Collaborative Research: On Some Fundamental Computational Issues in Simulating Interaction Models

本项目将研究科学和工程应用中产生的数学相互作用模型的一些基本计算问题。这些模型的例子包括物理学中的电磁和引力相互作用，生物学中分子或细胞之间的相互作用，以及材料科学、量子理论和社会科学中更一般的分数阶微分方程和网络模型。显然，现代交互模型正变得越来越复杂，需要更精确和高效的计算方法来处理高性能计算机上的大规模交互。该项目将(1)引入适合加速计算的交互模型的新表示，(2)设计有效的交互评估算法，(3)开发先进的开源工具，(4)将其应用于纳米光子器件，遥感和医疗成像设备。该项目还将培训研究生，包括来自STEM领域代表性不足群体的研究生。该项目每年将支持一名毕业生在一个校区学习3年，并每年支持一名毕业生在另一个校区学习2年和3年。在大多数相互作用模型中，核通常取决于空间或时间位置，这些位置可能包括来自不同材料之间界面的贡献。它们甚至可能取决于源和目标位置的给定密度函数。为了更容易分析和加速计算，找到合适的压缩核表示是至关重要的。该项目将从层状介质的声波和电磁波的格林函数的最佳表示开始，这些声波和电磁波由于层状界面的贡献而在空间上发生变化。这些将通过最优积分轮廓和相应的离散基函数得到。pi还将拉普拉斯层势的部分波和平面波框架表示推广到汤川、亥姆霍兹和层状介质势。结果将导致物理、生物学、材料科学、社会科学和图像分析中更具挑战性的非局部模型的更好的压缩密度、核和潜在表示。pi将开发有效的数值方案来计算可压缩特征，并通过利用多分辨率框架来识别不同尺度的交互核特征来加速其代数运算。该项目还旨在为常用的拉普拉斯方程、汤川方程和亥姆霍兹方程创建先进的开源软件包。最后，通过与应用领域的科学家和工程师的合作，数值工具将用于设计最佳的纳米光子器件，如被动冷却装置，这可能导致减少碳足迹，并帮助社会经济弱势社区降低他们的能源账单。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will investigate some fundamental computational issues for mathematical interaction models arising from scientific and engineering applications. Examples of these models include the electromagnetic and gravitational interactions in physics, interactions between molecules or cells in biology, and more general fractional differential equations and network models in material science, quantum theory, and social science. It is evident that modern interaction models are becoming more complex and demanding more accurate and efficient computational methods to handle the large scale interactions on high-performance computers. This project will (1) introduce novel representations of interaction models that are suitable for accelerated computation, (2) design efficient algorithms for interaction evaluations, (3) develop advanced open source tools, and (4) apply them to applications in nano-photonic devices, remote sensing, and medical imaging devices. This project will also train graduate students, including those from under-represented groups in STEM fields. This project will support one graduate per year for all 3 years on one campus and one graduate per year for years 2 and 3 on the other campus. In most interaction models, kernels usually depend on the spatial or temporal locations that may include the contributions from the interfaces between different materials. They may even depend on the given density functions at both the source and target locations. It is critical to find suitably compressed kernel representations for easier analysis and accelerated computation. This project will start from the optimal representations of the layered media Green’s functions for acoustic and electromagnetic waves that are spatially variant due to the contributions from the layer interfaces. These will be found through optimal integration contours and the corresponding discretized basis functions. The PIs will also generalize the partial-wave and plane-wave frame representations of the Laplace layer potentials to Yukawa, Helmholtz, and layered media potentials. The result will lead to better compressed density, kernel, and potential representations of more challenging non-local models in physics, biology, material science, social science, and image analysis. The PIs will develop effective numerical schemes for computing the compressible features and accelerating their algebraic operations by utilizing a multi-resolution framework to identify the interaction kernel features at different scales. The project also aims to create advanced open source software packages for the commonly used Laplace, Yukawa, and Helmholtz equations. Finally, through collaborations with application domain scientists and engineers, the numerical tools will be used in the design of optimal nano-photonic devices such as passive cooling devices, which may lead to a reduced carbon footprint and help socioeconomically disadvantaged communities lower their energy bills.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.