Modelling non-Gaussian uncertainties and the Karhunen–Loéve expansion within the context of polynomial chaos

• Gaussian random variables are used to solve problems involving non-Gaussian parameters via the Nataf transformation.
• The Nataf covariance functions are derived in detail for nine pdfs, including: uniform, Rayleigh, exponential, log-normal.
• The Methodology is applied to calculating the neutron or photon flux through a non-scattering slab for various types of pdf.

We examine the utility of a generalised, non-Gaussian Karhunen–Loéve expansion in nuclear engineering applications. This is useful because in many situations the joint probability distribution function of the random variables of interest is unobtainable, whereas the marginals and covariance functions can generally be found. Given these priors, we follow and expand upon the work of other authors to transform the priors into a Gaussian covariance suitable for the Karhunen–Loéve expansion; this is done using the Nataf formulation which is explained in some detail. We derive analytical solutions to fundamental marginal distributions of the same and mixed types and show how the effective correlation function used in the K–L integral equation is related to the correlation function of the non-linear process. Specifically, we consider the uniform, step, triangular, Rayleigh, exponential, log-uniform and log-normal pdfs for covariance problems and the uniform + log-normal pdfs for a cross-covariance problem. We also show how these modified K–L expansions can be used to solve some simple neutron transport problems involving spatially stochastic cross sections with given probability distributions and associated correlation functions. An outcome of the investigation is a numerical study of the sensitivity of the final result, e.g. average flux and variance, to the Nataf transformation. That is whether it is always necessary to use this somewhat convoluted approach. In general, for problems in which the overall fluctuations are small it may not be necessary but one can often only decide this after a full Nataf calculation has been made, This aspect of the work is highlighted by our studies of transmission of neutral particles through a slab.