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We study approximation of the embedding $\ell_p^m \hookrightarrow \ell_q^m$, $1 \leq p < q \leq \infty$, based on randomized algorithms that use up to $n$ arbitrary linear functionals as information on a problem instance where $n \ll m$. By analysing adaptive methods we show upper bounds for which the information-based complexity $n$ exhibits only a $(\log\log m)$-dependence. In the case $q < \infty$ we use a multi-sensitivity approach in order to reach optimal polynomial order in $n$ for the Monte Carlo error. We also improve on non-adaptive methods for $q < \infty$ by denoising known algorithms for uniform approximation.

We study approximation of the embedding $\ell_p^m \rightarrow \ell_{\infty}^m$, $1 \leq p \leq 2$, based on randomized adaptive algorithms that use arbitrary linear functionals as information on a problem instance. We show upper bounds for which the complexity $n$ exhibits only a $(\log\log m)$-dependence. Our results for $p=1$ lead to an example of a gap of order $n$ (up to logarithmic factors) for the error between best adaptive and non-adaptive Monte Carlo methods. This is the largest possible gap for linear problems.

We study the numerical integration of functions from isotropic Sobolev spaces $W_p^s([0,1]^d)$ using finitely many function evaluations within randomized algorithms, aiming for the smallest possible probabilistic error guarantee $\varepsilon > 0$ at confidence level $1-\delta \in (0,1)$. For spaces consisting of continuous functions, non-linear Monte Carlo methods with optimal confidence properties have already been known, in few cases even linear methods that succeed in that respect. In this paper we promote a new method called stratified control variates (SCV) and by it show that already linear methods achieve optimal probabilistic error rates in the high smoothness regime without the need to adjust algorithmic parameters to the uncertainty $\delta$. We also analyse a version of SCV in the low smoothness regime where $W_p^s([0,1]^d)$ may contain functions with singularities. Here, we observe a polynomial dependence of the error on $\delta^{-1}$ which cannot be avoided for linear methods. This is worse than what is known to be possible using non-linear algorithms where only a logarithmic dependence on $\delta^{-1}$ occurs if we tune in for a specific value of $\delta$., Comment: 22 pages, 2 figures