The Curse of Dimensionality for Numerical Integration of Smooth Functions II

TL;DR

This paper investigates the curse of dimensionality in numerical integration for smooth functions, establishing conditions under which high-dimensional integration becomes computationally infeasible, especially for functions with large derivative bounds.

Contribution

It provides necessary and sufficient conditions for the curse of dimensionality across various classes of smooth functions, including infinitely differentiable functions, and characterizes tractability.

Findings

Curse of dimensionality holds for large derivative bounds.

Volume estimates of neighborhoods of convex hulls are key to proofs.

Conditions for quasi-polynomial and weak tractability are identified.

Abstract

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth $d$-variate functions. Roughly speaking, we consider different bounds for the derivatives of $f \in C^k(D_d)$ and ask whether the curse of dimensionality holds for the respective classes of functions. We always assume that $D_d \subset \mathbb{R}^d$ has volume one and consider various values of $k$ including the case $k=\infty$ which corresponds to infinitely many differentiable functions. We obtain necessary and sufficient conditions, and in some cases a full characterization for the curse of dimensionality. For infinitely many differentiable functions we prove the curse if the bounds on the successive derivatives are appropriately large. The proof technique is based on a volume estimate of a neighborhood of the convex hull of $n$ points which decays exponentially fast if $n$ is small relative to $d$. For $k=\infty$, we also also study conditions for quasi-polynomial, weak and uniform weak tractability.