Mathematics Colloquium - Akil Narayan
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 case. The need for flexible multivariate quadrature rules will be motivated by solutions to parametric partial differential equations, and the efficacy of our approach will be shown on various test problems in scientific computing.