QUADRATURE-FORMULAS FOR MONOTONE-FUNCTIONS

We prove that adaptive quadrature formulas for the class of monotone functions are much better than nonadaptive ones if the average error is considered. Up to now it was only known that adaptive methods are not better in the worst case (for this and many other classes of functions) or in various average case settings. We also prove that adaptive Monte Carlo methods are much better than nonadaptive ones. This also contrasts with analogous results for other classes (Sobolev classes, Holder classes) where adaptive methods are only slightly better than nonadaptive ones.