Collaborative Research: PACE-Parallel Accelerated Cartesian Expansions with Application to Molecular Dynamics

计算成对势函数对于模拟许多领域的基础物理是至关重要的，尽管计算成本很高。为了降低这一成本，人们开发了几种势函数的快速近似势计算方法；例如，粒子网格方法，快速傅里叶变换，快速多极方法（FMM），以及限制邻域计算。这些方法在效率、准确性和适用性方面各不相同。其中一名pi最近的工作为使用加速笛卡尔展开框架快速计算非振荡势的统一、鲁棒、精确和并行方法的发展提供了基础。本文采用了双管齐下的方法，包括开发(i)翻译算子，以实现基于FMM的不同两两势的计算，包括汤川势、伦纳德琼斯势、高斯势、莫尔斯势和白金汉势，以及（ii）并行框架，用于同时计算单个和多个势。这些技术将被应用于一组实际系统，包括泊松、扩散、延迟和亥姆霍兹（亚波长）和Klein-Gordon方程，以及计算范德华（在介视系统中）。底层的方法要求只有转换运算符从势改变到势，并为上下遍历FMM树提供了一个精确的数学公式。对计算多个势的统一处理简化了并行代码开发，特别是在可伸缩性方面。为了确保广泛的影响，这项研究的部分内容将作为LAMMPS软件包的一部分提供，以确保广泛传播。研究生将接受跨多个学科的培训，并将访问对方的机构。利用现有渠道招收妇女和少数民族学生，并通过高级设计项目和潜在的REU补充项目招收本科生。

Computation of pairwise potential functions is crucial, albeit computationally expensive, to simulating the underlying physics in many fields. To mitigate this cost, fast and approximate potential computation methods have been developed for several potential functions; for example, particle-mesh methods, Fast Fourier Transforms, Fast Multipole Method (FMM), and limiting computation to neighborhoods. These methods differ in efficiency, accuracy, and applicability. Recent work by one of the PIs provides the foundation for the development of unified, robust, accurate and parallel methods for fast computation of non-oscillatory potentials using the Accelerated Cartesian Expansion framework. A two pronged approach undertaken herein involves the development of (i) translation operators to enable FMM based computation for different pairwise potentials, including Yukawa, Lennard Jones, Gauss, Morse, and Buckingham potentials, and (ii) parallel framework for computing individual and multiple potentials simultaneously. These techniques are to be applied to a set of practical systems involving the Poisson, diffusion, retarded and Helmholtz (sub-wavelength), and Klein-Gordon equations, and to computing van-der Waals (in mesoscopic systems). The underlying methodology requires that only translation operators change from potential to potential, and provides a mathematically exact formulation for traversal up and down the FMM tree. The unifying treatment for computing multiple potentials simplifies parallel code development, especially with regard to scalability. To ensure broad impact, portions of this research will be available as part of LAMMPS software package to ensure widespread dissemination. Graduate students will be trained across multiple disciplines, and will visit each other's institutions. Existing channels are utilized to recruit women and minorities and undergraduate students are involved through senior design projects and potential REU supplements.