Presidential Young Investigator Award: Rapid Numerical Algorithms for Scientific Computation

目标之一是将快速多极方法 (FMM) 纳入 Courant 研究所的 Charles Peskin 和 Tamar Schlick 正在开发的分子动力学模拟包中。 在此类粒子模拟中，最大的计算负担通常是评估由所有部分带电的单个原子引起的自洽静电场。 这正是 FMM 所要解决的问题。 希望能够消除分子动力学计算中的某些不准确之处，从而产生更物理上合理的结果。 第二个主要研究领域是热方程的势能理论。 早期与 John Strain 合作的工作已经产生了一种快速评估热势的方案。 计划在未来几年投入大量研究时间来审查该计划的扩展及其后果。 势论和积分方程的结合可以为热方程的求解方式带来根本性的改变，因为空间复杂和时间相关的域的处理没有额外的困难。 该方法也可能以自然的方式扩展到变系数和非线性抛物型偏微分方程。 第三个研究领域是材料科学中的相分离。 为多重连通域中的拉普拉斯方程构造了一种新的基于积分方程的方案。 计划将该方法纳入“奥斯特瓦尔德熟化”模型中，该模型遵循两相之间界面的动态运动。 这需要在每一步求解拉普拉斯方程。

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.