  The search for multivariate quadrature rules of minimal size with a specified
polynomial accuracy has been the topic of many years of research. Finding such
a rule allows accurate integration of moments, which play a central role in
many aspects of scientific computing with complex models. The contribution of
this paper is twofold. First, we provide novel mathematical analysis of the
polynomial quadrature problem that provides a lower bound for the minimal
possible number of nodes in a polynomial rule with specified accuracy. We give
concrete but simplistic multivariate examples where a minimal quadrature rule
can be designed that achieves this lower bound, along with situations that
showcase when it is not possible to achieve this lower bound. Our second main
contribution comes in the formulation of an algorithm that is able to
efficiently generate multivariate quadrature rules with positive weights on
non-tensorial domains. Our tests show success of this procedure in up to 20
dimensions. We test our method on applications to dimension reduction and
chemical kinetics problems, including comparisons against popular alternatives
such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud
rules. The quadrature rules computed in this paper outperform these
alternatives in almost all scenarios.
