Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond (Q2895760)

This article is a review of quasi-Monte Carlo methods for high-dimensional integration. Quasi-Monte Carlo (QMC) methods are equal-weight quadrature rules which are used for approximating the value of high-dimensional integrals over the unit cube. The authors of the paper mostly focus on a special class of QMC integration nodes, namely lattice point sets, however, large parts of the theory explained are applicable or can easily be adapted to other QMC methods as well.NEWLINENEWLINEThe paper first gives an overview of the ``standard setting'' of weighted Hilbert spaces of functions with square-integrable mixed first derivatives. In particular, the worst-case error of integration in such spaces is analyzed and it is discussed how one can construct lattice rules that achieve the optimal order of convergence. Furthermore, it is motivated why it is useful to go beyond the standard setting, and it is explained how to do so in two ways. On the one hand, the authors move away from Hilbert space settings to Banach space settings, and on the other hand, more general weights than product weights are considered. In particular, the paper includes new results on ``product and order dependent'' weights. It is shown that the fast component-by-component construction of lattice points achieving the optimal convergence order of the integration error can be extended to this type of weights.