Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness

TL;DR

This paper proves that higher order digital nets can achieve optimal convergence rates in quasi-Monte Carlo integration over weighted Sobolev spaces of arbitrary smoothness, improving previous bounds and matching theoretical lower limits.

Contribution

It establishes the optimal convergence rate for quasi-Monte Carlo integration using higher order digital nets with random shifts, improving the logarithmic factor exponent over prior results.

Findings

Achieves convergence rate of $N^{- ext{alpha}}( ext{log} N)^{(s-1)/2}$ for worst-case error.

Improves the logarithmic factor exponent from $s ext{alpha}/2$ to $(s-1)/2$.

Shows the existence of digital nets attaining the best possible convergence rate.

Abstract

We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $\alpha \in \mathbb{N}$, $\alpha \ge 2$, defined over the $s$-dimensional unit cube. We prove that randomly digitally shifted order $\beta$ digital nets can achieve the convergence of the root mean square worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ when $\beta \ge 2\alpha$. The exponent of the logarithmic term, i.e., $(s-1)/2$, is improved compared to the known result by Baldeaux and Dick, in which the exponent is $s\alpha /2$. Our result implies the existence of a digitally shifted order $\beta$ digital net achieving the convergence of the worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$, which matches a lower bound on the convergence rate of the worst-case error for any cubature rule using $N$ function evaluations and thus is best possible.