The Curse of Dimensionality for Numerical Integration of Smooth Functions

TL;DR

This paper demonstrates that multivariate integration of smooth functions suffers from the curse of dimensionality, requiring exponentially more function evaluations as the dimension increases, based on volume estimates and smoothing techniques.

Contribution

It establishes the curse of dimensionality for smooth functions in multivariate integration, extending previous results to higher smoothness classes using volume estimates and convolution smoothing.

Findings

Number of function evaluations grows exponentially with dimension

Established curse of dimensionality for C^r functions

Used volume estimates and smoothing to construct fooling functions

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.