Shape Aware Quadrature

摘要

With Shape Aware Quadratures (SAQ), for a given set of quadrature nodes, order, and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain (Omega(0)) that is a reasonable approximation of the original domain (Omega) that are then corrected for the deviation of the shape of Omega(0) from Omega via shape correction factors. This idea was already successfully utilized in the Adaptively Weighted Numerical Integration (AW) method [1 3], where moments were computed over simplified but homeomorphic domain and then corrected using first-order shape sensitivity, allowing efficient and accurate integration of integrable functions over arbitrary domains.