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- W2894519641 abstract "We study the complexity of approximating integrals of smooth functions at absolute precision $varepsilon > 0$ with confidence level $1 - delta in (0,1)$. The optimal error rate for multivariate functions from classical isotropic Sobolev spaces $W_p^r(G)$ with sufficient smoothness on bounded Lipschitz domains $G subset mathbb{R}^d$ is determined. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $delta$ in the complexity. In the limiting case $p = 1$ we see that deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the additional effort for diminishing the uncertainty. Finally, we add a discussion about this problem for function spaces with mixed smoothness." @default.
- W2894519641 created "2018-10-05" @default.
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- W2894519641 date "2018-09-26" @default.
- W2894519641 modified "2023-09-27" @default.
- W2894519641 title "Optimal confidence for Monte Carlo integration of smooth functions" @default.
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