Mathematical Sciences: Numerical Methods for Molecular Dynamics

这项工作涉及分析、综合和测试有效的数值时间步进方案，用于微分方程哈密顿系统的长时间模拟。这是出于提高模拟生物分子动力学的计算机程序性能的愿望，特别关注MDScope模拟程序NAMD/SIgMA。许多提出的工作是一般性的，并直接适用于由哈密顿系统建模的各种应用，如天体物理学、一些结构动力学问题和波传播。有些工作不太通用，但可以适用于其他应用。典型地，哈密顿系统的复杂动力学是许多简单相互作用的综合效应，通常具有一系列的时间尺度。该研究的一个重点是发展健全的“长时间步长”方法，这种方法利用了运动的许多组成部分中的缓慢性和其他组成部分中可能存在的规律性。更具体地说，这种方法在时间增量上评估一些相互作用，这些时间增量远远超过了系统快速模态周期的一半。另一个重点是时间步进方案的设计，它可以计算具有正确质量的长轨迹。一个特定的目标是构建一个多时间步进方案，该方案既可以用于长时间步进，也可以用于长时间模拟——现有方案在一个方面或另一个方面失败。另一个具体目标是通过在不牺牲定性性质的情况下降低某些计算的准确性来降低成本，例如反平方律力。人们正在设计数学公式，以预测蛋白质和DNA等大分子的典型运动。其目的是找到能够在大型计算机上进行快速计算的快捷方式。这些想法经过准确性和速度的测试，更有用的想法被放入生物学家和其他科学家用来研究分子行为的软件中。计算机模拟可以帮助科学家了解分子如何发挥生物功能，这很难通过实验观察到。这类研究在农业和医疗用途的分子设计中也很有用。许多提出的工作是一般性的，并扩展到耗散可以忽略不计的其他应用，如天体物理学、一些结构动力学问题和波传播。

9600088 Skeel The work involves the analysis, synthesis, and testing of efficient numerical time stepping schemes for the long-time simulation of Hamiltonian systems of differential equations. This is motivated by the desire to improve the performance of computer programs that simulate the dynamics of biomolecules, with particular attention to the MDScope simulation program NAMD/SIgMA. Much of the proposed work is general and applies directly to a variety of applications modeled by Hamiltonian systems, such as astrophysics, some structural dynamics problems, and wave propagation. Some of the work is less general but can be adapted to other applications. Typically, the complicated dynamics of Hamiltonian systems is a combined effect of many simple interactions, often having a range of time scales. A key emphasis of the research is the development of sound "long-timestep" methods, which exploit the slowness present in many components of the motion and the possible regularity present in other components. More specifically, such methods evaluate some interactions at time increments that well exceed half the period of the fast modes of the system. Another emphasis is the design of time stepping schemes that can compute long trajectories having the right qualities. A specific goal is the construction of a multiple time stepping scheme that can be used both with long timesteps and for long-time simulations--existing schemes fail in one respect or the other. Another specific goal is to reduce the cost of certain calculations, such as inverse-square-law forces, by reducing their accuracy without sacrificing qualitative properties. Mathematical formulas are being devised that can predict typical motions of large molecules such as proteins and DNA. The aim is find shortcuts that enable very fast calculations on large computers. These ideas are tested for accuracy and speed, and the more useful ones are put into the software that biologis ts and other scientists use to study the action of molecules. Computer simulations can help the scientist to understand how molecules perform biological functions, which can be very difficult to observe experimentally. Such studies are also useful in the design of molecules for agricultural and medical purposes. Much of the proposed work is general and extends to other applications in which dissipation is negligible, such as astrophysics, some structural dynamics problems, and wave propagation.