Efficient Monte Carlo Methods for Gaussian Random Fields

该奖项的研究目标是开发高效的蒙特卡罗方法，用于计算高斯随机场中罕见事件的概率和给定稀有事件的条件分布下场的几何性质。一个重要的方法论思想包括利用极限定理和渐近逼近来指导可应用于预极限的高效蒙特卡罗方法的构造。关于高斯随机场中某些罕见事件(如高层漂移)的概率的渐近性，已经有了丰富的文献。PI利用这种渐近性发展过程中隐藏的信息来开发蒙特卡罗方法，该方法在大多数情况下是基于重要性抽样的。这项研究还包括利用有效的马尔可夫链蒙特卡罗技术结合凸优化算法和重要抽样思想来研究PI计划攻击的一类高维#P-Hard问题。这项研究的动机来自环境科学、图像分析、统计应用、风险管理等领域。如果成功，这一成果可能会对广泛的科学领域产生高度和积极的影响。例如，在与城市发展相关的环境研究中，在发生高污染的情况下，有效地评估一个地理区域不同地区的污染水平变化的能力，将在制定政策和决策过程中增加实质性的价值。这项研究的结果也可能对处理高斯随机场及其在克里格法和最优化中的应用的其他模拟领域有潜在的帮助。

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.