Linear Monte Carlo quadrature with optimal confidence intervals
Abstract
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, nonlinear 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 nonlinear algorithms where only a logarithmic dependence on $\delta^{1}$ occurs if we tune in for a specific value of $\delta$.
 Publication:

arXiv eprints
 Pub Date:
 September 2023
 DOI:
 10.48550/arXiv.2309.09059
 arXiv:
 arXiv:2309.09059
 Bibcode:
 2023arXiv230909059K
 Keywords:

 Mathematics  Numerical Analysis;
 65C05;
 65D30;
 65J05
 EPrint:
 22 pages, 2 figures