Multidimensional Numerical Integration Under Strong Convexity

Abstract

This paper introduces and studies a new class of multidimensional numerical integration, which we call `` \emph{strongly positive definite cubature formulas}''. We establish, among others, a characterization theorem providing necessary and sufficient conditions for
the error estimates (based on such cubature formulas) to be always controlled by the Lipschitz constants of the gradients, the strong convexity parameter and the error associated when using the quadratic function. This result is obtained as a consequence of two characterization results for linear functionals, which hold in large generality and is of independent interest.

Thus, this paper extends some result previously obtained in [2,5] when the
convexity in the classical sense is only imposed.

We also show that the centroidal Voronoi diagrams give access to efficient
algorithms for constructing such cubature formulas. Numerical results for the two-dimensional test functions are given to illustrate the efficiency of our resulting cubature formulas.