Presidential Young Investigator Award: Rapid Numerical Algorithms for Scientific Computation

一个目标是将快速多极子方法（FMM）结合到 分子动力学模拟软件包由 Charles Peskin和Tamar Schlick在Courant研究所。 在 这种粒子模拟，最大的计算负担， 经典的方法是评估 静电场感应的所有部分带电 单个原子。 这正是 FMM。 希望分子动力学中的某些不准确性 计算可以被消除，产生更多的物理 合理的结果。 第二个主要的研究领域是 热方程 早期的工作，与约翰应变， 已经产生了一个快速评估热势的方案。 计划在下一年投入大量的研究时间， 还有几年时间来审查该计划的扩展情况， 作为其后果。 势理论与 积分方程可以提供一个根本性的变化， 热方程求解，因为空间复杂和时间- 处理从属域没有额外的困难。 它 这种方法也很可能以自然的方式延伸， 变系数非线性抛物型偏微分方程 微分方程 第三个研究领域是材料中的相分离 科学 提出了一种新的基于积分方程的方法， 多连通域上的拉普拉斯方程构造 域. 计划将这一方法纳入模型 “奥斯特瓦尔德成熟”，它遵循的动态运动的 两个阶段之间的接口。 这就需要解决 每一步的拉普拉斯方程。

One goal is to incorporate the Fast Multipole Method (FMM) into the molecular dynamics simulation package being developed by Charles Peskin and Tamar Schlick at the Courant Institute. In such particle simulations, the greatest computational burden has classically been that of evaluating the self-consistent electrostatic field induced by all of the partially charged individual atoms. This is precisely the problem addressed by the FMM. The hope is that certain inaccuracies in molecular dynamics calculations can be eliminated, yielding more physically reasonable results. The second major area of research is potential theory for the heat equation. Earlier work, in collaboration with John Strain, has yielded a scheme for the rapid evaluation of heat potentials. It is planned to devote substantial research time over the next few years to the examination of extensions of this scheme as well as its ramifications. The combination of potential theory and integral equations may provide a fundamental change in the way heat equations are solved, since spatially complex and time- dependent domains are handled without additional difficulty. It also appears likely that the method will extend in a natural way to variable coefficient and nonlinear parabolic partial differential equations. The third area of research is phase separation in material science. A new integral equation-based scheme has been constructed for the Laplace equation in multiply connected domains. It is planned to incorporate this method into a model of "Ostwald ripening," which follows the dynamic motion of the interface between two phases. This requires the solution of Laplace's equation at each step.