Foundations of Computational Mathematics

, Volume 14, Issue 5, pp 1027–1077 | Cite as

Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher-Order Convergence

  • Josef Dick
  • Michael Gnewuch


We study the numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing-dimension algorithms. More precisely, the spaces of integrands we consider are weighted, reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst-case error. We study two cost models for function evaluations that depend on the number of active variables of the chosen sample points, and we study two classes of weights, namely product and order-dependent weights and the newly introduced finite projective dimension weights. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing-dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter \(\alpha \) and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing-dimension algorithms based on higher-order polynomial lattice rules.


Path integration Multilevel algorithms Changing-dimension algorithms  Quasi-Monte Carlo methods Polynomial lattice rules Reproducing kernel Hilbert spaces 

Mathematics Subject Classification

Primary 65C05 65D30 Secondary 11K38 



We want to thank Michael Griebel for suggesting that we study algorithms for infinite-dimensional integration of higher-order convergence. We are grateful for the opportunity to work at the Hausdorff Institute in Bonn, where the work on this paper was initiated. Furthermore, we want to thank Greg Wasilkowski, Henryk Woźniakowski, and two anonymous referees for valuable comments. Josef Dick is supported by an ARC Queen Elizabeth II Fellowship. Michael Gnewuch was supported by the German Science Foundation DFG under Grant GN 91/3-1 and by the Australian Research Council.


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Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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