Section 2.5 and 2.7 describe methods for uncertainty propagation (forward problem) and statistical inference (inverse problem) respectively. While sampling methods and analytical methods are discussed in the former case, the latter case considers only Markov Chain Monte Carlo-based sampling methods.
very meaningful becauseyis single-valued given x, and hencefY(y|x) is nothing but a Dirac delta function. Alternatively, the PDF can be calculated by differentiating the CDF, as:
fY(y) = dFY(y)
dy (2.13)
Two types of methods – sampling-based and analytical methods – are be used to calculate the PDF and CDF of Y.
2.5.1 Monte Carlo Sampling
The most intuitive method for uncertainty propagation is to make use of Monte Carlo simulation (MCS). In this method, several random realizations of X are gen- erated based on CDF inversion, and the corresponding random realizations of Y are computed. Then the CDFFY(y) is calculated as the proportion of the number of real- izations where the output realization is less thany. The generation of each realization requires one evaluation/simulation of the computational model. Several thousands of realizations may often be needed to calculate the entire CDF, especially for very high/low values of y. Error estimates for the CDF, in terms of the number of simu- lations, are available in the literature [27, 28]. Once the samples of Y are obtained, then a histogram can be drawn easily or the empirical PDF of Y can be calculated using kernel density estimation [29].
Advanced MCS methods such as importance sampling, stratified sampling, latin hypercube sampling, etc. are also available to aid in the reduction of computational effort [28, 30]. The basic underlying concept of these methods is to generate pseudo- random numbers which are uniformly distributed on the interval [0, 1]; then the CDF of X is inverted to generate the corresponding realization of X. Therefore, these methods are applicable only when the CDF ofX is fully known and can be inverted.
This is the case in uncertainty propagation because the PDFs of X are assumed to
be known and the goal is to propagate them through the model Y = G(X). The topic of uncertainty propagation will be considered in all of the forthcoming chapters in this dissertation, and the above sampling methods will be repeatedly used for this purpose, and kernel density estimation will be used to construct the PDF based on Monte Carlo samples.
There is another class of sampling methods, collectively referred to as MCMC sampling methods, which are used to draw samples from a probability distribution whose CDF cannot be inverted or whose PDF is known only up to a proportionality constant. This is often used to solve statistical inverse problems (Bayesian inference), and hence discussed later in Section 2.7.
2.5.2 Analytical Methods
A new class of methods was developed by reliability engineers in order to facilitate efficient, quick but approximate calculation of the CDFFY(y); the focus is not on the calculation of the entire CDF function but only to evaluate the CDF at a particular value of the output, i.e. FY(Y =yc); the value ofyc is chosen so that the CDF value, i.e. the probability P(Y ≤yc) is the failure probability of the system represented by the modelY =G(X).
The basic concept is to “linearize” the model G so that the the output Y can be expressed as a linear combination of the random variables. Further, the random variables are transformed into uncorrelated standard normal space and hence, the output Y is also a normal variable (since the linear combination of normal variables is normal). Therefore, the CDF value FY(Y =yc) can be computed using the stan- dard normal distribution function. The transformation of random variables X into uncorrelated standard normal space (U) is denoted byU =T(X), and the details of the transformation can be found in [28].
Since the modelGis non-linear, the failure probability depends on the location of
“linearization”. This linearization is done at the so-called most probable point (MPP) which is the shortest distance from origin to the limit state, calculated in the U— space. Then, the failure probability is calculated asPf = Φ(−β), where Φ denotes the standard normal CDF function, andβ denotes the aforementioned shortest distance.
The MPP and the shortest distance are estimated using the well-known Rackwitz- Fiessler algorithm [31], which is based on a repeated linear approximation of the non- linear constraint G(x)−yc = 0. This method is popularly known as the first-order reliability method (FORM). There are also several second order reliability methods (SORM) based on the quadratic approximation of the limit state [28, 32–34].
The entire CDF can be calculated using repeated FORM analyses by considering different values of yc; for example, if FORM is performed at 10 different values of yc, the corresponding CDF values are calculated, and an interpolation scheme can be used to calculate the entire CDF, which can be differentiated to obtain the PDF. This approach is difficult because it is almost impossible to choose such multiple values of yc, because the range (i.e. extent of uncertainty) of Y is unknown. This difficulty is overcome by the use of an inverse FORM method [35] where multiple CDF values are chosen and the corresponding values of yc are calculated. This approach is simpler because it is easier to choose multiple CDF values since the range of CDF is known to be [0, 1].
In this dissertation, FORM and inverse FORM are not implemented for the pur- pose of uncertainty propagation. However, Chapter VII will require the use of FORM for the numerical implementation of a new methodology for uncertainty quantifica- tion in multi-disciplinary analysis. Hence, details regarding the Rackwitz-Fiessler algorithm (identification of MPP and calculation ofFY(yc)) will be provided in Chap- ter VII.