Fredholm Alternative Quadrature: A Novel Framework for Numerical Integration Over Geometrically Complex Domains

集成是科学、工程、医学和经济学中许多过程的数学建模的基础。例如，积分用于以数学方式表示给定空间区域（或域）上某种物质（如危险化学品）的总量。然而，这些模型中的积分问题很少能通过纸笔技术来解决，因此研究人员必须采用数值积分或正交方法。这个项目引入了一个全新的框架，Fredholm替代正交（FAQ），用于执行正交的基本任务。因此，它为研究人员解决积分问题提供了新的有效选择，特别是那些涉及几何复杂域和不规则采样数据的问题。这个框架也为这个已经存在了几千年的经典学科提供了一种新的方法，提供了新的见解和教学机会。该项目将支持最近创建的计算数学、科学与工程（CMSE）项目的一名博士生，这也将有助于加强该项目的研究组合。基于从代表性不足的群体中招募计算数学研究生的成功记录，研究者将继续与包容性和变革性奖学金研究所合作，帮助确定该项目的潜在候选人。通过垂直整合项目（VIP）的发展，整合项目中的主题，还将为本科生、硕士和博士学生创造新的教育机会。常见问题解答框架是基于泊松方程的连续Fredholm替代定理（FA）和线性系统的离散FA之间的关系，这些线性系统是由泊松方程离散化产生的。它不使用积分而是需要在积分域上的一组给定点上离散某些拉普拉斯算子并求解一个特征值问题。为了最大限度地提高FAQ的灵活性和实用性，将使用无网格径向基函数有限差分（RBF-FD）方法对拉普拉斯算子进行离散化。这导致了一种方法，1)不需要显式或隐式积分基函数，2)可用于几何复杂的域（偶数曲面），3)可以实现对被积体的分散样本而不进行网格划分，4)可以产生高阶精度的光滑函数，5)可以高效地计算。在数值和理论方面也将取得一些进展，包括产生高阶精确RBF-FD离散化的技术、计算FAQ公式的高效无网格多层方法、提高FAQ公式稳定性的最小二乘技术、分析FAQ近似性质的工具以及对经典正交公式的新见解。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。

Integration is fundamental to the mathematical modeling of many processes in science, engineering, medicine, and economics. For example, integration is used to mathematically express the total quantity of a substance, such as a hazardous chemical, over a given spatial region (or domain). However, the integration problems in these models can rarely be solved by pen and paper techniques, so researchers must employ numerical integration, or quadrature, methods. This project introduces an entirely new framework, Fredholm Alternative Quadrature (FAQ), for performing the essential task of quadrature. It thus gives researchers new effective options for tackling integration problems, especially those involving geometrically complicated domains and irregularly sampled data. The framework also offers a new approach to a classical subject that has been around for millennia, providing fresh insights and pedagogical opportunities. The project will support one Ph.D. student in the recently created computational math, science, & engineering (CMSE) program, which will also help bolster the research portfolio of this program. Building from a successful track record of recruiting graduate students in computational mathematics from underrepresented groups, the investigator will continue working with the Institute for Inclusive and Transformative Scholarship to help identify potential candidates for the project. New educational opportunities for undergraduate, master's, and Ph.D. students will also be created through the development of a Vertically Integrated Project (VIP) that incorporates topics from the project.The FAQ framework is based on a relationship between the continuous Fredholm Alternative (FA) theorem for Poisson's equation and the discrete FA for linear systems that arise from discretizing this equation. It does not employ integration but instead requires discretizing certain Laplace operators at a given set of points over the integration domain and solving an eigenvalue problem. To maximize the flexibility and practicality of FAQ, the mesh-free radial basis function finite difference (RBF-FD) method for discretizing the Laplace operators will be used. This results in a method that 1) does not require explicitly or implicitly integrating basis functions, 2) can be used on geometrically complicated domains (even surfaces), 3) can be implemented for scattered samples of the integrand without meshing, 4) can yield high orders of accuracy for smooth functions, and 5) can be computed efficiently. Several numerical and theoretical advancements will also be made, including techniques for producing high-order accurate RBF-FD discretizations, efficient meshfree multilevel methods for computing the FAQ formulas, least squares techniques for enhancing the stability of FAQ formulas, tools for analyzing FAQ approximation properties, and new insights on classical quadrature formulas.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.