Integration error for multivariate functions from anisotropic classes

The exact order of deterministic, stochastic and average error bounds of multidimensional quadrature formulas for anisotropic Sobolev class W r pðIÞ; Nikolskii class H pðIÞ and their periodic analogue is obtained. It is proved that if p41; then the stochastic and average error bounds are essentially smaller than the deterministic error bounds. Nonlinear methods, adaptive methods, or even methods with varying cardinality are not significantly better than the simplest linear method s̃ðf Þ 1⁄4 Pn i1⁄41 cif ðaiÞ: r 2003 Elsevier Science (USA). All rights reserved.