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(p)(r) (I(d)), Nikolskii class H(p)(r)(I(d)) and their periodic analogue is obtained. It is proved that if p > 1, 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 (s) over tilde (f) = Sigma(i=1)(n) c(i)f (a(i)). (C) 2003 Elsevier Science (USA). All rights reserved.
anisotropic quadrature formula error estimates stochastic case setting average case setting sobolev class of functions anisotropic nikolskii class of functions