Randomized Smolyak algorithms based on digital sequences for multivariate integration

摘要

In this paper, we consider Smolyak algorithms based on quasi-Monte Carlo rules for high-dimensional numerical integration. The quasi-Monte Carlo rules employed here use digital (t, α, β, σ, d)-sequences as quadrature points. We consider the worst-case error for multivariate integration in certain Sobolev spaces and show that our quadrature rules achieve the optimal rate of convergence. By randomizing the underlying digital sequences, we can also obtain a randomized Smolyak algorithm. The bound on the worst-case error holds also for the randomized algorithm in a statistical sense. Further, we also show that the randomized algorithm is unbiased and that the integration error can be approximated as well.