Abstract

Numerical quadrature rules that use point values are ubiquitous tools for approximating integrals. Some of the most popular rules achieve accuracy by enforcing exactness for integrands in a finite-dimensional polynomial space. When the integration domain is one-dimensional, classical rules are available and plentiful. In multidimensional domains with non-standard polynomial spaces and weights, the situation is far more complicated.

We will present a general methodology for numerically generating approximate polynomial quadrature rules in multidimensional situations. Our technique is based on recent advances in feature selection algorithms, allowing rigorous guarantees on feasibility and computability of the selection problem. We show that this approach allows one to generate excellent quadrature rules in an automated way, and demonstrate the accuracy of these rules in applications.