Efficiency of Quasi-Monte Carlo Algorithms for High Dimensional Integrals

Gowri Suryanarayana

Kinjal Basu

Emanouil I. Atanassov, Ivan Dimov, Mariya K. Durchova

Frances Y. Kuo, Ian H. Sloan

Ian H. Sloan, Henryk Woźniakowski

Recently, quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimensiond, 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 larged. It is done by identifying classes of functions for which the effect of the dimensiondis negligible. These areweightedclasses in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimalworst caseerror of quasi-Monte Carlo algorithms does not depend on the dimensiondiff 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 byCε−p, where the exponentp∈ [1, 2], andCdepends 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 theworst casesetting is of order ε−pwith the exponentp= 2. The deterministic construction of strongly tractable quasi-Monte Carlo algorithms as well as the minimal exponentpis open.

Qin Li

This paper introduces the fundamental ideas of Monte Carlo methods and quasi-Monte Carlo methods and analyzes the relations between them from the perspective of algorithms, estimation of errors and convergence rates. In the paper, the author puts great emphasis on quasi-Monte Carlo methods' application on high-dimensional integration to certify the superiority of the quasi-Monte Carlo methods.