We focus in this work on demonstrating the approach and defer application to specific problems to a future investigation. Numerical results for the mean and variance in the scalar flux and leakage currents are obtained for weak and strong random variations. Woodcock Monte Carlo (WMC) is then used to simulate transport using random sampling of cross sections and deterministic sampling based on a stochastic collocation technique.
Specifically, we represent the cross sections as a lognormal spatial random process with specified mean, variance and covariance function and use a Karhunen-Loeve (KL) decomposition to generate cross section realizations that are strictly positive. Here we apply these techniques to radiation transport in media with spatially randomly varying cross sections without restriction on fluctuation amplitudes. Advances in UQ techniques have tended to focus on efficiently handling large numbers of uncertain variables but the rigorous stochastic basis of the approach also promotes its use in situations where stochasticity is due to spatial heterogeneity and the associated uncertainty is large. Recently, more » stochastic spectral methods such as polynomial chaos and stochastic collocation have been developed for aleatoric uncertainty quantification and sensitivity analysis, and successfully applied in radiation transport work. Attempts at developing approximate closures that yield only low order statistical information (e.g., mean and variance of the flux) have proved to be highly restrictive under real physics conditions or rely on techniques that require fluctuation amplitudes to be small for robustness. Neutral and charged particle transport computations in such media have relied heavily on formulating transport equations with spatially random coefficients (physical data) and developing solution methods to deal with the additional stochastic dimensions. Strongly heterogeneous media arise in several applications that include radiation shields, nuclear fuel, BWR moderators, clouds, planetary and stellar atmospheres, turbulent gases and plasmas. The theory and simulation of random variables and vectors is also reviewed for = , Second, we provide simple algorithms that can be used to generate independent samples of general stochastic models. First, we provide some theoretical background on stochastic processes and random fields that can be used to model phenomena that are random in space and/or time. While numerous algorithms and tools currently exist to generate samples of simple random variables and vectors, no cohesive simulation tool yet exists for generating samples of stochastic processes and/or random fields. The use of Monte Carlo simulation requires methods and algorithms to generate samples of the appropriate stochastic model these samples then become inputs and/or boundary conditions to established deterministic simulation codes. Mathematical models for these random phenomena are referred to as stochastic processes and/or random fields, and Monte Carlo simulation is the only general-purpose tool for solving problems of this type. Examples are diverse and include turbulent flow over an aircraft wing, Earth climatology, material microstructure, and the financial markets.
Many problems in applied science and engineering involve physical phenomena that behave randomly in time and/or space.