Mathematical Sciences: Numerical Methods for Molecular Dynamics

9600088 Skeel这项工作涉及分析、综合和测试用于哈密顿微分方程组的长期模拟的有效的数值时间推进格式。这是由于希望改善模拟生物分子动力学的计算机程序的性能，尤其是MDScope模拟程序NAMD/Sigma。许多拟议的工作是一般性的，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。有些工作不那么通用，但可以适用于其他应用程序。通常，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，通常具有一定的时间尺度。研究的一个重点是开发合理的“长时间步长”方法，这种方法利用运动的许多部分存在的缓慢和其他部分存在的可能的规律性。更具体地说，这种方法以远远超过系统快速模式周期的一半的时间增量来评估一些交互作用。另一个重点是时间步进方案的设计，该方案可以计算具有正确质量的长轨迹。一个具体的目标是构建一种既可用于长时间步长又可用于长时间模拟的多时间步进方案--现有方案在这方面或那方面都失败了。另一个具体目标是在不牺牲定性属性的情况下，通过降低某些计算的精度来降低某些计算的成本，例如平方反比力。正在设计的数学公式可以预测蛋白质和DNA等大分子的典型运动。其目的是找到在大型计算机上实现非常快速计算的捷径。这些想法的准确性和速度都经过了测试，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作用的软件中。计算机模拟可以帮助科学家了解分子如何执行生物功能，而这可能是很难通过实验观察到的。这类研究对农业和医疗用途的分子设计也很有用。许多建议的工作是一般性的，并扩展到其他应用，其中耗散可以忽略不计，如天体物理学，一些结构动力学问题，和波的传播。

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.