Effective numerical integration on complex shaped elements by discrete signed measures

We discuss a cheap and stable approach to polynomial moment-based compression of multivariate measures by discrete signed measures. The method is based on the availability of an orthonormal basis and a low-cardinality algebraic quadrature formula for an auxiliary measure in a bounding set. Differently from other approaches, no conditioning issue arises since no matrix factorization or inversion is needed. We provide bounds for the sum of the absolute values of the signed measure weights, and we make two examples: efficient quadrature on curved planar elements with spline boundary (in view of the application to high-order FEM/VEM), and compression of QMC integration on 3D elements with complex shape.