance in specific high impact applications. A second is to abstract more general principles and procedures for multiscale simulation that may be used for other applications beyond those for which they were originally developed. Such procedures include error estimation and uncertainty quantification tools. At the same time, it is worth noting to keep in mind the need for appropriate computations level, as emphasized in the observation made recently by the late John Pople in the quote below ([14], page 34) ... 

Even though Bayesian inference is useful for uncertainty quantification that fulfills the need in civil engineering, the literature shows that developments and applications of this powerful tool in civil engineering are still at an early stage. Therefore, there is plenty of room to be explored for Bayesian applications in civil engineering. This book introduces some recently developed Bayesian methods and applications to a number of areas in civil engineering. The main concern here is on the identification of dynamical systems, but some of the methods are also applicable to static problems. Two types of problems in system identification are... 

In this chapter, the Bayesian spectral density approach, which is a frequency-domain approach, for modal/model updating using wide-band response data is presented. It utilizes the statistical properties of the spectral density estimator to obtain not only the optimal values of model parameters but also their associated uncertainty by means of the updated probability distribution of the uncertain parameters. Uncertainty quantification is important for many applications, such as damage detection and reliability analysis. 

Reagan MT, Naim HN, Debusschere BJ, Le Maitre OP, Knio OM, Ghanem RG (2004) Spectral stochastic uncertainty quantification in chemical systems. Combust Theory Model 8(3) 607-632... 

In this work, the aim is to quantify and propagate uncertainty to eventually use it to evaluate the confidence in the extrapolated dynamic coupling response of a prediction of a full system model. To demonstrate the reliability methodologies developed in this work, a problem developed at a two axis position mechanical is used. The following section gives a brief description of a Bayesian Network (DBN) to uncertainty quantification and its implementation to the selected problem. 

Angel Urbina, Sankaran Mahadevan., 2009. Uncertainty Quantification in Hierarchical Development of Computational Models. 50th AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics, and Materials Conference 17th 4—7 May, Palm Springs, California. C. Andrieu, N. de Freitas, A. Doucet, M. Jordan, 2003. An introduction to MCMC for machine learning. Machine Learning, 50, 5—43. 

For the RISMC project, we employed RAVEN and RELAP5-3D to analyze a Station Black-Out (SBO) accident scenario for a Boiling Water Reactor (Mandelli, Smith, Alfonsi, Rabiti, et al. 2014). In particular, we employed both classical statistical tools, i.e. MC, and more advanced machine learning based algorithms to perform uncertainty quantification in order to quantify changes in system performance and limitations as a consequence of power uprate. 

III Probability of frequency Uncertainty quantification Model seen to describe true risk Expert judgment is truth-approaching Uncertainty is quantified Judgment aiming at truth Risk is a property of the world. Based on hard evidence and judgment Uncertainty related to impredsion of underlying true risk... 

Kaplan (1997) proposes the so-called probability of frequency approach to risk assessment, based on a risk concept in line with risk definition C6 (R = P C), where subjective probabilities are used to express uncertainty about true frequen-tist probabilities. The assessment thus focuses on quantifying uncertainty about an underlying true risk, which is estimated. Kaplan s view is strongly tied to realism, as the risk description focuses on a true risk as determined by experts. Closely related perspectives are those where uncertainty is quantified around a true risk, such as in the traditional Bayesian perspective where uncertainty is quantified in relation to model parameters (Aven Heide 2009). Such uncertainty quantification can also be done using non-probabilistic representations of epistemic uncertainty (Helton Johnson 2011). These methods typicdly consider a risk problem in a highly mathematized form. 

Bae, H.R., Grandhi, R.V. Canfield, R.A. 2004. An approximation approach for uncertainty quantification using evidence theory. Reliability Engineering ... 

Dakota, 2011. A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis. Version 5.2 Reference Manual. SANDIA Report SAND2010 2184. 

Eldred, M.S. et al., 2011. Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval, estimation, Reliability Engineering and System Safety 96, 1092-1113. 

Tanaka, M., et al., 2016. Development of V2UP (V V plus uncertainty quantification and prediction) procedure for high cycle thermal fatigue in fast reactor — framework for V V and numerical prediction. Nuclear Engineering and Design 299 (2016), 174—183. 

Tanaka, M., et al., 2015. Numerical simulation of thermal striping phenomena in a T-junction piping system for fundamental vahdation and uncertainty quantification by GCI estimation. Bulletin of the JSME, Mechanical Engineering Journal 2 (5) (2015). Paper No.15-00134. 

Wang, H., Sheen, D.A. Combustion kinetic model uncertainty quantification, propagation and minimization. Prog. Energy Combust. Sci. 47, 1-31 (2015)... 

Cheng, H.Y., Sandu, A. Uncertainty quantification and apportionment in air quality models using the polynomial chaos method. Environ. Model. Software 24, 917-925 (2009)... 

Prager, J., Najm, H.N., Sargsyan, K., Safta, C., Pitz, W.J. Uncertainty quantification of reaction mechanisms accounting for correlations introduced by rate rules and fitted Arrhenius parameters. Combust. Flame 160, 1583-1593 (2013)... 

Reagan, M.T., Najm, H.N., Ghanem, R.G., Knio, O.M. Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection. Combust Flame 132(3), 545-555 (2003)... 

Ruscic, B. Uncertainty quantification in thermochemistry, benchmarking electronic structure computations, and Active Thermochemical Tables, hit. J. Quantum Chem. 114, 1097-1101 (2014)... 

Sheen, D.A., Wang, H. The method of uncertainty quantification and minimization using polynomial chaos expansions. Combust. Flame 158, 2358-2374 (2011b)... 

Russi, T., Packard, A., Frenklach, M. Uncertainty quantification making predictions of complex reaction systems reliable. Chem. Phys. Lett. 499, 1-8 (2010)... 

The objective of model updating (often also referred to as parameter estimation) is to calibrate unknown system properties which appear as parameters in numerical models, based on actually observed behavior of the system of interest. In Bayesian model updating, this is performed in a probabilistic uncertainty quantification framework PDFs representing the uncertainty on the model parameters are updated through the experimental data this procedure is described briefly below. 

The sum of both errors is the difference between the model predictions and the observed quantities and is defined as the total observed prediction error rj. The above expression serves as a starting point for the Bayesian uncertainty quantification method. 

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