CAREER: Multiscale stochastic processes, Monte Carlo Methods and Irreversibility

当今应用数学和概率学所面临的挑战之一是获得准确且被证明有效的方法来使用概率模型来近似和模拟一系列复杂系统。概率模型通常用于表示物理、生物和金融现象，然而这些现象往往过于复杂，无法解决、近似甚至模拟。这项研究的主要目的是严格研究与多尺度系统、罕见事件和相关的蒙特卡罗模拟方法有关的问题。我们感兴趣的是研究可能具有不同时间尺度的随机动力系统，以及研究和量化在给定时间尺度上可能很少发生的相关事件，但这些事件可能会对系统本身产生重要影响。精确的解通常是不可能的，所以人们必须依赖于模拟，因此，严格开发可证明有效的近似方法和模拟蒙特卡罗方法是我们分析的核心。这个项目中感兴趣的问题既有基本的数学问题，也有其他科学分支的广泛问题。例子包括化学物理中罕见事件概率的估计、生物学中的遗传开关模型、工程中的跟踪回路问题、复杂金融系统中的网络故障和巨额投资组合损失，在这些系统中，多尺度特征和罕见事件是核心问题。此外，该研究项目还与一个教育项目相结合，该项目旨在帮助本科生和研究生在罕见事件的探索、多尺度过程及其分析、校准和模拟方面进行应用数学、物理、工程和化学方面的培训。该研究项目导致了一个严格的数学框架的发展，该框架允许设计与罕见事件的建模和估计以及统计校准方法相关的被证明有效的算法。我们对大偏差区域(尾部或罕见事件)和中等偏差区域(典型分布中心和尾部之间的差距)都感兴趣。我们发展了可证明有效的蒙特卡罗方法，如重要抽样和统计校准方法。在一个密切相关的方向上，我们严格研究了蒙特卡罗方法设计中违反时间可逆性的后果，如稳态模拟和加速收敛到平衡的算法。该研究项目试图澄清一些不太清楚的概念和方法，如亚稳性对蒙特卡罗方法的影响，多尺度对大偏差和中偏差的影响，对蒙特卡罗方法和统计估计方法的影响。其目的是为复杂随机多尺度动力系统的模拟和统计过程提供一种有用和可靠的方法，该方法可被广泛的学科集合使用，并为研究开辟前沿。亚稳行为和多尺度现象是人们关注的中心问题。

One of the challenges facing today applied mathematics and probability is to obtain accurate and provably efficient methods to approximate and simulate a range of complex systems using probabilistic models. Probabilistic models are commonly used to represent physical, biological and financial phenomena that are often however too complex to solve, approximate or even simulate. The primary purpose of this research is to rigorously investigate problems related to multiscale systems, rare events and related Monte Carlo simulation methods. We are interested in studying stochastic dynamical systems that may have different time scales and understudying and quantifying related events that may be rare in a given time scale, but can have important consequences for the system itself. Exact solutions are typically impossible, so one has to rely on simulation and for this reason rigorous development of provably-efficient approximation methods and simulation Monte Carlo methods is in the core of our analysis. The problems of interest in this project are motivated both by fundamental mathematical questions and by a broad array of questions in other branches of science. Examples range from estimation of rare event probabilities in chemical physics, genetic switch models in biology, tracking loop problems in engineering to network failure and large portfolio losses in complex financial systems, where multiscale features and rare events are core issues. In addition, the research project is integrated with an educational program that is designed to help in the training of undergraduate and graduate students in applied mathematics, physics, engineering and chemistry in the exploration of rare events, multiscale processes and their analysis, calibration and simulation.This research project leads to the development of a rigorous mathematical framework that allows to design provably efficient algorithms associated to modeling and estimation of rare events as well as statistical calibration methods. We are interested in both the large deviations regime (tail or rare events) as well as in the moderate deviations regime (the gap between the typical center and the tail of the distribution). We develop provably-efficient Monte Carlo methods such as importance sampling and statistical calibration methods. In a closely related direction we rigorously investigate the consequences of violation of time-reversibility in the design of Monte Carlo methods such as steady state simulation and algorithms for acceleration of convergence to equilibrium. The research project attempts to crystallize concepts and methods that are not well understood such as the effect of metastability on Monte Carlo methods and of multiple scales on large and moderate deviations, on Monte Carlo methods and on statistical estimation methods. The goal is to provide a useful and reliable methodology for simulation and statistical procedures for complex stochastic multiscale dynamical systems that can be used by a wide collection of disciplines and also open frontiers for research. Metastability behavior and multiscale phenomena are of central interest.