Efficient Monte Carlo Methods for Gaussian Random Fields

该奖项的研究目标是发展有效的蒙特卡罗方法，用于计算高斯随机场中罕见事件的概率，以及给定稀有事件的条件分布下的场的几何性质。一个关键的方法论思想涉及利用极限定理和渐近逼近，以指导有效的蒙特卡罗方法的构建，可以应用于前极限。关于高斯随机场中某些罕见事件（如高能级漂移）概率的渐近性，已有丰富的文献。PI利用隐藏在这些渐近发展中的信息，以发展蒙特卡罗方法，在大多数情况下，这是基于重要抽样。本研究还包括对一类高维#P-hard问题的调查，PI计划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攻击这些问题。这项研究的动机是各种各样的应用，从环境科学，图像分析，统计应用，风险管理等。如果成功，产出可能会对广泛的科学领域产生高度和积极的影响。例如，在与城市发展有关的环境研究中，在发生高污染的情况下，有效评价一个地理区域内不同地区污染程度变化的能力将大大有助于制定政策和决策过程。这项研究的结果也可能潜在地帮助其他模拟领域处理高斯随机场及其在克里格和优化中的应用。

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.