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，积分用于在给定的空间区域(或区域)上以数学方式表示一种物质(如危险化学品)的总量。然而，这些模型中的积分问题很少用纸笔技术来解决，所以研究人员必须使用数值积分或求积方法。这个项目引入了一个全新的框架，FredholmAlternative Quadrature(FAQ)，用于执行求积的基本任务。因此，它为研究人员提供了处理集成问题的新的有效选择，特别是涉及几何复杂区域和不规则样本数据的集成问题。该框架还为一个存在了数千年的经典主题提供了一种新的方法，提供了新的见解和教学机会。该项目将支持一名博士生参加最近创建的计算数学、科学和工程(CMSE)项目，这也将有助于支持该项目的研究组合。在从代表性不足的群体中招聘计算数学研究生的成功记录基础上，调查员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。本科生、硕士生和博士生的新教育机会也将通过开发垂直集成项目(VIP)来创造，该项目结合了项目中的主题。常见问题框架基于泊松方程的连续Fredholm替代(FA)定理和线性系统的离散FA之间的关系，这些FA是通过离散化该方程而产生的。它不使用积分，而是要求在积分区域上的给定点上离散化某些拉普拉斯算子，并求解特征值问题。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)可用于几何复杂的区域(偶数曲面)，3)可用于被积函数的离散样本而无需网格化，4)可产生光滑函数的高阶精度，5)可高效地计算。还将在数值和理论上取得一些进展，包括产生高精度RBF-FD离散化的技术，计算FAQ公式的高效无网格多层方法，增强FAQ公式稳定性的最小二乘技术，分析FAQ近似性质的工具，以及对经典求积公式的新见解。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。

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.