Strong tractability of quasi-Monte Carlo quadrature using nets for certain Banach spaces

摘要

We consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in L-p norms and the weighted coefficients introduced via l(q) norms, where p, q is an element of [1, infinity]. The integration domain may be bounded or unbounded. The worst-case error and randomized error are investigated for quasi-Monte Carlo quadrature rules. For the worst-case setting the quadrature rule uses deterministic ((T-u), s)-sequences in base b, and for the randomized setting the quadrature rule uses randomly scrambled digital ((Tu), m, s)-nets in base b. Sufficient conditions are found under which multivariate integration is strongly tractable in the worst-case and randomized settings, respectively. Similar results hold for the Banach spaces of finite-order weights. Results presented in this article extend and improve upon those found previously.