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

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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.