# Mathematics of Probabilistic Uncertainty Modeling

D. Datta (Bhabha Atomic Research, India)
DOI: 10.4018/978-1-4666-4991-0.ch009

## Abstract

This chapter presents the uncertainty modeling using probabilistic methods. Probabilistic method of uncertainty analysis is due to randomness of the parameters of a model. Randomness of parameters is characterized by specified probability distribution such as normal, log normal, exponential etc., and the corresponding samples are generated by various methods. Monte Carlo simulation is applied to explore the probabilistic uncertainty modeling. Monte Carlo simulation being a statistical process is based on the random number generation from the specified distribution of the uncertain random parameters. Sample size is generally very large in Monte Carlo simulation which is required to have small errors in the computation. Latin hypercube sampling and importance sampling are explored in brief. This chapter also presents Polynomial Chaos theory based probabilistic uncertainty modeling. Polynomial Chaos theory is an efficient Monte Carlo simulation in the sense that sample size here is very small and dictated by the number of the uncertain parameters and by choice of the order of the polynomial selected to represent the uncertain parameter.
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## Background

Monte Carlo simulation is the basic instrument for quantifying uncertainty of any system having random parameters (Mckay et al., 1979; Steck et al., 1976). Sampling scheme to generate random numbers is described in detail in (Iman et al., 1981a; Iman et al., 1981b; Iman et al., 1980). The Latin hypercube sampling technique has been applied to many different computer models since 1975. Uncertainty quantification in this mode is expressed in terms of percentiles computed from the cumulative distribution of the output of the system under quest. Polynomial Chaos theory, an efficient Monte Carlo method used to quantify the probabilistic uncertainty by various researchers (Wiener, 1938; Ghanem & Spanos, 1991; Xiu & Karniadakis, 2002; Wan et al., 2004; Isukapalli, 1999; Papoulis, 1991). Uncertainty associated with any differential equation can be quantified by Polynomial Chaos theory (Wan et al., 2004; Isukapalli,1999).

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