(6/2/00)

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.

Postscript (75K)

Last modified: Fri Dec 10 14:24:51 GMT 2004