Algorithms for Complex Systems

复杂系统的算法

• 批准号: 1522398
• 负责人: David Aristoff
• 金额: $ 17.98万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-15 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522398&HistoricalAwards=false
• 关键词: Algorithms; Complex; Systems

Complex systems arise in many scientific problems. They are high dimensional structures, comprised of many locally interacting agents, with emergent phenomena often occurring at multiple length and time scales. Such systems have global structural and dynamical properties that are usually impossible to determine exactly. Instead, these properties must be estimated by computer experiment and simulation. Unfortunately, due to the size and multiscale effects in complex systems, straightforward algorithms are often too slow. Thus, fast algorithm design, along with a rigorous study of accuracy, is crucial. This research project focuses on designing, improving, and quantifying the error of state-of-the-art algorithms for complex systems. Potential applications include a wide range of problems arising in materials science, computational chemistry, and solid-state physics. In particular, the methods studied could help pave the way for cheap and efficient in silico drug design. The principal investigator will analyze state-of-the-art algorithms using a mixture of rigorous mathematical analysis and computer experiment. An important application will be the efficient simulation of metastable systems, in which the system dynamics tend to remain for very long times in certain subsets of state space. Such systems are widespread in molecular dynamics, an increasingly important tool in computational chemistry. In molecular dynamics, metastability arises from the well-known time scale problem: atomic vibrations occur on a time scale much smaller than that of thermally activated reactions and other interesting dynamical events. For this reason, it is usually impossible to observe the most interesting aspects of the dynamics by direct atomistic simulations. For metastable dynamics, many approximate simulation methods serve as alternatives to direct atomistic simulation, but they are limited by applicability and accuracy. The project will focus on several methods for overcoming metastability, including the parallel replica method, milestoning, and kinetic Monte Carlo. The PI will introduce a mathematical framework for generalizing these algorithms, based largely on the quasi-stationary distribution, a mathematical object that encodes metastability. The PI will show how the more general framework leads to new applications, including more efficient simulation of Markov State Models and glasses. Moreover, the PI will use this framework to pursue rigorous error estimates, which are crucial for extracting quantitative information from simulations.

复杂系统出现在许多科学问题中。它们是高维结构，由许多局部相互作用的主体组成，突发现象通常在多个长度和时间尺度上发生。此类系统具有通常无法准确确定的全局结构和动力学特性。相反，这些特性必须通过计算机实验和模拟来估计。不幸的是，由于复杂系统的尺寸和多尺度效应，简单的算法通常太慢。因此，快速的算法设计以及对准确性的严格研究至关重要。该研究项目的重点是设计、改进和量化复杂系统最先进算法的误差。潜在的应用包括材料科学、计算化学和固态物理学中出现的广泛问题。特别是，所研究的方法可以帮助为廉价且高效的计算机药物设计铺平道路。首席研究员将结合严格的数学分析和计算机实验来分析最先进的算法。一个重要的应用是亚稳态系统的有效模拟，其中系统动力学往往会在状态空间的某些子集中保持很长时间。此类系统在分子动力学中广泛存在，分子动力学是计算化学中日益重要的工具。在分子动力学中，亚稳态源于众所周知的时间尺度问题：原子振动发生的时间尺度远小于热激活反应和其他有趣的动力学事件的时间尺度。因此，通常不可能通过直接原子模拟来观察动力学最有趣的方面。对于亚稳态动力学，许多近似模拟方法可以作为直接原子模拟的替代方法，但它们受到适用性和准确性的限制。该项目将重点研究克服亚稳态的几种方法，包括并行复制法、里程碑法和动力学蒙特卡罗法。 PI 将引入一个数学框架来推广这些算法，该框架主要基于准平稳分布（一种编码亚稳态的数学对象）。该 PI 将展示更通用的框架如何带来新的应用，包括更有效地模拟马尔可夫状态模型和眼镜。此外，PI 将使用该框架进行严格的误差估计，这对于从模拟中提取定量信息至关重要。