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.
Abstract The deterministic sparse grid method, also known as Smolyak’s method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about randomized versions of the sparse grid method. In this paper we analyze randomized sparse grid algorithms, namely randomized sparse grid quadratures for multivariate integration on the
D-dimensional unit cube
[0,1)D. Let
d,s∈N be such that
D=d⋅s. The
s-dimensional building blocks of the sparse grid quadratures are based on stratified sampling for
s=1 and on scrambled
(0,m,s)-nets for
s≥2. The spaces of integrands and the error criterion we consider are Haar wavelet spaces with parameter $\alpha $ and the randomized error (i.e., the worst case root mean square error), respectively. We prove sharp (i.e., matching) upper and lower bounds for the convergence rates of the
Nth minimal errors for all possible combinations of the parameters
d and
s. Our upper error bounds still hold if we consider as spaces of integrands Sobolev spaces of mixed dominated smoothness with smoothness parameters
1/2<α<1 instead of Haar wavelet spaces.