The curse of dimensionality for numerical integration on general domains

TL;DR

This paper demonstrates that the difficulty of multivariate numerical integration increases exponentially with dimension for certain smooth functions on convex domains, extending previous results to a broader class of domains and using advanced geometric analysis techniques.

Contribution

It extends the curse of dimensionality results to volume-normalized _p^d-balls for all p in [2, ], unifying previous work and applying geometric analysis tools.

Findings

Curse of dimensionality proven for smooth functions on convex bodies.

Extension of results to entire range 2  for p in [2, ].

Uses deep geometric concentration results in proofs.

Abstract

We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal $\psi_2$-estimate. In particular, we obtain the result for the important class of volume-normalized $\ell_p^d$-balls in the complete regime $2\leq p \leq \infty$. This extends a result in a work of A. Hinrichs, E. Novak, M. Ullrich and H. Wo\'zniakowski [J. Complexity, 30(2), 117-143, 2014] to the whole range $2\leq p \leq \infty$, and additionally provides a unified approach. The key ingredient in the proof is a deep result from the theory of Asymptotic Geometric Analysis, the thin-shell volume concentration estimate due to O. Gu\'edon and E. Milman. The connection of Asymptotic Geometric Analysis and Information-based Complexity revealed in this work seems promising and is of independent interest.