CAREER: Multiscale stochastic processes, Monte Carlo Methods and Irreversibility

当今应用数学和概率所面临的挑战之一是获得准确且可证明有效的方法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，这些现象通常过于复杂，无法解决，近似甚至模拟。本研究的主要目的是严格调查多尺度系统，罕见事件和相关的蒙特卡罗模拟方法相关的问题。我们有兴趣研究可能具有不同时间尺度的随机动力系统，并研究和量化在给定时间尺度内可能罕见但可能对系统本身产生重要影响的相关事件。精确的解决方案通常是不可能的，所以必须依赖于模拟，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。在这个项目中感兴趣的问题都是由基本的数学问题和其他科学分支的广泛的问题。例子包括化学物理学中的稀有事件概率估计、生物学中的遗传开关模型、工程学中的跟踪环路问题，以及复杂金融系统中的网络故障和大量投资组合损失，其中多尺度特征和稀有事件是核心问题。此外，该研究项目与一个教育计划相结合，该计划旨在帮助培养应用数学，物理，工程和化学的本科生和研究生，以探索罕见事件，多尺度过程及其分析，校准和模拟。这个研究项目导致了一个严格的数学框架的发展，允许设计可证明有效的算法，稀有事件的建模和估计以及统计校准方法。我们对大偏差区（尾部或罕见事件）和中等偏差区（分布的典型中心和尾部之间的差距）都感兴趣。我们开发了可证明有效的蒙特卡罗方法，如重要性抽样和统计校准方法。在一个密切相关的方向，我们严格调查违反时间可逆性的后果，如稳态模拟和算法加速收敛到平衡的蒙特卡罗方法的设计。该研究项目试图使尚未很好理解的概念和方法具体化，例如亚稳态对蒙特卡罗方法的影响，以及多尺度对大偏差和中等偏差，蒙特卡罗方法和统计估计方法的影响。我们的目标是为复杂的随机多尺度动力系统的模拟和统计过程提供一个有用的和可靠的方法，这些方法可以被广泛的学科和开放的前沿研究所使用。亚稳态行为和多尺度现象是中心的兴趣。

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.