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

这个学院早期职业发展(CALEAR)项目的研究目标是研究和开发一个框架，该框架利用粗略表示的渐近分析，系统地生成复杂随机系统的有效罕见事件模拟算法，这些算法必须在精细规模上实现。其目的是研究在罕见事件模拟中尚未被很好地研究的展现风格化特征的五种环境类型，即a)具有重尾的随机递归(用于模拟保险风险和水库过程)，b)重尾队列(出现在数据库和联网应用中)，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)沉浸在随机介质中的物体的位置(特别强调其中需要找到长期逃脱检测的目标的军事应用)，以及e)随机场(出现在诸如海洋学的环境中，环境研究和医学成像)。该策略包括将大偏差分析与高效模拟估值器的算法设计相结合。我们在算法的设计和性能分析中使用的一个关键工具是系统地使用马尔可夫链的Lyapunov界，并结合重要抽样分布的参数族。环境或自然灾害、重大市场崩盘、养老金和保险崩溃以及恐怖袭击等事件很少见，但后果重大。如果成功，建议的研究计划将为这类事件的风险评估提供有效的计算工具，这些事件表现出重尾、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量化政策评估指标和相应的洞察力。例如，计算目标能够躲避一组探测器的概率及其条件最有可能的位置，以及为了调整保险和金融公司的资本储备而评估破产概率。

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.