Quasi-Monte Carlo integration over Rs based on digital nets

Abstract

This paper discusses $\phi$-weighted integration of functions over the $s$-dimensional Euclidean space using quasi-Monte Carlo (QMC) rules combined with an inversion method, where the probability density function (PDF) $\phi$ is of product form, i.e., a product of uni-variate PDFs ${\phi }_i$ for each coordinate $i$ in $\{1,\dots ,s\}$.

The space of integrands is specified by means of a $\gamma$-weighted $p$-norm, $p\ge 1$, which involves coordinate weights $\gamma$, the partial derivatives of order up to one of the integrands as well as additional weight functions ${\psi }_i$ and the PDFs ${\phi }_i$. The coordinate weights $\gamma$ model the importance of different coordinates or groups of coordinates in the sense of Sloan and Woźniakowski, and the weight functions ${\psi }_i$ are additional parameters of the space which describe the decay of the partial derivatives of the integrands. Fast decaying weights ${\psi }_i\left(x\right)$ for $x\to \pm \infty$ enlarge the space of functions with finite norm, but decrease the convergence rate of the worst-case error of the proposed algorithms.

Our algorithms for integration use digitally shifted digital nets in combination with an inversion method. We study the (root) mean squared worst-case error with respect to random digital shifts. The obtained error bounds depend on the choice of weight functions ${\psi }_i$ and coordinate weights $\gamma$. Under certain conditions on $\gamma$, these bounds hold uniformly for all dimensions $s$.

Numerical experiments demonstrate the effectiveness of the proposed algorithms.