Deriving Quadrature Rules from Gaussian Processes

Quadrature rules are often designed to achieve zero error on a small set of functions, e.g. polynomials of specified degree. A more robust method is to minimize average error over a large class or distribution of functions. If functions are distributed according to a Gaussian process, then designing an average-case quadrature rule reduces to solving a system of 2n equations, where n is the number of nodes in the rule (O’Hagan, 1991). It is shown how this very general technique can be used to design customized quadrature rules, in the style of Yarvin & Rokhlin (1998), without the need for singular value decomposition and in any number of dimensions. It is also shown how classical Gaussian quadrature rules, trigonometric lattice rules, and spline rules can be extended to the average-case and to multiple dimensions by deriving them from Gaussian processes. In addition to being more robust, multidimensional quadrature rules designed for the average-case are found to be much less ambiguous than those designed for a given polynomial degree.