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

几何空间的算法问题在科学和工程的许多领域都是普遍存在的。例如，在数据分析中，一个中心任务是提取数据的重要特征，在许多情况下是一个高维点集。提取重要特征之后，就可以使用可视化数据或以有意义的方式对数据进行聚类等应用程序。另一个例子是在科学计算中，对几何空间内的热、声或电等物理现象的行为进行建模是一个中心问题。这些问题可以用偏微分方程（PDEs）来表示，因此需要有效的求解器。这些几何问题中的许多都可以被视为线性代数中的问题：关于拉普拉斯-德-拉姆算子的计算问题，这是一个简洁地表示几何空间的矩阵序列。因此，使用线性代数的强大工具，可以为这些问题开发更快更简单的方法，这是本项目的重点。图拉普拉斯矩阵（拉普拉斯-德拉姆序列中的第一个矩阵）研究的最新进展已经产生了一系列简单而优雅的算法，这些算法有助于计算机科学和数据分析中的重要应用。相比之下，拉普拉斯-德拉姆算子的高维变体，尽管在科学和工程中发挥了关键作用（例如，在求解物理方程，在数据可视化和统计排序中执行亥姆霍兹/霍奇分解，或在揭示隐藏空间的有效几何或拓扑特性方面），但在很大程度上仍未得到充分的探索。这个项目将填补这一空白，解决关于拉普拉斯-德朗算子的重要开放问题，用于两种不同类型的应用，在两个高级目标下：(1)为科学计算中由偏微分方程（PDEs）提出的重要问题（如Navier-Stokes或Maxwell方程中的向量拉普拉斯算子）中出现的拉普拉斯-德朗矩阵设计高效的求解器；(2)发现并利用拉普拉斯-德朗矩阵及其谱的有效性质进行数据分析。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。

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.