Rare-event Analysis and Computational Methods for Stochastic Systems Driven by Random Fields

: Random fields constitute a corner stone of many probability models to characterize spatially varying uncertainties. This research develops asymptotic theories and numerical methods for computing rare-event probabilities associated with random fields and the associated random elliptic differential equations. The equation has been employed to describe physics systems in areas as diverse as material science, fluid dynamics, neuroscience, fiber optics, astronomy, further civil engineering, engineer design, ocean-earth sciences, and so forth. We perform risk analysis of such systems by investigating the asymptotic behavior of certain interesting rare events. For instance, we provide assessment and efficient numerical methods for the risk of material's not being able to support a certain amount of external force and of predicting the major causes of material failure. Such an analysis will add substantial value in the development of policies and decision making processes. The output of this research has positive impacts on those areas and also aids other areas of asymptotic analysis and simulation development of random fields and the associated stochastic systems.