Fundamentals of Uncertainty

Uncertainty is an important aspect in experiment as well as modelling in various science and engineering problems. It has become one of the popular paradigmatic changes in recent decades. These changes have been demonstrated by a gradual transition from the traditional outlook, which suggested that uncertainties were undesirable in science and should be avoided by all possible means. Traditionally, science strived for certainty in all its manifestations (precision, specificity, sharpness, consistency, etc.); thus, uncertainties (impreciseness, non-specificity, vagueness, inconsistency, etc.) may have been regarded earlier as unscientific. In the modern view, uncertainties have been considered as an essential part of science, and it is the backbone of various real-world problems.

Generally, we deal with systems that are constructed as models of either some facet of reality or some desirable artificial objects. The need for constructing models is to understand some phenomenon of reality, be it natural or artificial, making decent predictions, learning to control the phenomenon in any desirable way and utilizing all these capabilities for various ends; models of the latter type are constructed for the purpose of prescribing operations by which a conceived artificial object can be constructed in such a way that desirable objective criteria are satisfied within given constraints.

As such, uncertainty plays an important role in various branches of engineering and science. These uncertainties may be quantified by two approaches, viz. parametric and non-parametric. In the parametric approach, uncertainties may be associated with system parameters, such as Young's modulus, mass density, Poisson's ratio, damping coefficient and geometric parameters, for structural dynamic problems. Various authors have quantified these parameters using statistical methods, for example the stochastic finite element method. On the other hand, uncertainties that occur due to incomplete data or information, impreciseness, vagueness, experimental error and different operating conditions influenced by the system may be handled by non-parametric approaches.

In recent years, the probabilistic approach has been adopted to handle the uncertainties involved in the systems. Probability distributions have been used in traditional deterministic model parameters that account for their uncertainty. Different frameworks have been proposed to quantify the uncertainties caused by randomness (aleatory uncertainty) as well as lack of knowledge (epistemic uncertainty), along with probability boxes (p-boxes) (Ferson and Ginzburg 1996), Bayesian hierarchical models (Gelman 2006), Dempter- Shafer's evidence theory (Dempster 1967; Shafer 1976) and parametric p-boxes using sparse polynomial chaos expansions (Schobi and Sudret 2015a). These frameworks are generally referred to as imprecise probabilities. The uncertainty propagation of imprecise probabilities leads to an imprecise response. Input uncertainties are characterized and then propagated through a computational model. To reduce the computational effort, a well-known meta-modelling technique such as polynomial chaos expansions (Ghanem and Spanos 2003; Sudret 2014; Schobi and Sudret 2015b) can be used. An algorithm has been proposed by Schobi and Sudret (2015b) for solving imprecise structural reliability problems. This algorithm transforms an imprecise problem into two precise structural reliability problems, which reveals the possibilities for using traditional structural reliability analyses techniques.

Although the uncertainties are handled by various authors using probability density functions or statistical methods, these methods need plenty of data and also may not consider the vague or imprecise parameters. Accordingly, one may use the interval/fuzzy computation in the analysis of the problems. In this context, a few authors have used finite element method when the uncertain parameters are in terms of interval/fuzzy and it is called the interval/fuzzy finite element method (I/FFEM). Accordingly, a new computation method (Chakraverty and Nayak 2013; Nayak and Chakraverty 2013) with interval/ fuzzy values was developed for reducing the computational effort. Applying the I/FFEM, we get either interval/fuzzy system of equations or eigenvalue problems. The solutions for the interval/fuzzy system of linear equations are studied by various researchers. A few authors have also discussed the method of the uncertain bound of eigenvalues. Sevastjanov and Dymova (2009) investigated a new method for solving both the interval/fuzzy equations for linear case. Friedman et al. (1998) used the embedding approach to solve the n x n fuzzy linear system of equations. Some authors (Abbasbandy and Alvi 2005; Allahviranloo et al. 2008; Li et al. 2010; Senthilkumar and Rajendran 2011) proposed various other methods for finding the uncertain solutions of fuzzy system of linear equations. They have considered the coefficient matrix as crisp. This literature shows that the uncertain systems may also be handled by using the non-probabilistic approach. Accordingly, both probabilistic and non-probabilistic approaches may be considered for investigating uncertain systems.

Both approaches, viz. probabilistic and non-probabilistic, have become popular among researchers in the field. They are discussed in the detail next. * 1 2 3

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