CAREER: Multiscale stochastic processes, Monte Carlo Methods and Irreversibility

职业：多尺度随机过程、蒙特卡罗方法和不可逆性

• 批准号: 1550918
• 负责人: Konstantinos Spiliopoulos
• 金额: $ 48万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1550918&HistoricalAwards=false
• 关键词: CAREER; Multiscale; stochastic; processes; Monte

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.

当今应用数学和概率论面临的挑战之一是如何利用概率模型获得精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，然而这些现象往往太复杂而无法解决、近似甚至模拟。本研究的主要目的是对多尺度系统、罕见事件及相关的蒙特卡罗模拟方法进行严谨的研究。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，并对相关事件进行研究和量化，这些事件在给定的时间尺度内可能是罕见的，但可能对系统本身产生重要影响。精确的解通常是不可能的，因此必须依靠模拟，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。本项目中感兴趣的问题是由基础数学问题和其他科学分支中的广泛问题引起的。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大型投资组合损失，其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计划相结合，旨在帮助培养应用数学、物理、工程和化学领域的本科生和研究生，帮助他们探索罕见事件、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，允许设计与罕见事件建模和估计以及统计校准方法相关的可证明的有效算法。我们既对大偏差区（尾部或罕见事件）感兴趣，也对中等偏差区（典型中心和分布尾部之间的间隙）感兴趣。我们开发了可证明高效的蒙特卡罗方法，如重要抽样和统计校准方法。在一个密切相关的方向，我们严格研究违反时间可逆性的后果在蒙特卡罗方法的设计，如稳态模拟和算法加速收敛到平衡。该研究项目试图使尚未得到很好理解的概念和方法结晶化，例如亚稳态对蒙特卡罗方法的影响以及对大偏差和中等偏差的多重尺度的影响，对蒙特卡罗方法的影响以及对统计估计方法的影响。目标是为复杂随机多尺度动力系统的模拟和统计程序提供一种有用和可靠的方法，可以被广泛的学科集合使用，也为研究开辟了前沿。亚稳态行为和多尺度现象是中心兴趣。

项目成果

Mean Field Limits of Particle-Based Stochastic Reaction-Diffusion Models

基于粒子的随机反应扩散模型的平均场极限

• DOI: 10.1137/20m1365600
• 发表时间: 2022
• 期刊: SIAM journal on mathematical analysis
• 影响因子: 2
• 作者: Isaacson, Samuel A.;Ma, Jingwei;Spiliopoulos, Konstantinos
• 通讯作者: Spiliopoulos, Konstantinos

Moderate deviations for systems of slow–fast stochastic reaction–diffusion equations

• DOI: 10.1007/s40072-022-00236-y
• 发表时间: 2020-12
• 期刊: Stochastics and Partial Differential Equations: Analysis and Computations
• 作者: Ioannis Gasteratos;M. Salins;K. Spiliopoulos

Online Adjoint Methods for Optimization of PDEs

偏微分方程优化的在线伴随方法

• DOI: 10.1007/s00245-022-09852-5
• 发表时间: 2022
• 期刊: Applied Mathematics & Optimization
• 影响因子: 1.8
• 作者: Sirignano, Justin;Spiliopoulos, Konstantinos
• 通讯作者: Spiliopoulos, Konstantinos