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，利用在粗尺度上表达的渐近分析，系统地为复杂的随机系统生成有效的罕见事件模拟算法，这些算法必须在精细尺度上实现。目标是研究五种类型的环境，这些环境表现出在罕见事件模拟中尚未得到很好研究的风式化特征，即a)带有重尾的随机递归（用于模拟保险风险和水库过程），b)重尾队列（出现在数据库和网络应用中），c)组合结构的计数问题和推理（出现在社会学和生物学中），d)浸没在随机介质中的物体的位置（特别强调需要寻找长期未被探测到的目标的军事应用），以及e)随机场（在诸如海洋学、环境研究和医学成像等环境中出现）。该策略是将大偏差分析与高效仿真估计器的算法设计相结合。我们在算法的设计和性能分析中利用的一个关键工具是系统地使用马尔可夫链的李雅普诺夫界，结合重要抽样分布的参数族。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很少发生，但后果重大。如果成功，该研究计划将为此类事件的风险评估提供有效的计算工具，这些事件表现出诸如重尾、复杂依赖和组合对象合并等特征。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伴随的见解。例子包括计算目标能够逃避一组检测器的概率，以及它的条件最可能位置，以及评估破产概率，以确定保险和金融公司的资本储备规模。

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.