When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?

Abstract

Recently, quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimension d, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasi-Monte Carlo algorithms does not explain this phenomenon. This paper presents a partial answer to why quasi-Monte Carlo algorithms can work well for arbitrarily large d. It is done by identifying classes of functions for which the effect of the dimension d is negligible. These are weighted classes in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimal worst case error of quasi-Monte Carlo algorithms does not depend on the dimension d iff the sum of the weights is finite. We also prove that the minimal number of function values in the worst case setting needed to reduce the initial error by ε is bounded by Cε −p , where the exponent p ∈ [1, 2], and C depends exponentially on the sum of weights. Hence, the relatively small sum of the weights makes some quasi-Monte Carlo algorithms strongly tractable. We show in a nonconstructive way that many quasi-Monte Carlo algorithms are strongly tractable. Even random selection of sample points (done once for the whole weighted class of functions and then the worst case error is established for that particular selection, in contrast to Monte Carlo where random selection of sample points is carried out for a fixed function) leads to strong tractable quasi-Monte Carlo algorithms. In this case the minimal number of function values in the worst case setting is of order ε −p with the exponent p = 2. The deterministic construction of strongly tractable quasi-Monte Carlo algorithms as well as the minimal exponent p is open.

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