Grady Wright: A New Quadrature Framework for Geometrically Complex Domains

Abstract

The dominant approach to constructing quadrature (or cubature) formulas is to select a “nice” vector space of functions for which the formulas are exact, such as algebraic or trigonometric polynomials. For one dimensional integration, this leads to classical Newton–Cotes and Gaussian quadrature rules. However, in higher dimensions and for geometrically complex domains, this exactness-based approach can be challenging or even infeasible, since it requires exact integration of basis functions over the domain or over suitable subdomains. Additional challenges arise when the integrand is known only through samples at predefined, possibly “scattered” points (i.e., a point cloud), as is common in applications involving experimental data or when integration is a secondary step in a larger computational process.