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

积分是科学、工程、医学和经济学中许多过程的数学建模的基础。例如，积分用于在给定空间区域（或域）上数学地表达物质（例如危险化学品）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求积方法。该项目引入了一个全新的框架，Fredholm替代求积（FAQ），用于执行求积的基本任务。因此，它为研究人员提供了新的有效的选择来解决集成问题，特别是那些涉及几何复杂的区域和不规则的采样数据。该框架还提供了一种新的方法来处理一个已经存在了几千年的经典主题，提供了新的见解和教学机会。该项目将资助一名博士。最近创建的计算数学，科学，工程（CMSE）计划的学生，这也将有助于加强该计划的研究组合。从招聘计算数学研究生从代表性不足的群体的成功记录的基础上，调查员将继续与包容性和变革奖学金研究所合作，以帮助确定该项目的潜在候选人。为本科生、硕士和博士提供新的教育机会。学生也将通过一个垂直整合项目（VIP）的发展，结合项目的主题创建。常见问题的框架是基于连续的Fredholm替代（FA）定理泊松方程和离散FA的线性系统，从离散化这个方程之间的关系。它不采用积分，而是需要在积分域上的给定点集上离散某些拉普拉斯算子，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）方法来离散拉普拉斯算子。这导致了一种方法，该方法1）不需要显式或隐式地积分基函数，2）可以在几何复杂的域（甚至表面）上使用，3）可以在没有网格的情况下对被积函数的分散样本实施，4）可以产生光滑函数的高阶精度，以及5）可以有效地计算。一些数值和理论上的进步也将取得，包括技术产生高阶精度的RBF-FD离散，有效的无网格多层方法计算的FAQ公式，最小二乘法提高稳定性的FAQ公式，工具分析FAQ近似性能，该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准。

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.