AF: Small: Laplace-de Rham Operators in Scientific Computing and Data Analysis

几何空间的数学问题在科学和工程的许多领域中是普遍存在的。 例如，在数据分析中，中心任务是提取数据的重要特征，在许多情况下是高维点集。 然后，重要特征的提取导致应用程序，例如可视化数据或以有意义的方式对数据进行聚类。 另一个例子是在科学计算中，在几何空间内对物理现象（如热、声或电）的行为进行建模是一个核心问题。 这些问题可以用偏微分方程（PDE）来描述，对于偏微分方程，高效的求解器是必要的。 这些几何问题中的许多都可以转换为线性代数中的问题：拉普拉斯-德拉姆算子的计算问题，一系列简洁地表示几何空间的矩阵。 因此，更快，更简单的方法可以开发这些问题使用强大的工具，从线性代数，这是这个项目的重点。最近的进展，在研究图拉普拉斯算子（第一矩阵的拉普拉斯-德拉姆序列）已经导致了一系列简单而优雅的算法，有助于重要的应用程序在计算机科学和数据分析。相比之下，拉普拉斯-德拉姆算子的高维变体，尽管它们在科学和工程中的关键作用（例如，在求解物理方程中，在执行亥姆霍兹/霍奇分解以用于数据可视化和统计排名中，或者在揭示隐藏空间的有效几何或拓扑性质中），但在很大程度上仍然未被探索。该项目将填补这一空白，解决两个高层次目标下的两种不同类型的应用程序的拉普拉斯-德拉姆算子的重要开放问题：（1）设计拉普拉斯-德拉姆矩阵的有效求解器，该拉普拉斯-德拉姆矩阵出现在由偏微分方程（PDE）引起的重要问题中，例如，在科学计算中，Navier-Stokes或麦克斯韦方程中的向量拉普拉斯算子，以及（2）发现和利用拉普拉斯-德拉姆矩阵及其光谱的有效性质进行数据分析。该奖项反映了NSF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。

Algorithmic problems about geometric spaces are ubiquitous in many fields of science and engineering. For example, in data analysis, a central task is to extract important features of data, a high-dimensional point set in many cases. The extraction of the significant features then leads to applications such as visualizing the data or clustering the data in a meaningful way. Another example is in scientific computing where modeling the behavior of a physical phenomenon such as heat, sound or electricity within a geometric space is a central problem. These problems can be specified by partial differential equations (PDEs), for which efficient solvers are necessary. Many of these geometric problems can be cast as problems in linear algebra: computational problems on Laplace-de Rham operators, a sequence of matrices that succinctly present geometric spaces. Thus, faster and simpler methods can be developed for these problems using powerful tools from linear algebra, which is the focus of this project.Recent advances in the study of graph Laplacians (the first matrix in the Laplace-de Rham sequence) have resulted in a series of simple and elegant algorithms that helped important applications in computer science and data analysis. In contrast, higher dimensional variants of Laplace-de Rham operators, in spite of their key roles in science and engineering (e.g. in solving physical equations, in performing Helmholtz/Hodge decomposition for applications in data visualization and statistical ranking, or in revealing effective geometric or topological properties of a hidden space) have remained largely under-explored. This project will fill this gap by addressing important open problems about Laplace-de Rham operators for two different types of applications under two high-level aims: (1) design efficient solvers for Laplace-de Rham matrices that appear in important problems raised from partial differential equations (PDEs), e.g. vector Laplacians in Navier-Stokes or Maxwell equations, in scientific computing, and (2) discover and harness effective properties of Laplace-de Rham matrices and their spectra for data analysis.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.