Algorithms for Complex Systems

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作用物组成，经常在多个长度和时间尺度上发生紧急现象。这样的系统具有全球结构和动力学特性，通常不可能准确确定。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，由于复杂系统的规模和多尺度效应，简单的算法往往太慢。因此，快速的算法设计以及对精度的严格研究是至关重要的。本研究项目致力于设计、改进和量化用于复杂系统的最新算法的误差。潜在的应用包括材料科学、计算化学和固体物理中出现的一系列问题。特别是，所研究的方法可以帮助为廉价和高效的硅基药物设计铺平道路。首席研究员将使用严格的数学分析和计算机实验相结合的方法来分析最先进的算法。一个重要的应用将是亚稳态系统的有效模拟，其中系统动力学往往在状态空间的某些子集中保持非常长的时间。这样的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，亚稳性产生于众所周知的时间尺度问题：原子振动发生的时间尺度远远小于热激活反应和其他有趣的动力学事件的时间尺度。出于这个原因，通常不可能通过直接的原子模拟来观察动力学中最有趣的方面。对于亚稳态动力学，许多近似模拟方法是直接原子模拟的替代方法，但它们的适用性和准确性受到限制。该项目将专注于克服亚稳性的几种方法，包括并行复制法、里程碑和动力学蒙特卡罗。PI将引入一个数学框架来推广这些算法，主要基于准静态分布，准静态分布是编码亚稳态的数学对象。PI将展示更通用的框架如何产生新的应用，包括更有效地模拟马尔可夫状态模型和眼镜。此外，PI将使用这个框架来追求严格的误差估计，这对于从模拟中提取定量信息至关重要。

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.