Speaker
Grady Wright, Boise State University
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.
In this talk we introduce a new framework for generating quadrature formulas that bypasses these challenges. The framework relies on numerical approximations of certain elliptic operators and on linear algebra. We show how several classic univariate quadrature formulas naturally arise from this framework and demonstrate its ability to produce accurate quadrature formulas for geometrically complex domains, including point cloud discretizations of surfaces and unfitted discretizations of Euclidean domains.