Efficient Monte Carlo Methods for Gaussian Random Fields

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The research objective of this award is for the development of efficient Monte Carlo methods for computing probabilities of rare events in Gaussian random fields and geometric properties of the fields under the conditional distribution given rare events of interest. A crucial methodological idea involves taking advantage of limit theorems and asymptotic approximations in order to guide the construction of efficient Monte Carlo methods that can be applied in the pre-limit. There is a rich literature on asymptotics for the probabilities of certain rare events in Gaussian random fields (such as high level excursions). The PI's exploit the information hidden in the development of such asymptotics in order to develop the Monte Carlo methodology, which in most cases is based on importance sampling. This research also includes the investigation of a class of high dimensional #P-hard problems that the PI's plan to attack by taking advantage of efficient Markov chain Monte Carlo techniques combined with convex optimization algorithms and importance sampling ideas.This research is motivated by a variety of applications, ranging from environmental sciences, image analysis, statistical applications, risk management and so forth. If successful, the output may highly and positively impact a wide range of scientific areas. For instance, in environmental studies tied to urban development, the ability of efficiently evaluating changes in contamination levels in different areas of a geographic region given that high contamination occurs will add substantial value in the development of policies and decision making processes. The output of this research may also potentially aid other simulation areas dealing with Gaussian random fields and their applications to kriging and optimization.