Generalized Barycentric Coordinates and Sharp Strongly Negative Definite Multidimensional Numerical Integration

Abstract

This paper is devoted to study and construct a family of multidimensional numerical integration formulas (cubature formulas), which approximate all strongly convex functions from above. We call them strongly negative definite cubature formulas (or for brevity snd-formulas). We attempt to quantify their sharp approximation errors when using continuously differentiable functions with Lipschitz continuous gradients. We show that the error estimates based on such cubature formulas are always controlled by the Lipschitz constants of the gradients and the error associated with using the quadratic function. Moreover, assuming the integrand is itself strongly convex, we establish sharp upper as well as lower refined bounds for their error estimates. Based on the concepts of barycentric coordinates with respect to an arbitrary polytope P, we provide a necessary and sufficient condition for the existence of a class of snd-formulas on P. It consists of checking that such coordinates exist on P. Then, the Delaunay triangulation is used as a convenient partition of the integration domain for constructing the best piecewise snd-formulas in L1 metric. Finally, we present numerical examples illustrating the proposed method.