Presidential Young Investigator Award: Rapid Numerical Algorithms for Scientific Computation

其中一个目标是将快速多极方法(FMM)纳入到库兰特研究所的Charles Peskin和Tamar Schlick正在开发的分子动力学模拟程序包中。在这样的粒子模拟中，最大的计算负担通常是评估由所有部分带电的单个原子诱导的自洽静电场。这正是FMM解决的问题。希望可以消除分子动力学计算中的某些不准确，产生更合理的物理结果。第二个主要研究领域是热势理论。早期的工作，与约翰·斯特拉特合作，已经产生了一个快速评估热势的方案。当局计划在未来数年投入大量研究时间，研究这项计划的扩展范围及其影响。位势理论和积分方程组的结合可能会从根本上改变求解热方程的方法，因为处理空间复杂和依赖时间的区域不会有额外的困难。这种方法也有可能以自然的方式推广到变系数和非线性抛物型偏微分方程组。第三个研究领域是材料科学中的相分离。构造了多连通区域上拉普拉斯方程的一种新的基于积分方程组的格式。计划将这种方法结合到一个“奥斯特瓦尔德成熟”模型中，该模型跟踪两相之间界面的动态运动。这就要求每一步都要解拉普拉斯方程。

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.