The Curse of Dimensionality for Numerical Integration of Smooth Functions
Numerical Analysis
2017-06-22 v2
Abstract
We prove the curse of dimensionality for multivariate integration of C^r functions: The number of needed function values to achieve an error \epsilon\ is larger than c_r (1+\gamma)^d for \epsilon\le \epsilon_0, where c_r,\gamma>0 and d is the dimension. The proofs are based on volume estimates for r=1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for r>1.
Keywords
Cite
@article{arxiv.1211.0871,
title = {The Curse of Dimensionality for Numerical Integration of Smooth Functions},
author = {Aicke Hinrichs and Erich Novak and Mario Ullrich and Henryk Wozniakowski},
journal= {arXiv preprint arXiv:1211.0871},
year = {2017}
}
Comments
15 pages, minor revision