Very Low Truncation Dimension for High Dimensional Integration Under Modest Error Demand

TL;DR

This paper demonstrates that high-dimensional integrals in weighted Sobolev spaces can be approximated efficiently using low-dimensional quadratures, with the truncation dimension depending only on the desired accuracy and not on the total dimension.

Contribution

It proves constructively that for large or infinite dimensions, integrals can be approximated by low-dimensional quadratures with a dimension depending solely on the error tolerance.

Findings

Truncation dimension is surprisingly small for large s.

The truncation dimension depends only on the error tolerance, not on the specific function.

The method applies even for s=∞ with arbitrary error demand.

Abstract

We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of $s$-variate functions. Here $s$ is large including $s=\infty$. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions $f_k$ with only $k$ variables, where $k=k(\varepsilon)$ depends solely on the error demand $\varepsilon$ and is surprisingly small when $s$ is sufficiently large relative to $\varepsilon$. This holds, in particular, for $s=\infty$ and arbitrary $\varepsilon$ since then $k(\varepsilon)<\infty$ for all $\varepsilon$. Moreover $k(\varepsilon)$ does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.