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

这个项目将研究在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组和网络模型。很明显，现代的交互模型变得越来越复杂，需要更准确、更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)引入适合于加速计算的新的交互模型表示，(2)设计高效的交互评估算法，(3)开发先进的开源工具，(4)将其应用于纳米光子器件、遥感和医学成像设备中。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。该项目将在一个校区的三年内每年资助一名毕业生，在另一校区的二年级和三年级每年资助一名毕业生。在大多数相互作用模型中，核通常取决于空间或时间位置，其中可能包括不同材料之间界面的贡献。它们甚至可能取决于源位置和目标位置处的给定密度函数。为了便于分析和加速计算，找到适当压缩的核表示是至关重要的。这个项目将从分层介质格林函数的最佳表示开始，声波和电磁波的格林函数由于层界面的贡献而在空间上是不同的。这些将通过最优积分等值线和相应的离散化的基函数来找到。PI还将拉普拉斯层势的分波和平面波框架表示推广到Yukawa、Helmholtz和分层媒质势。其结果将导致更好的压缩密度、核和更具挑战性的非局部模型在物理、生物学、材料科学、社会科学和图像分析中的潜在表示。通过利用多分辨率框架来识别不同尺度上的相互作用核特征，PI将开发有效的数值方案来计算可压缩特征并加速其代数运算。该项目还旨在为常用的Laplace、Yukawa和Helmholtz方程创建高级开源软件包。最后，通过与应用领域的科学家和工程师的合作，数值工具将被用于设计最佳的纳米光子设备，如被动冷却设备，这可能会减少碳足迹，并帮助处于社会经济不利地位的社区降低他们的能源账单。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.