CAREER: Efficient Monte Carlo Methods in Engineering and Science: From Coarse Analysis to Refined Estimators

这个教师早期职业发展（CAREER）项目的研究目标是研究和开发一个框架，利用渐近分析，在粗尺度上表示，系统地生成有效的稀有事件模拟算法的复杂随机系统，这必须在一个精细的规模实施。我们的目标是研究五种类型的环境，表现出程式化的特点，还没有得到很好的研究，在罕见事件模拟，即：a）随机递归与重尾（用于对保险风险和水库过程进行建模），B）重尾队列（出现在数据库和网络应用中），c）计数问题和组合结构的推断（产生于社会学和生物学），d）浸没在随机介质中的物体的定位（特别强调军事应用，其中需要找到长时间逃避检测的目标），以及e）随机场（其产生于诸如海洋学、环境研究和医学成像的设置中）。该策略包括连接大偏差分析与算法设计的有效模拟估计。一个关键的工具，我们利用我们的算法的设计和性能分析是一个系统的使用马尔可夫链的李雅普诺夫界，结合参数家庭的重要性抽样分布。 环境或自然灾害、重大市场崩溃、养老金和保险崩溃以及恐怖袭击等事件虽然罕见，但后果严重。如果成功的话，拟议的研究计划将提供有效的计算工具，这些事件的风险评估表现出的功能，如重尾，复杂的依赖性和组合对象的合并。对罕见事件概率的有效评估可以为决策者提供关键的定量政策评估指标和相应的见解。例子包括计算目标能够避开一组检测器的概率以及其条件最可能的位置，并评估破产概率以确定保险和金融公司的资本储备。

The research objective of this Faculty Early Career Development (CAREER) project is to investigate and develop a framework that exploits asymptotic analysis, expressed at a coarse scale, to systematically generate efficient rare-event simulation algorithms for complex stochastic systems, which must necessarily be implemented at a fine scale. The objective is to study five types of environments that exhibit stylized features that have not been well studied in rare-event simulation, namely, a) Stochastic recursions with heavy-tails (which are used to model insurance risk and reservoir processes), b) Heavy-tailed queues (which arise in database and networking applications), c) Counting problems and inference for combinatorial structures (arising in sociology and biology), d) Location of objects immersed in a random medium (with particular emphasis on military applications where one needs to find targets that have eluded detection for long time), and e) Random fields (which arise in settings such as oceanography, environmental studies and medical imaging). The strategy consists in connecting large deviations analysis with algorithmic design of efficient simulation estimators. A key tool that we exploit in the design and performance analysis of our algorithms is a systematic use of Lyapunov bounds for Markov chains, combined with parametric families of importance sampling distributions. Events such as environmental or natural disasters, major market crashes, pension and insurance breakdowns and terrorist attacks are rare but consequential. If successful, the proposed research program will provide efficient computational tools for risk assessment of such events which exhibit features such as heavy-tails, complex dependence and incorporation of combinatorial objects. Efficient evaluation of rare-event probabilities can provide decision makers with key quantitative policy assessment metrics and accompanying insights. Examples include computing the probability that a target is able to evade a set of detectors as well as its conditional most likely location, and assessing ruin probabilities for purposes of sizing the capital reserve of insurance and financial companies.