Intractability Results for Integration and Discrepancy

We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight #j moderates the behavior of functions with respect to the jth variable. For #j #1, we obtain the classical Sobolev spaces whereas for decreasing #j 's the weighted Sobolev spaces consist of functions with diminishing dependence on the jth variables. We study the minimal errors of quadratures that use n function values for the unit ball of the space Fd . It is known that if the smoothness parameter of the Sobolev space is one, then the minimal error is the same as the discrepancy. Let n(=, Fd) be the smallest n for which we reduce the initial error, i.e., the error with n=0, by a factor =. The main problem studied in this paper is to determine whether the problem is intractable, i.e., whether n(=, Fd) grows faster than polynomially in = or d. In particular, we prove intractability of integration (and discrepancy) if lim supd j=1 #j ln d= . Previously, such results were known only for restricted classes of quadratures. doi:10.1006 jcom.2000.0577, available online at http: www.idealibrary.com on