Product rules are optimal for numerical integration in classical smoothness spaces

TL;DR

This paper establishes explicit error bounds and optimal algorithms for numerical integration of smooth functions on the unit cube, showing product rules are optimal and error depends only on the number of points and smoothness.

Contribution

It provides explicit error bounds without hidden constants and proves product rules are optimal for smooth function integration in high dimensions.

Findings

Optimal error rate is min{1, d n^{-r/d}}

Product rules achieve the optimal order of convergence

Lower bounds are established for arbitrary domains

Abstract

We mainly study numerical integration of real valued functions defined on the $d$-dimensional unit cube with all partial derivatives up to some finite order $r\ge1$ bounded by one. It is well known that optimal algorithms that use $n$ function values achieve the error rate $n^{-r/d}$, where the hidden constant depends on $r$ and $d$. Here we prove explicit error bounds without hidden constants and, in particular, show that the optimal order of the error is $\min \bigl\{1, d \, n^{-r/d}\bigr\}$, where now the hidden constant only depends on $r$, not on $d$. For $n=m^d$, this optimal order can be achieved by (tensor) product rules. We also provide lower bounds for integration defined over an arbitrary open domain of volume one. We briefly discuss how lower bounds for integration may be applied for other problems such as multivariate approximation and optimization.