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Tractability of infinite-dimensional integration in the worst case and randomized settings. (English)
J. Complexity 27, No. 6, 505-518 (2011).
The authors consider the approximation of integrals of functions $f$ with infinitely many variables in worst case and randomized settings. Here the class of $\infty$-variate functions $f$ is build as an infinite sum of weighted tensor products of a reproducing kernel Hilbert space of scalar functions. The authors extend results from {\it F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski} and {\it H. Woźniakowski} [J. Complexity 26, No.~5, 422‒454 (2010; Zbl 1203.65057)] to more general function spaces. Mainly, the complexity and polynomial tractability of approximating integrals of such $\infty$-variate functions is studied. Polynomial tractability means that the $ε$-complexity (that is the minimal cost of all algorithms whose worst case errors are $\le ε$) is ${\cal O}(ε^{-p})$, where the smallest $p\le 0$ is the exponent of tractability. The authors determine upper bounds of the tractability exponent and show that some bounds are sharp.
Reviewer: Manfred Tasche (Rostock)